Efficient Graph Matching for Correlated Stochastic Block Models
Authors: Shuwen Chai
Miklos Z. Racz
Venue: NeurIPS 2024 poster (2024)
Keywords: Graph matching
correlated random graphs
stochastic block model
community recovery
subgraph counting
Primary area: learning_theory
TL;DR: We give the first efficient algorithm for graph matching for correlated stochastic block models with two balanced communities.

Abstract: We study learning problems on correlated stochastic block models with two balanced communities. Our main result gives the first efficient algorithm for graph matching in this setting. In the most interesting regime where the average degree is logarithmic in the number of vertices, this algorithm correctly matches all but a vanishing fraction of vertices with high probability, whenever the edge correlation parameter $s$ satisfies $s^2 > \alpha \approx 0.338$, where $\alpha$ is Otter's tree-counting constant. Moreover, we extend this to an efficient algorithm for exact graph matching whenever this is information-theoretically possible, positively resolving an open problem of Rácz and Sridhar (NeurIPS 2021). Our algorithm generalizes the recent breakthrough work of Mao, Wu, Xu, and Yu (STOC 2023), which is based on centered subgraph counts of a large family of trees termed chandeliers. A major technical challenge that we overcome is dealing with the additional estimation errors that are necessarily present due to the fact that, in relevant parameter regimes, the latent community partition cannot be exactly recovered from a single graph. As an application of our results, we give an efficient algorithm for exact community recovery using multiple correlated graphs in parameter regimes where it is information-theoretically impossible to do so using just a single graph.

Efficient Graph Matching for Correlated Stochastic Block Models
Shuwen Chai
Miklós Z. Ráez Northwestern University
Northwestern University Evanston, IL 60208
Evanston, IL 60208
shuwenchai2027@m
wow.:aaiieeeee....oooooe..eeeeee www.nnotttttterre edu
miklos .iw.os.raczsooreeeeoeeoooo..cc www..rrarssoootwveteee.eeddd
Abstract
We study learning problems on correlated stochastic block models with two bal- anced communities. Our main result gives the first efficient algorithm for graph matching in this setting. the most interesting regime where the average degree is logarithmic in the number of vertices, this algorithm correctly matches all but vanishing fraction of vertices with high probability, whenever the edge correlation parameter satisfies s²
0.338, where is Otter's tree-counting constant. Moreover, we extend this to an efficient algorithm for exact graph matching when- ever this is iiformatiion-hhooetiiin of Rácz and Sridhar (NeurIPS 2021). Our algorithm generalizes the recent break- through work of Mao, Wu, Xu, and Yu (STOC 2023), which is based centered subgraph counts of a large family of trees termed chandeliers. A major technical challenge that overcome is dealing with the additional estimation errors that are necessarily present due to the fact that, in relevant parameter regimes, the latent community partition cannot be exactly recovered from single graph. As an application of our results, we give an efficient algorithm for exact commu- nity recovery using multiple correlated graphs parameter regimes where information-theorrtiialll impossible to do so using just a single graph.
possible, positively resolving an open problem
1 Introduction
The proliferation of network data has highlighted the ubiquity and importance of graph matching in machine learning, with applications in variety of domains, including social networks [51, 56], com- putational biology [61] J. and computer vision 36). While the graph matching task-recovering the latent node alignment between two networks- is known proximate in general [38, 52], in practice it is often possible to solve well, such as in the works cited above. This has motivated an exciting recent line of work studying average-case graph match- ing [14, 15, 65, 16, 23, 27, 24, 49, 5, 21, 22, 39, 40, 41, 17, 20, 18], focusing on correlated Erdós- Rényi random graphs [56]. These papers culminated in recent breakthrough works which developed efficient graph matching algorithms in the constant noise regime [40, 41J.
ap- be NP-hard solve or
However, real- world networks not modeled well by Erdós Rényi random graphs. which in turn has motivated a growing line of recent work studying graph matching beyond Erdös-Rényi [8, 32, 11, 54, 58, 70, 57, 25, 63, 59, 19, 67, 66, 60, 10). In particular, graph matching in correlated stochastic block models (correlated SBMs) [35, 54, 34] (see Section for definitions), since community structure is prevalent in many networks and the community recovery problem is a fundamental inference task that is often starting point for deeper analyses. Recent work of Rácz and Sridhar [57] determined the fundamental information-theoreti limits for exact graph matching in correlated SBMs; however, the underlying algorithm used to achieve this limit is inefficient (that is, not polynomial time). Rácz and Sridhar [57] posed the open problem of finding an efficient algorithm for (exact) graph matching whenever this is information information-theoretcalll thworetically feasible.
important problem in this vein is study
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Our main contribution positively resolves this open problem of Rácz and Sridhar [57], giving the first efficient algorithm for graph matching for correlated SBMs with two balanced communities, under a condition on the correlation strength that is conjectured to be necessary. Specifically, we give an efficient algorithm that, in the most interesting regime where the average degree is logarithmic in the number of vertices, achieves almost exact recovery of the latent matching, whenever the edge correlation parameter satisfies 8² > a R 0.338, where a is Otter's tree-counting constant. Moreover, we extend this to an efficient algorithm for exact graph matching whenever this is anformationttheoretically possible. See Section and Theorem I for details.
In addition, our results on graph matching directly imply novel efficient algorithms and results for community recovery. Specifically, combining- -in a black-box fashion -ur (exact) graph matching algorithm with existing community recovery algorithms, we give an efficient algorithm for exact community recovery using multiple correlated graphs in parameter regimes where it is information- theoretically impossible to do so using just a single graph. See Section 3 and Theorem 2 for details.
Our algorithm generalizes the recent breakthrough work of Mao, Wu, Xu, and Yu [41], which based on centered subgraph counts of a large family of trees termed chandeliers, to the setting of correlated SBMs. A major technical challenge that we overcome is dealing with the additional estimation errors that are necessarily present due to the fact that, in relevant parameter regimes, the latent community partition cannot be exactly recovered from a single graph, and thus the edge-indicator variables in the centered subgraph counts cannot be precisely centered. Our technical contributions highlight the interplay between graph matching and community recovery in ways that are complementary to the recent work of Gaudio, Rácz, and Sridhar [25].
2 Models and problems
In this section we describe the setting of the paper by introducing the stochastic block model (SBM), correlated SBMs, and the community recovery and graph matching tasks.
The stochastic block model (SBM) is the canonical probabilistic generative model for a network with
latent community structure. The SBM was first introduced by Holland, Laskey, and Leinhardt [29] and has been widely studied over the past decades [1]. In general, an SBM may consist of a number of communities, with distinct vertices connected randomly with probability that depends on their community memberships.
In this work, we focus the simplest setting of the balanced two-community SBM. Given E Z+ and p. q E [0, 1]. we construct G SBM(n, p, q) as follows. The graph G has n vertices, with vertex labels given by V -In] = {1, ................................ .n]. Let o {+)( " be the vector of community labels, where each entry o. E +1) is drawn independently and uniformly at random. Then, given the community labels a., for any pair of vertices i * [m]. edge (i.j) is in G with probability pl (or + (o_ That is, two different vertices are connected with probability if they are from the same community and connected with probability q otherwise.
Correlated SBMs are multiple SBMs where the corresponding edge variables are correlated [35, 54, 34]. Specifically, we construct two correlated SBMs (G1, G2) ~ CSBM(n, p, s) using natural subsampling procedure as follows. Let G ~ SBM(n, p. q) be a parent graph with community labels o.. Next, given G, we construct G1 by random sampling of the edges: each edge of G is included in G1 with probability s, independently of everything else, and non-edges of G remain non-edges of G1. We then do the edge sampling independently again to obtain G²2 in the same way. The child graphs G1 and G² inherit both the vertex labels (given by (r)) and the community labels o. from the parent graph G. Finally, let I, be uniformly random permutation of [n] := 2.. n.) and generate G2 by relabeling the vertices of G²2 according to т, (e.g., vertex i in G²2 is relabeled as T. (1) in G2). This last step reflects the fact that in practice often the correspondence between the two vertex sets is unknown, We denote the adjacency matrices of G₁ and G2 as A and B, respectively, and note that the community labels of the two graphs are o := o, and o == o. o न. respectively. See Figure I for an illustration.
Marginally, G1 and G2 identically distributed SBMs: we have G1, G2 SBM(n, ps. gs). Moreover, G₁ and G2 are correlated. Specifically, for every pair of distinct vertices {i.j}. the edge- indicator random variables Ar,, and (z) (i) correlated Bernoulli random variables. A simple calculation shows that if oi( o Oi) then the correlation coefficient of A and B (0.m.) is
2

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subsample
subsample
18,2, 12, 7, 11,0 10,3,4, 1,99
G SBM(n, p, ②)
(20
8 (12)
(3)
(4)
2
G1
C22
222
(10)
(5)
12 (20
(0)
5 (11)
11 (1)
Figure 1: Schematic illustrating two-community correlated SBMs; see the text for details. (Figure
reproduced from [57] with permission.)
equal to P+ == :=-T-s I-pss whereas if (.((i +f (.)) then this correlation coefficient is p- 8 I- Our focus will be on the sparse setting where p. o(1) (as + oo). in which case both p+ and p- are asymptotically (1 (())),, and hence we can regard s as the edge correlation parameter.
1-p
Community recovery. The goal of community recovery is to recover the latent community labels given some (graph) data, such as an SBM G or correlated SBMs (G1, G2). There are various notions of community recovery, depending
how close estimate is to the ground truth o... In this work, we focus on exact community recovery, defined as follows: an estimator ō achieves exact community (i) (i) = 1) 1. The absolute value is present in the previous
recovery if -o (I Σi= / expression since we can only hope to recover the community labels up to a global sign flip; in other words, our goal is recover the partition of the graph into two communities. A slightly weaker notion, which also appears throughout our work, is almost exact community recovery, which holds if lim [1 Σ (i)or- (i) = 1- (1)) 1; in other words, this notion tolerates a vanishing fraction of errors. Further weaker notions include partial recovery and weak recovery; since these are not the focus here, we refer to [1] for details.
Different parameter regimes give rise to different challenges and different notions of recovery become most relevant. In the constant average degree regime, that is, when p = a and q = for some constants and b, it is impossible to recovery the communities exactly. Prior works (46. 48, 43] threshold and developed efficient algorithms for partial
have characterized the information-thoomm recovery in this regime. On the other hand, if the vertices have polynomially growing degrees, that is, when p = n a-+ and q = n b+o(() for some constants a, b E [0, 1), then community recovery is easy as long as inm Pn/ pn/qp - 1| > (see [47]).
In this work, we focus on the "bottleneck regime" of logarithmic average degree, which is the bare minimum for the graph to be connected, and which is when exact community recovery is most interesting. In most of the paper we assume that p a log and q 6 blog for some positive constants a, b. For an SBM with two balanced communities and these parameters, there is a sharp information-
x
threshold for exact community recovery, which given by (a, 1, where (a. == (Va- vb)² /2 (see [47, 2, 4, I)). This quantity is known as the Chernoff-Hellingge divergence in the general k-community SBM setting with linear size communities [4] and simplifies to the above form in our setting. In other words, when D+ (a, b) > 1, there exists an estimator ō that is computable in polynomial-time and which achieves exact community recovery with high probability. On the other hand, when D+ (a, b) < 1, exact community recovery is impossible, the sense that for all estimators ã have that 1im [ Σ (i)) = 1) O. Moreover, when it is possible to achieve exact community recovery on a single graph, several polynomial -orttiimll-eee algorithms have been studied by previous works (e.g.. [47, 2)).
Community recovery and graph matching. What are the information-theorete limits for exact n
CSBM(n, a Jogn flggn ss)? This question was initiated and partially solved by Rácz and Sridhar [57], and subsequently fully solved by Gaudio, Ráez, and Sridhar [25]. Without yet going into the details, these works highlight the importance of graph matching, that is, the task of recovering the latent matching n, given the two correlated graphs (G1, G2). In brief, when n. can be perfectly recovered from (G1, G2), then one can take the union graph G1 Va. G2 of G and G2, which is also SBM, but with larger edge density,
community recovery given two correlated SBMs (G1,
3

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which makes community recovery easier. This shows that exact community recovery is possible from (G,, G2) even in parameter regimes where this is impossible from just single graph G₁ (see [57]).
Graph matching. Motivated by the above discussion, we now discuss average-case graph matching and notions of recovery. In general, suppose that (G1, G2) are two correlated random graphs with n vertices each and that т. is the underlying latent vertex matching. The goal of graph matching to output an estimator म ( G2) that is close to Te. There are various notions of recovery depending on how close से is to T the two most relevant notions are the following, We say that an estimator achieves exact graph matching if lim F( T.) = 1. We say that an estimator achieves almost exact graph matching if with high probability there exists subset I S [n]] with [ = (1 - (1)) such that म| = n. 1z. where क|; denotes the restriction of π to I. In words, almost exact graph matching allows the estimator to make vanishing fraction of errors.
The graph matching problem has been widely studied, with applications computer vision [13, computational biology [61], and social networks [56]. In particular, de-anonymizing social networks possible with graph matching algorithms, which implies that anonymity not equivalent to privacy [51]. That said, studying the limits of graph matching algorithms-inclue potential information-computttionn gaps, as we shall discuss-can help guide data regulators on when to take more actions with regards to data protection, in addition to anonymity.
As discussed in the introductory paragraphs, there has been large body of recent work average- case graph matching, both studying correlated Erdös-Rényi random graphs [56, 14, 15, 65, 16, 23, 27, 24, 49, 5, 21, 22, 39, 40, 41, 17, 20, 18] and more general models of correlated random graphs [8, 32, 54, 58, 77, 25, 63, 59, 19, 67, 66, 60, 10]. In particular, Rácz and Sridhar [57] cooomationntheorete limits for exact graph matching in correlated SBMs:
determined the fundamental
in the logarithmic average degree regime discussed above, this threshold is s aath 1. However, this result is information commattion-hhortttcc... and the authors posed the open problem of finding an efficient algorithm for exact graph matching, whenever this is information iifommation-thhoretiaalll theoretically possible.
Our main contribution positively resolves this open problem, giving the first efficient algorithm for graph matching for correlated SBMs with two balanced communities. Our algorithm generalizes the recent breakthrough work of Mao, Wu, Xu, and Yu [41] that developed an efficient graph matching algorithm for correlated Erdös-Rény graphs. As an application, our results imply novel efficient algorithms and results for community recovery. We now turn to describing our results.
3
Main results: Graph matching
Our main theorem for graph matching on correlated SBMs is that there exists a polynomial time algorithm that can achieve exact matching if s² > a, where is Otter's tree counting constant' [55].
Theorem 1. Fix constants a + b > 0 and s E [o, 1]. Let (G1, G2) ~ CSBM(n, a logn For any E > 0, if 2 a + E, then the following holds.
log s).
n
n
(a) (Almost exact matching) There exists a polynomial-time algorithm that outputs a subset 1 E [n] and a mapping मे I I [n] such that गे = Jxl1 and [7] = (1 - o(1))n with high probability.
(b) (Exact matching) If, in addition, (a + /)/2 > 1, then there exists a polynomial-time algorithm that ouputs a mapping such that lim P( m.) 1.
Several remarks now in order about the tightness of the main result, overview of the chandelier counting algorithm when a = b, and the main challenge of our analysis.
Tightness. This result is tight whenever s > a, because sa /2/2 = 1 is the information-theoreet threshold of exact graph matching given two correlated SBMs (GI, G2) [57]. When s² < a, is conjectured that an infornation-cooppuatioo gap exists for the correlated Erdös-Rényi graphs [41]. Specifically. by assuming that a = b. it is Erdōs Rényi graphs exactly if s²a > 1 [14, 15, 65]. However, it is believed hard to find a polynomial- time algorithm to do this. In our model with SBMs, which is an extension from Erdōs Rénii graphs, it is also likely hard to find polynomial
-hhorttiiit possible to match correlated
algorithm when 5² a.
'This constant captures the base of the exponential growth of unlabeled rooted trees: the total number of unlabeled rooted trees with N vertices is (a + o(1))

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(a) $=0.6
(b) 0=0.6
(c) s=0.8
Figure 2: Phase diagram for graph matching on (G1, G2) ~ CSBM(n, alog blog s). The red diagonal line depicts b, which is an Erdós Rényi graph. Black regions: exact graph matching is possible and can be done efficiently for each community separately by applying the graph matching algorithm for correlated Erdōs-Rece graphs: Green regions: exact graph matching is possible and can be done efficiently; Light green regions: exact graph matching is impossible, but almost exact graph matching possible and can be done efficiently; Cyan regions: exact graph matching is possible and can be done efficiently by first recovering the community labels almost exactly; Yellow regions: exact graph matching is impossible but almost exact graph matching can be done efficiently by first recovering the community labels almost exactly.
n n
Erdōs-Rényi
Signed chandelier counts. Our theorem extends from the main theorem in Mao al. [41], which proposed a polynomial-time algorithm that matches the correlated Erdós-Rényí graphs exactly. The algorithm has two main steps: First, construct signature vectors 8; and t, for vertices i E [m] in G and [n]] in G2 by the signed subgraph counts of a specially designed graph class-termed Chandeliers-and calculate the weighted inner product of pairs of signature vectors (st, and match vertices if the inner product value is large enough; Second, use a seeded graph matching algorithm boost the almost exact graph matching algorithm to exact graph matching. It natural to adapt this algorithm from correlated Erdös-Rénn. Rényi graphs to correlated SBMs but the details present non-trivial challenges, as we explain below.
Main challenge. The main challenge on correlated SBMs is that signed subgraph counts no longer free lunch. Signed subgraph counts is counting the subgraphs centralized adjacency matrix, which is first proposed by Bubeck et al. [9] and later commonly used to control the variance of counting statisties. The success of the chandelier counting method relies the sufficient separation of the two inner product distributions of true and false vertex correspondence. We want to find a way that keeps doing the adjacency matrices centralization possible. Recall that explore the graph matching motivated by community recovery. Interestingly, the solution to this centralization problem is now the other way around-using matching. Our main technical contributions are first showing that when there are no eттог occurs in the community label estimate, the signed chandelier counting can be generalized to correlated SBMs and then show that when the exact community recovery is not possible, the errors introduced in the community label estimation, which is polynomial in n, are actually tolerable for the whole algorithm.
rough community label estimate help the graph
The analysis falls in two cases. If a sD (a, b) > 1, then we can achieve exact community recovery on each of the graphs by applying the community recovery algorithm from [47]. In addition, if 22 > 1, then it suffices to look at each community individually. This is easy and follows in a black-box fashion from [41] (See Section 7). However, on the other side, s2g < 1, one still needs to use information the community information. Therefore, we need go through the whole algorithm analysis again in this case. Note that the analysis would work for both regimes with constraint on 2 1. We plot the black-box regime in black and the non-black-be regime in green in Figure 2.
The second case is even trickier. If sD (a, b) < 1, by the same algorithm, we can only obtain almost exact correct community labels ã a and o on graph G1 and G2, respectively. We perform adjacency matrix centralization based on ã. and θn and show that the emror introduced in this step is negligible
For a SBM(n, alog ò blog ), each community, conditioned on its size N n/2,, is Eédös-Rényi graph
G(N, long R G(N. log N
5

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(a) $=0.6
(b) $=0.7
(c) $=0.8
Figure 3: Phase diagram for exact community recovery with fixed & on correlated SBMs. Green regions: exact community recovery is possible from G1 alone and be done efficiently; Lightgreen regions: exact community recovery is possible from (G1, G2) but impossible from G1 alone, exact graph matching can be done efficiently and therefore exact community recovery can be done effi- ciently; Violet regions: exact community recovery is impossible from G1 alone, impossible from (G1, G2) if (=th)) + s(1 s) D+ (a, b) < 1 and possible if s a+bb + s(1 - Ds) (a, b) > 1 [25]. It is unknown whether there exists efficient algorithm for exact community recovery in this regime.
in the sense that the inner-product scores remains sufficiently distinguishable between true pairs T. (i)) and fake pairs (j (i)).
Application: Community recovery
Once matching up the vertices on two correlated graphs, we can combine the information of them onto union graph and then immediately have application community recovery. Our result for community detection that there exists a polynomial-tinn algorithm for exact community recovery correlated SBMs when the squared edge correlation parameter satisfies s²
Theorem 2. Fix constants a + > and s E [0, 1]. Let (G1, G2) ~ CSBM(n, a log log s). For any 0, if
(² + b
5² ta,
1,
and (1-(1-s))
then, there exists an estimator GG1, G2) that can be computed in polynomial such that lim,-+oo [[ Σ @(ioo_ (00oo..1. (i) =
Theorem 2 is direct application of our Theorem 1. The proof mainly follows the Theorem 3.3 in [57], which gives exact community recovery on the union graph of G₁ and G2 regarding to the permutation म., G1 Vş G2. The key difference is that we substitute the maximum posterior estimator used in the first step with the T(G Gz) output by the algorithm used to prove Theorem Figure is summary of the phase diagram for community recovery determined by this work along with previous works [57, 25), focusing on the exact community recovery and efficiency.
Remark 1. Consider a more general correlated SBMs with K correlated graphs (G1, Gr....,, for exact community recovery above the exact graph matching threshold for K correlated SBMs when D4 (a. b) I-[1-s]
"2b.....Gr, Gk) ~ CSBM(n, a logn loog ss, K). Theorem also implies an efficient algorithm n a
(Theorem 3.6 in [57]).
5 Algorithm and the overview of the proofs
In this section, we define few key concepts, give a brief overview of the algorithm, and briefly discuss the proof of Theorem and challenges. See Appendix C for full version.
Chandelier. An (I,, M, K, R, D)-chandelier a rooted tree with branches, each of which consists of path with M edges (M-wwrr)) followed by rooted tree with K edges (K-bulb) the K X-bulbs

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non-isomorphic to each other, each of them has at most R automorphisms, and the maximum degree most D. For each chandelier H. let K(H) denote the set of bulbs of H.
For a rooted tree T, let aut(H denote the number of rooted automorphisms of T throughout this paper. We abbreviate rooted automorphism as automorphism when it is clear that we are applying it to chandelier. The number of automorphisms of И is determined by the automorphisms of its bulbs.
M, Let T denote the family of non-isomorphic (L, M, K, R, Dh-chnddelier. The family size of chande-
lier is |T| = (l)). where edges, at most R automorphisms, and maximum degree D. Otter [55] showed that the number of unlabeled rooted trees with K edges (and no constraint on the automorphisms and vertex degrees) is [J(K, 0o, [ (a + o(1)) where z 0.338. We show that under proper choices of R and D, we have [J(K, D) = (a o(1)) - K in Section C.
J(K, R. D) denotes the collection of unlabeled rooted trees with K
Algorithm overview. Given (G,, ~ CSBM P. q, Our algorithm contains mainly three steps. Firstly, we apply the Algorithm 3 (discussed in Section D.3) by Mossel, Neeman, and Sly [47] to obtain almost exact community label estimates for each single graph. Secondly, we calculate the signed chandelier counting [41] based similarity score give an almost exact graph matching (Algorithm 1 and Algorithm 4). Lastly, we boost the almost exact matching to exact matching by extending the seeded graph matching algorithm [41] on Erdös-Rényi graphs SBMs (Algorithm 2).
Subgraph counts. For an arbitrary weighted adjacency matrix M of some adjacency matrix A. vertex E [n], and rooted graph H, we define the weighted subgraph counts on M as
Wi, (M)
Σ Ms, where Ms = II Ma.
S(+)=
CE
and S(i) enumerates subgraphs of the complete graph Kn, rooted at i, that are isomorphic to H.
When M is the adjacency matrix itself, (M) is the usual subgraph count, representing the number of subgraphs rooted at i in M that are isomorphic to H. When M is the centralized adjacency matrix A := A- - E[A], we call Wa (MM) a signed subgraph count following [9]. However, we do not have access to [[A] in many cases. Specifically for SBM,
estimate E[A] through estimating the community labels. We define the approximately centralized adjacency matrix regarding to community
label estimate o denoted as A entry-wise as A.j Ai, plaji -
ql [])
Given a family Ht of non-isomorphic rooted graphs, we define the subgraph count signature of vertex i WH (M) == (Wa,u (WV, HEH
Similarity score. Given a pair of correlated SBMs (G1, G2), we define the similarity score between vertex i on graph Gi and vertex j on graph G2 as weighted inner product between two signatures:
b == (W7 (A), WJ (B) = Σ (A (B).
HET
When we do not have access to A and B, we use the approximately centralized adjacency matrices A and B We define the similarity score in a slightly different notation:
T = (W? (A A), W (B "²)))=
=))= = Σ aut(H) 3 A.AAAW,,,.
HET
Almost exact graph matching. The first part in the analysis is to show that by calculating this similarity score, with an appropriate thresholding strategy, we match up (1 - o(1))n vertices correctly (Theorem 3). The high-level idea is to show that the similarity score distributions are well-separated between true pairs and fake pairs. We expect the similarity score having the following properties, under event H. := { n < VV+ z
+ +
} (to be discussed in Section D.2);
For true pairs j = T(i):
El (0) (0) 1x] > 0,
Varl
i3 1x] = (E( (1) 1xl²)
(1)
For fake pairs j a i():
El Sn. (6) 1x)
Ei kj 1x] = [100)(()))))))
V = o
(2)
n²
7

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Precisely forming bounds for these moments constitutes the main bulk of the paper. We provide results from Proposition I to Proposition Followed by these moment bounds, we have Theorem
Theorem 3. Fix a * b 0 and 8 E [0. Let p = a logn ,9 = blogn and (G1, G2) ~ CSBM(1, P. 8). For any > O, suppose 5² 2 a
There exists positive constants C2, C3. Ca. C3 > O such that the following holds. Pick K, M, N, Das
C1 E
CsK
log n
L
K = C2lonn M
R Rxxxp(Cak),
D Cs
log(ns(p Aq))))
(log log n)²
Pick an arbitrary e. E (0,1) and set |T o 2N where o := - (02+02 Then, Algorithm I outputs set with size (1 - o(1)) and a mapping such that 7|, = T. with high probability.
off
Proof challenge. regime sD+ (a. b) > the probability of existing one vertex being classified incorrectly is vanishing for Algorithm 3. Therefore, with high probability, A A. If sD (a. b) < 1, then the recovered â contains errors (polynomial in n), which will cause some edges being centralized incorrectly and thereby affect the moments calculation.
This phenomenon poses a challenge to the algorithm analysis. We highlight some key points in the context of second moment calculation. For simplicity, we ignore event H. here. From definition,
Var [
Σ
aut (H)aHuu
Σ
E
[Eo.
3 [E[ASs
Bs ATi
3 T²AT" BTz
a.1]
HJIET
(a))),())2n Ti (1)
Er.
[[As, θ
B
E..
の사 [ [[AAT.
B [o.]]
Let us define the union graph U S, UTUTT If sD 1, we view A and B" as A and B, respectively. E[As, Bs, Bs, Ar, o. * 0 only if there exists no edge E(U) such that it occurs only once among ($), S2, Tú, T2). This is because different edges are independent and centralized conditioned on
US
34 A BT, a.] + 0 even when edges are not However, in regime (a, < 1, E[A
occurring multiple times as we have the expectation of A can be non-zero when conditioning on o. and an estimate A that disagrees with o.. the edge type (i.e., for e (u, v), o, (u)or, (v) o d
This not only causes this mrossmmoments calculation being more complicated, but significantly increasing the possibility of the combinations between S₁, S,, Tı, and T2.
The most important high-level idea to properly bound the moments is: the momenttoment conditioning on specific x = Ao σ would be non-trivial if and only if all edges occurring only once are centralized incorrectly. Assuming there are edges occurring once, we show that the probability
that ō satisfying this property is no greater than n (a.)_ logia//)//a for any E > 0, by using
the definition of T and Lemma 8 later in Section K and Section L. It turns out that we require s(uD (w.0) lox(te/0/2))
n
D
to be of !o for some positive constant C so that (1) and (2) are satisfied.
Efficient algorithm. Calculating D. exactly (Algorithm takes quasi polynomill time as searching for all S > H has time complexity n ((N), Algorithm 4 in Section F computes an approximated score in polynomial time. Specifically, we follow the color-coding -bsed similarity score approximation from [41]. The basic idea is coloring the stochastic block model using N + colors uniformly at random. Then, we only do signed counts on vertex sets that are colorful with N 1 distinct colors. We show that this is an unbiased estimator of th and only potentially increase the variance by an additional constant factor in Section F. The result is stated as in Theorem
Theorem 4. Theorem continues to hold with the color-coding sampled estimation Vi in place of s Moreover, by Algorithm ( 1 (.jmm) can be computed in Onn)) for some constant C C(s) depending only on E, where e is from (3).
i
Note that S₁ and Ti are rooted at i, while S2 and T2 are rooted at j. For = (u,v) v) E E(S)), we say it occurs in Ti if E(Ti) and it occurs in Sa (resp. 7b) if (o. (u), (v)) E E(S)) (resp. Th).
8

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Exact graph matching, The final step of the algorithm is boosting the almost exact matching to a exact matching. The key idea is exploring the number of common neighbors.
Denote (i(i, j) as the number of common neighbors of i and j under correspondence π. The high-level idea is that if i and j form true correspondence, then for correct partial matching on (1 - m())) vertices, with high probability, j) ž p² 2 + 2n under the nice event H. Therefore, we match up i and j if they have more common neighbors than this threshold. Define h(x) = x: log x - I + 1. We give the following guarantee.
2
Theorem 5. Fix a + b > 0 and s [0, Let p logn .9 blogn and (G1, G2) n CSBM(n, p, 8, Suppose 8² (²()))) 2 + and x² 2 a e, for some & > Lel Y be the unique solution in (1, 00) to h(()) = 3log
[n-22ppgsz- Then, the seeded matching Algorithm 2 with input ā and an index set I C [n] |4| = =(--oL))n 2 (1-e/16) such that म|r тN outputs an exact matching से = n. in On 3 (p + 9)²) time with probability 1 (1))
Putting these pieces together implies Theorem 1. From Theorem 4, we know that for (G1, G2) ~ CSBM(n, a loggn , loggm s), a o b, if s² 2 a+ for some = > 0 then there we can match (1 - (1))m vertices correctly and efficiently with high probability. Take the returned from Algorithm 4 as input of Algorithm 2, then Theorem 5 guarantees that the final output ã. = T, with high probability.
6 Related work
Graph Matching, Correlated Frdös-Rényi graphs were first studied in [56] for social network de- anonymization. The cnformation-theoretii threshold for partial matching was determined by [24, 27] and the iioommation-thooreee threshold for exact graph matching was determined by [14, 15, 65].
Mao et al. [40] proposed the first efficient algorithm that achieves exact graph matching for correlated graphs with average degree (1 ++)) e) log n S nq S n ottog (n) and constant noise. This algorithm only requires a constant edge correlation (sufficiently close to 1) rather than converging to I, which represents a perfect correlation. Mao et al. [41] followed up with an improved efficient algorithm that achieves exact graph matching for any correlation P satisfying p² when nq(4 + p[1 - 4)) 2 (1 + E) log n. It is conjectured that for random graphs of logarithmic average degree, p² a is the computational threshold [41]. Muratori and Semerjian [50] added small constant constraint on the maximum vertex degree a chandelier to improve the runtime, at the expense of having slightly larger constant â the minimum squared correlation requirement.
In the denser regime where p n = E (0, 1]. Ding and Du [17] established a sharp cntrmatiiontthooeefee threshold for matching a positive fraction of vertices. Ding and Li [20] also developed an efficient algorithm for exact graph matching whenever the edge correlation is non- vanishing, which goes beyond the Otter's 's tree counting constant.
Several recent works also go beyond correlated Erdōs-Rényi graphs. Wang et al. [64] studied the exact graph matching with additional attribute information on vanishing edge correlation.
Closely related to our work, Yang et al. [67] adopted the binary tree counting algorithm [40] to give an efficient graph matching algorithm for correlated SBMs. However, [67] makes several significant assumptions (which we do not). For one, the algorithm in [67] assumes that the community labels are known. This is strong assumption which may be unrealistic in practice; moreover, this precludes using graph matching as a tool for improved community recovery. In contrast, we do not assume that community labels are known; in fact, significant part of our technical work is devoted to dealing with the errors arising from estimating the community labels. Moreover, our graph matching algorithm can be directly applied to improve community recovery, discussed in Theorem 2 and Section 3. In addition, [67] makes strong assumptions on the parameters, assuming that (1) the average degree is at least (log 1.1, (2) the SBM has least 3 communities, and (3) the correlation parameter satisfies > - so for some unspecified (small) Eg. In contrast, our results hold in the most interesting regime of logarithmic average degree and the most natural setting of two balanced communities; moreover, our assumption on s is also weaker.
Community recovery with side information Beyond correlated SBMs, there are some other mod- els utilizing side information, from multiple networks [28, 62, 33, 30, 68, 71], additional covariates

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[7], or both [44, 37J. Multi-layer SBM is first mentioned in [29], which is generated as following: first, generate the community labels for all vertices and fix them for all layers; second, form edges on each layer based on the community labels. Typically, different layers in multi-layer SBM are conditionally independent given the shared community labels. addition, several works [44, 37] also encode community membership correlated covariates onto each node. Aside from the multi-layer SBM, Braun and Sugiyama [7] recently studied community detection on a novel variation of SBM whose edges are attached with vectorial covariates.
Discussion and future work
Our main contribution in this paper is give the first efficient algorithm for exact graph matching for correlated SBMs with two balanced communities, as well as a rigorous proof of its correctness (Theorem 1). We also discuss novel applications to community recovery (Theorem 2). At the same time, our work raises many interesting questions for future research, which we discuss here.
Optimal runtime. While our graph matching algorithm is efficient, it would be desirable to under- stand the optimal running time that can be achieved. Mao, Rudelson, and Tikhomirov [40] gave an efficient algorithm for matching correlated Erdös-Rényi graphs with runtime n²*o(11;; the main drawback is that this algorithm requires the correlation parameter to satisfy 8 > I en for some unspecified (small) Eo > 0. Nonetheless, it would be interesting to generalize this algorithm to correlated SBMs and the techniques developed in work may be useful to do so. In very recent (and concurrent) work, Muratori and Semerjian [50] gave faster algorithms for matching correlated Erdōs Erdōs-Rényi graphs by introducing constraint on the maximum degree of chandelier, at the expense of strengthening the condition s² > a to x > ā for some ā > a. Exploring the connections between our work and theirs, and generalizing their ideas to correlated SBMs, are of interest.
information-coompuaaiio gap. An important assumption throughout this work is that the correlation parameter satisfies s² > a. We believe that this is inherently necessary and that there is no efficient algorithm (in the logarithmic average degree regime) when 8² < ax At the same time, exact graph matching is
possible whenever (a + b)/2 >> 1, so there is a conjectured information compuutatinn gap. This mirrors the conjecture in [41] for Erdös Rénts graphs; see also [18] for the low-degree hardness results the correlation detection and [10] for the very recent low-degree hardness results on testing a pair of correlated stochastic block models against a pair of independent Erdös-Rese Rényi graphs.
Efficient exact community recovery when exact graph matching is not possible. Gaudio, Rácz, and Sridhar [25] determined the information -hhoooii threshold for exact community recovery on correlated SBMs, in particular showing that there a regime when this is possible even though (1) this is impossible with single graph and also (2) exact graph matching is impossible. It remains unknown whether this can be done efficiently in this regime. We believe that this is possible, and our work is an important starting point for this question, yet additional ideas are needed to understand the subtle interplay between graph matching and community recovery in this regime.
Sparser and denser regimes. Our work focuses on the most interesting regime where the average degree is logarithmic in n; is worth understanding other regimes too. In particular, the chandelier counting algorithm by Mao et al. [41] gives almost exact graph matching whenever the average degree diverges. In our Theorem we require that the average degree diverges logarithmically for the corresponding result, so that the error rate for community recovery estimate is polynomially small n. would interesting to overcome this technical barrier and extend the analysis to this sparser regime. Denser regimes are easier to understand. A close inspection of our analysis shows that it also works when the average degree diverges as a (small) polynomial in n; in even denser regimes, the community partition can be recovered exactly and efficiently whenever lim inf [pn/q,, - 1/ > 0 (see [47)) and then the graph matching algorithm in [41] can be applied in a black-box fashion.
General block models. We focused here on the simplest case of SBMs with two balanced com- munities. It is of great interest to develop efficient graph matching algorithms in the general block model with & communities, whenever this is possible. Recently, Yang and Chung [66] determined the cotrmattion-hheoeettee threshold for exact graph matching in the k-community symmetric SBM, extending the results of Rácz and Sridhar [57]. We conjecture that substituting the community recov- ery algorithm used in our work with the degree-profine profiliig algorithm by Abbe and Sandon [4] gives an efficient algorithm for graph matching in this more general setting, assuming again that s² > a.
10

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Acknowledgements
We thank Julia Gaudio, Anirudh Sridhar, Yihong Wu, and Jiaming Xu for helpful discussions. We also thank anonymous reviewers for constructive feedback. S.C. was supported in part by the Institute for Data, Econometrics, Algorithms, and Learning (IDEAL), funded through the National Science Foundation TRIPODS Phase II program (NSF grant ECCS 2216970).
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A Organization
The rest of this paper is organized as follows. In Section B, we give some notations used throughout. In Section C, we define the similarity score of a pair of vertices and give the formal proof of Theorem I based on Theorem 3 (Almost exact graph matching), Theorem 4 (Efficient algorithm for almost exact graph matching), and Theorem (Exact graph matching by seeded graph matching). In Section D, we talk about some preliminaries: tail bounds, nice events, tree node assigning sub- sub-problem, an automorphism inequality for trees, and the calculation for cross-moments. These results will be repeatedly used in the following sections. In Section E, Section F. and Section G, we present the proofs for Theorem 3. Theorem 4, and Theorem 5, respectively. Section F and Section G are self-contained, while Section E. contains several propositions whose proofs are deferred and which make up the remainder of the paper.
In Section E, we introduce six additional Propositions to show that under two different cases- namely, sD+ b), 2 and (a.b) < the mean and variance of the similarity score are properly controlled. Specifically, if i sD+ (a, b) 2 1, then Proposition I gives the mean calculation of the similarity score, while Proposition and Proposition are about the variance calculation of the similarity score for true pairs and fake pairs of vertices. If sD+ (a. b) < 1, then Proposition 4 gives the mean calculation of the similarity score, while Proposition 5 and Proposition 6 are about the variance calculation of the similarity score for true pairs and fake pairs of vertices. The proofs of Proposition 1, Proposition 2, Proposition 3, Proposition 4, Proposition 5, and Proposition are presented in Section H, Section I, Section J, Section K, Section I., and Section M, respectively.
B Notation
For any graph G (V, E), we denote E[G) as the edge set and V(G) as the vertex set. We let e(G) : [[G] denote the number of edges of graph G and v(G) == [V(G) denote the number of vertices in graph G. We define the excess of graph G as e(G) (G). the difference between the number of edges and the number of vertices of G.
Consider an arbitrary graph where vertices are equipped with two possible community labels (+1,-1), we denote / as the set of vertices with community label +1, V- as the set of ver- tices with community label (v) the set vertices that are neighbors of
Let x be a permutation on [ı:]. (G1, G2) ~ CSBM(n, p, g.s). Let A (resp. B) be the adjacency = A E|A] (resp. B) be the centralized adjacency matrix. We further define the approximately centralized adjacency matrix with respect to community label estimate ō
matrix of Gi (resp. G2). Let :=
as A A := A - EA, where EA is an n X n matrix whose (i, entry is p if (i) = ā(j) and otherwise."
We denote G1 V2 G2 as the union graph with respect to x, such that (i,7) E E(G, V~ G2) if and only if (i.j) E E(G1) or (ii), a(j)) E(G2). We denote G1 Ax G2 the intersection graph with respect to म. such that (i, E E(GI Ar G2) and only if (i, E E(G1) and (I(1), π (i)) E E(G2).
We denote the variance of in-community edges of = sp(1 (p) and the variance of cross-community edges as x² == sq(1 ss). Denote the correlation for in-community edges and cross- coss-commmuity edges as 0+ and p-. respectively. In addition, we define p = titp_ Consider the average degree
being logarithmic in the number of vertices, then for some constants a, b > 0.p n 4 Mog a
alog
P (1 + O
((oog log ))++ (1 O log ))pp (1 + O "og"))s ss.5
Throughout the paper, we use standard asymptotic notation O(-), ((), 0(.))))))))) ww)). Any limitation is for n + do without special explanations. For real numbers 2, y. we define x V y max sr, y} and x A y == min(æ; (z. y). In this paper, log is natural logarithmic function (with base e). Further notation is introduced in the following section which details the algorithms.
'See Section K for illustrations and more discussions on approximately centralized adjacency matrices.
For arbitrarily small constant e > 0. there exists n large enough, such that p² 2 + e if and only if 2 a +
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C Proof overview and proof of Theorem 1
C.1 Chandelier
line of works convert the graph matching problem from quadratic assumption to linear assignment by creating a signature vector 8; for each vertex i E [v]] followed by calculating the similarity score B (s { of all possible pairs of signatures two graphs. Recently, Mao et al. [41] proposed a special tree family T, chandelier, that shows result of efficient graph matching under constant correlation.
Definition 1 ((L, M, (R--chandelie((11).. An (L, M, K. R)-chandelier is a rooted tree with L branches, each of which consists of a path with M edges (M-wire), followed by rooted tree with K edges (K -buub;;; the K K-bulbs are non-isomorphic to each other and each of them has at most R automorphisms.
In this paper, we give an alternative definition of chandelier with five tuple. The first four parameters remain the same (I., M, K, Rh-chaddelier. The last parameter D) stands for the maximum degree of vertices on this chandelier. We explain the necessity of controlling D in the proof challenge.
Definition 2 ((L, K, R. channdelel,,, An (L, M, K, R, chchndellirr is rooted tree with branches, each of which consists of path with edges (M-wire), followed by a rooted tree with K edges (K bulb);; the K-bulba are non-isomorphic each other, each of them has at most R automorphisms, and the degree of each vertex at most D.
For each chandelier H, let K(H) denote the set of bulbs of H. For a rooted tree T, let aut ()) denote the number of rooted automorphisms of T throughout this paper. We abbreviate rooted automorphism as automorphism when it is clear that we are applying of automorphisms of H is determined by the automorphisms of its bulbs. Because all bulbs are non-isomorphic to each other,
chandelier. The number
aut (H)
II aut
BEK(H)
Let T denote the family of non-isomorphic (L, M, R, Dy-chandelier. The family size of chande- lier is IT| = (l7) where J = J(K, R, D) denotes the collection of unlabeled rooted trees with K edges, most R automorphisms, and maximum degree D.
Otter [55] showed that the number of unlabeled rooted trees with K edges (and no constraint on the automorphisms and vertex degrees) is [K, 00, xc) = (a o()) where 0.338. We show that under proper choices of R and D, we have 1 [K, R, D) (a + o(1)) -K through the following two Lemmas.
Lemma 1. Let K be the number of vertices on unlabeled rooted tree, C' log(1/a) 2 0.9227,
D 2 C" log K. As K- + 00,
o,, =1 =1 o(1. 20,0000,0)
(5)
[J( "p" where ap is the radius of convergence Proof. Otter [55] characterized that [J[K,o, [0] a
for the generating function of the number of unlabeled rooted trees whose maximum vertex degree less than equal to D. Goh and Schumutz [26] (Theorem 7) showed the following property: as D + D0, for some constant C > 0, &D = a + Ca + o(a Immediately we can see
that [J(,,o, [X,
(1 O)aP))
o(1 if Ko - 0. Let C' >
log(/aj choosing
D 2 C" log K satisfies Ko +
Lemma 2. Let K be the number of vertices on a unlabeled rooted tree, C. be a constant and choose R = exp(CK). For sufficiently large C and K - XC,
[J(K, , R, oc) [[(K 20, 00)
= =1-
(6)
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L-
Ma-
dege-D
Figure 4: A chandelier.
Proof. Olsson and Wagner [53] (Theorem 2) showed central limit theorem result for the number of automorphism on unlabeled rooted trees: 1
vK (log aut(Hx) =K) - NO, o²) as K + oo, where Hx is uniform random unlabeled rooted tree with K edges and μ 0.137, o² 0.197. This implies that for some constant C > μ and R = exp(CK). au)(Hx) < with high probability. D
Putting together Lemma and Lemma and choosing R and D as specified, we have that [J(K R, D) = (1 o(1))] co, oc) 00,00) (a o(1)) Let 8 denote universal constant
such that [T] S 3K We take B
C.2 Algorithm overview
Given (G1, G2) ~ (S Our algorithm contains mainly three steps. Firstly, we apply the algorithm by Mossel, Neeman, and Sly [47] to obtain almost exact community label estimates for each single graph. Secondly, we calculate the signed chandelier counting [41] based similarity score to give an almost exact graph matching. Lastly, we boost the almost exact matching to exact matching by extending the seeded graph matching algorithm [41] on Erdös Rényyi graphs stochastic block models.
Almost exact community recovery. Obtaining community label estimates for both graph G1 and G2 is the first step of our algorithm. This is necessary for centralizing the adjacency matrices for the signed subgraph counts afterwards. We expect the following properties from the community recovery algorithm:
(a) gives almost exact recovery (down to the information- threshold, which s² + 6) = 1 in the correlated SBMs with two balanced communities [57]);
(b) gives rate for each vertex of unverse-polynooee
(c) gives error rates on different vertices that are approximately independent.
The community recovery algorithm in [47] (described as Algorithm 3) has been shown with property (a) by [47] and property (b) by [25] with error rate of n - (a.b), In this paper, we show that property (c) satisfied (See Lemma 8).
Subgraph counts. For an arbitrary weighted adjacency matrix M of some adjacency matrix A, vertex (n). and rooted graph H, we define the weighted subgraph counts on M as
(M) = Σ Ms, where Ms
II Mc. CE
S(OCH
and S(i) enumerates subgraphs of the complete graph Kn, rooted at i, that are isomorphic to H.
When M is the adjacency matrix itself, Wi, (M) is the usual subgraph count, representing the M that are isomorphic to H. When M is the centralized adjacency matrix Ā. A F[A], we call W. (MM signed subgraph count following [9]. However, do not have access to E[A] in many cases. Specifically for SBM, we can estimate E[A] through estimating the community labels. We define the approximately centralized adjacency matrix regarding
number of subgraphs rooted
to community label estimate o, denoted as A" entry-wise as A As.s pla||==(1) [1]1))
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Using A in (7) yields the weighted subgraph counts for approximately centralized adjacency matrix. We also refer to this as signed subgraph count, though errors may exist.
Given family H of non-isomorphic rooted graphs, define the subgraph count signature of vertex i as
(M) = (Wi, (M)) MEH.
(8)
Similarity score. Given a pair of correlated SBMs (G1, G2). we define the similarity score between vertex i on graph G1 and vertex j on graph G2 weighted inner product between two signatures:
2i = (A), (B)) = E
aut(H)W..rAAWW (B),
HET
where T is the family of chandelier.
When we do not have access to the centralized adjacency matrices À and B3 we use community label estimates A and Ø for G1 and G2 correspondingly. We define the similarity score with a slightly different notation:
4 (W? A W
B
B)
Σ NET
ant(H)W.,HH (A WW, B
(10)
Almost exact graph matching. The first part in the analysis is to show that by calculating this similarity score, with an appropriate thresholding strategy, we can match up (1 - o(1))n vertices correctly (Theorem 3). The high-level idea is to show that the similarity score distributions are well-separated between true pairs and fake pairs. We expect the similarity score having the following properties, under event H:
For true pairs j - T Ti)):
1ul 0, Varlø (((E (1) 1nl?)
For fake pairs j # T(I):
E in, (1) 1v
E 1x] (E[4
Var
n²
Precisely forming bounds for these moments constitutes the main bulk of the paper. Aside from proving the desired properties of the first and second order moments, we follow the color-coding- based similarity score estimation idea from [41] to analyze an efficient algorithm. The basic idea is color the vertices of SBMs using N + 1 colors uniformly random. Then, we only do signed counts on vertex sets that are colorful with N + colors. We show that this is an unbiased estimator and only potentially increase the variance by an additional constant factor in Section The result stated formally as in Theorem 4.
Algorithm Almost Exact Graph Matching for CSBM
Input: Adjacency matrices A and for (G1, G2) ~ CSBM a loga log ,3), constant c, and mean value μ.
Output: A mapping मे - [n].
1: Run community recovery Algorithm [47] on A and B separately and get label vector θ θ
2: For each pair of vertex in A and vertex j in R compute their similarity score as in [41]:
के 1j = (W7 (A AAa) WT B B²²))) Σ aut(H)W,,H)) (A WH,"BB""
(11)
HET
3: Let T = cμ4, output I == {ili [m]. E [n]. s.t. $7, 2T, and 4k [m] (5}, TT T).
Proof challenge. In regime s sD (a, b) > the probability of existing vertex being classified
incorrectly is vanishing. Therefore, with high probability, A A. If sD+(a,b) 1, then
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the recovered ō contains enrors (polynomial in n), which will cause some edges being centralized incorrectly and thereby affect the moments calculation. For example, let i. E [n] be two vertices on G1 who has the same community label (1) =0 (j). If only one of i, is labeled incorrectly by
Algorithm 3, then the expectation of A conditioned on o. and ē is
This phenomenon poses challenge to the algorithm analysis. We highlight some key points in the context of second moment calculation. For simplicity, we ignore event H. here.
Var 1
Σ (ut(H)aut(I)
Σ S (1t)
Σ
Lo.
[[AS 3 D',
Ari B12
o.]]
HJIET
Ti (1)
-Ea.
[[As, a 8
B
Er.
[[AT]] a oy [
B
[o.]]
Let us define the union graph U= S1 US UT. If sD b) > 1, we view A and Bd as A and B, respectively. E[As, Bs, Ar, o.] * only if there exists no edge e E(U) such that it occurs only once among (S,. S2, To.Ti, T2). This is because different edges are independent and centralized conditioned on a.. Without loss of generality, we assume there exists an edge e E (A))
occur only on S1, thus E[Ae o.] 0 and also E[As, Bs, Ar, Ar,
Io. =0=
However, in regime (a, b) < 1, E[AS] 82 AT, A
BT o.] + 0 even when edges are not
have the expectation of A² can be non-zero when conditioning on o. and an estimate ō that disagrees with a on the edge type (i.e., for e = (u, u). ax (u) of σ (u)ō (e)). This not only causes this cross- moments calculation being more complicated, but significantly increasing the possibility of the combinations between S Ti, and 72. The maximum vertex in the union graph grows from 2N to 4N, squaring up the trivial bound on the number of subgraphs on the complete graph K₂ that is isomorphic to U.
occurring multiple times
The most important high-level idea to properly bound the moments is: the cross-moment conditioning on a specific = (ō δ) would be non-trivial if and only if all edges occurring only once centralized incorrectly. This is because, conditioning on ō satisfying the above property, the expectation of A. takes either p q or q p for all e E E(U) that occurs only once. Assuming there are = edges occurring once, we show that the probability that ā satisfying this property is no greater stx D2 (a.8) log(2/5)22)
for any E > 0. Intuitively, this is saying that incorrectly centralize edges at the same time, we expect the Algorithm 3 to label at least 1 vertices incorrectly. It turns out that we require a(oD, (6.0) logle/0)/2) be ol log : for some positive constant C so that (1)
than n
and (2) are satisfied.
Theorem 3. Fix # b 0 and s E [0.1]. Let a logn ,9 blog and (G1, G2) n
CSBM(n, p. 8). For any > suppose 82 a "
There exists positive constants C2, Co. C., C5 > O such that the following holds. Pick K, M, I,, N, D as
L Ci. 3
C2K
log
K= Colon, n, M log(ns(p ()))
R e=xxp(C.K), D= Ca
(log log n)²
(12)
Pick an arbitrary E 10,1) and set μ = |7|n
2N where o² := (= +o Then, Algorithum 1 outputs set I with size (1 o(1))n and a mapping such that 7|1 = # with high probability.
N PM off
Algorithm 1 takes quasi-polynomial time. Algorithm 4 in Section F computes an approximated score in polynomial time and satisfies the following Theorem.
Theorem 4. Theorem 3 continues to hold with the color-coding sampled estimation bj in place of & Moreover, by Algorithm 4, {
if (.jem]] can be computed in Onn ) for some constant = C(E) depending only E, where is from (3).
Exact graph matching. The final step of the algorithm is boosting the almost exact matching to a exact matching. The key idea is exploring the number of common neighbors for two unmatched vertices with regard to the current matching.
'Note that S₁ and Ti are rooted at i, while S2 and T22 are rooted at j. For e = (u, v) E E(S)), we say it occurs in Ti if E(Ti) and it occurs in Sa (resp. 7b) if (o. (u), (v)) E E(S)) (resp. Th).
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Denote N, (1. as the number of common neighbors of and j under correspondence я. In another word, N₂ (i. is the number vertex E such that v is a neighbor of in G1 and x(v) is neighbor of in G2. The high-level idea is that if i and j form true correspondence, then for correct partial matching т on (1 - o(1))m vertices, with high probability, (i, 2 p² s² s + 2n under the nice event H. Therefore, we match this threshold. In addition, we can show that the remaining vertices will be matched up with high probability. Formally, define h(x)) = log x and the guarantee as Theorem 5.
and j if they have more common neighbors than
1, we summarize the algorithm as Algorithm 2
Theorem 5. Fix a * b > 0 and & E [0. Let = a log , = log a and (G1, G2) ~ CSBM(n, p. p.9.8). Suppose
$21 (( 2 221+ and 8² 2 a t,
for some С > O. Let y be the unique solution in (1, 00) to h(7) (n 2)pqs2 Then, the seeded
log
matching Algorithm 2 with input ã and an index set C [n], ||| = (1 o(1)) 2 (1 S n such
that 1 TT outputs an exact matching nu in O( (p q) time with probability 1 o(1).
Algorithm 2 Seeded Graph Matching [41]
Input: Adjacency matrices and for (G1, G2) ~ CSBM(n, p. q: s). p a lug some a > 0, b >> 0. A mapping π - [n] with []] = (1 m())),, parameters p, q, 8, and "
a : b log for
(o) such that h(()) (n=2)pqs²*
3 llg
1: Let J = I, and π = म.
2: while there exists i J and & #(J) such that N*(i,)) 2 p2 2 (n+ 2n+) do
3: Add i to J and let ((1)= =
4: end while
Output: R.
C.3 Putting things together: Proof of Theorem 1
The proof of Theorem follows from Theorem 4 and Theorem 5.
Proof of Theorem I. From Theorem 4, we know that for (G1, G2) CSBM(n, a logn blogn s). + b, x² > efficiently with high probability.
for some > then there we can match (1 11-o(1) o(1))n vertices correctly and
Take the returned ît from Algorithm 4 as input of Algorithm 2, then Theorem 5 guarantees that the final output ãे Te with probability o(1). This completes the proof of Theorem 1.
D
Preliminaries
D.1 Tail bounds
Lemma 3 (Chernoff Bound, Theorem 2.1 of [31]). Let Binom p) be a binomial random variable. Then, for all t 2 0,
P(X 2 EX EX S exp
2(EX + +t/3)
t2
P(X < EX
t) S exp
2EX
Lemma (Multiplicative Chernoff Bound, Theorem 4.4 and Theorem 4.5 of [45]). Let X ~ Binoun(,, p) be binomial random variable, denote μ nip as the mean. Let h((x) log x
Then, for all 7 E (1,00),
P(X 7p) S exp(-uh(2))
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For y E (0,1).
P(X S Tp) I exp( (o((-uh(()).
Lemma 5. [47, 25] Suppose that a Let Y Binom m+, elog and Binom (m-, b log be independent. m = (1 o(1)) 2. (1 + o(1)) then,
n
P(Y < 2) = n2 D+ (=,b))++((1)
For any 0,
P(Y I E llgg n) S n n (a, -l +o(1)
This result has been proved in Lemma 8 in [3] and Lemma 3.3 in [25].
D.2
Nice events
When (G1, G2) CSBM(n, loog blog s), there some events that happen with high proba- bility and our following analysis intuitively relies on the happening of these nice events.
(Balanced Communities) We denote H == { n < -V-1.V-1 observe that [V+] ~ Binom(n, 2) and |V |V+1. By Chernoff bound,
We ~ 5 +
P(H*) P([V+ 2 + + P(V+ S n e(1-o(1)\v
(Reasonable Large Neighborhood) Let max (a. b), we also denote that F E [n], (W)) < 100 max(1, 7) log n).
Lemma 6. As - 0o, we have
P(F) n O(log" n)
Proof. Let X ~ Binom(n, loga
Fix i E [n]. conditioned on any o.. [W(i)) stochastically
dominated by X. Therefore,
(7 log n)²
P(((()) 2 7og n (100)) lo..) SPX 2 7log n) S exp
27 log 2/37 log
S <xpp( (exp(>>lop log n) = n ~ logg
where the second line uses Bernstein's inequality and the third line holds for any n such that log² n > 6. From an union bound, we have
= E[((oo) 2 -Ollog a)
D.3 Community recovery
We make slight change on the choice of the partition number of the community detection algorithm proposed by Mossel, Neeman, and Sly [47]. This algorithm gives almost exact recovery with n! I-sD (a.b) log(a/b) vertices labeled incorrectly. After community recovery, need to match the two communities in G1 and G2 by applying the community matching Algorithm 3.
Consider SBM(n, log blog)). Define 7 max(a, b}. For any vertex v E [n]. we define x the signed neighbor counts of v in G as
maig(w) = a. (v)
Σ a.(.).
(E)(2)
For any > 0, define set of vertices:
(G) = (u [n] maja(v) S elogn n or ((v)) 2 y ylog n).
Previous results (Lemma 5.1 [25], Proposition 4.3 [47]) have shown that Algorithm 3's correctness on [n] (G) with different choice of m and the lower bound of [N(u)) in the bad vertices set Te(G). our work, we first demonstrates that Algorithm on input (G, a, b, correctly classifies all vertices in [n] (G).
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Algorithm 3 Almost-exact Community Recovery [47]
Input: Adjacency matrix A vertices; parameters a, b, > 0.
Output: A community label estimate ō E (-1,+1}"
1: Choose a positive integer m satisfying (log(em(2 max(a, b) log* n) n)) 1)e/2 1. Initialize two empty sets, W+ and W-.
2: Using the spectral method of [3] to find community partition of [n]. denoted as (U+, U-).
3: Partition [r]] into {U1...., Um} uniformly at random.
4: for E [m]] do
5: Using the spectral method of [3] to find community partition (U,,+, U.-- of G[[n)] \ U.}. If
AU+++ 2 n/2, then swap Us+ and U,,..
6: For v E Uú, insert into W+ W- according to its neighborhood majority (resp., minority)
U.++ U if (resp. a < b).
7: end for
8: For E W+, set (i) 1, and for E W_, set (i) = -1. Return ō.
Lemma 7. Algorithm on input (n, a, b, s) classifies all vertices in [n] \. (G) correctly with high probability.
The proof directly follows from the proof of Proposition 5.1 in [25], with two remarks. First, although the maximum size of neighbors (N() we consider here is enlarged from 100 (1) log y log" n, adjust the condition of m accordingly such that the tail bound holds. Secondly, we need to justify that with high probability, all partitions done by the spectral method are still almost exactly correct under the new choice of m, which no longer a constant independent of n. Theorem 3.2 of [3] showed that the vanilla spectral method achieved optimal error rate, in the sense that El! Ei= 1 (o. S n (1-(++(1))) (a.b)_ This implies that with probability 1 - -1 the spectral method labels all but o(n) vertices correctly. Furthermore, for all i E [n]. U,, matches with V+ \ U, and U,-- matches with V U; on all but o(n) vertices after step (5) with high probability.
In this work, we further determine the probability generalization of the result on the P(u E I2) for an arbitrary E [m] (Lemma 5.3 of [25]).
vertices being in the set Le (G), which is
Lemma 8. Given a random graph SBM(n, log ab logg and a fixed subgraph induced by vertex set S E [m].
n
1s|s Ollogn,, then for any 0,8 0,
P((Vi E S,i E 1e) H) = 0(n (0 - 206(1-((1)) log n).
+
|f = o(log n). thr any E 0,6 >>00000 0,
P( E [2] n H) 00000
Proof. The main idea is considering the intersection of the interested event (Vi S, e 13) n with G = (Vi e S, (Ns(t)) & æō logn), where (v) denotes the set neighbors of u restricted on the vertex set S.
P([Vie 1) 02) & P( S,i ( P(G)" P(F).
(13)
Firstly, we give the upper bound of (P o.)lx. Assume that |S| 2 εō log without loss of generality. By using an union bound,
P(G) S |S| x E(P((Ns()) > ed slggn a..)]
[S]
|S|
a log n ye8 log n( blog n
= |S| X
|st 16 log
Ed eglogn
à= log
12
log loan log
Ca log? n jey log
O(n --505-11-- -05(-501-0()) -01))) ((()) loR
(14)
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where the second equation holds from the assumption of [S] = Olo Secondly, we study this event F (Vi S, E nH
P(E) = [[P S, m S E elogn} [., For any E S, maii)w)),ss( (f) is the difference of two independent binomial random variables Y, and [[ logg) With H happening.
[[[(([V E S, maiiin + maia;s (v) S e llg o.)1æl
I E[P(( S, majomn (())))))))))))))))))))) - <(1 6) logn) o.)1ul-
Zo where Y, ~ Binomm( ( (v)
([(a))ss) sa log ) Z. Binom((V (a.o))
we have [V [] (1 o ( (1 - o(1)) Since Y, and Zu do not take into account v E 5, they are independent for all V S. Therefore.
P(F) E[TTues ((Y, - Z S e(1 + 5) logn] ]_)1,] S (n - (a,b) - | I0(1))1s1
|S](>D) (a.b) -000)-((+)) Iog(a/b))),
(15)
where the second line holds by Lemma and the last line holds for sufficiently large n. We conclude with by Lemma 6, Inequality (14), and Inequality (15) into Inequality (13).
Remark 2. For arbitrary ε > O, we can find e' > 0 and ō > such that E' + 5) = c. For the sake of convenience, we also denote D+ (,,b,, 8,E) as sD (a, b) - al log(a b)] and this is equivalent as the (e, 8) D-parameterization in Lemma 8, We mainly use the s sD (a, b, 8, e) notation in the analysis throughout this paper.
D.4 Tree node assigning
Before getting to the first moment calculation of the similarity score, we introduce a sub-pr problem, named in-community edge counting for node assignment on Irees.
Assume that we have a random graph G with n vertices, which are labeled by a community label vector σ.. We are also given rooted tree T(2) with N vertices other than the root, where i specifies the root node of T on G. Planting T(1) onto G' has at least (x) possible positions. If G is a stochastic block model, different planted positions of T would contain different numbers of in-community edges. We are interested in the distribution of the number of in-community edges when planting T(i) onto G uniformly at random.
Lemma 9. Let Kn, be the complete graph of a stochastic block model G on n vertices. Let a.
for o..() E {11 drawn independently and uniformly at random as the vector of community labels. Let E [n] be an arbitrary node of Kn. Let be an arbitrary rooted tree with N vertices other than the root. Consider a uniformly random injective function V V(K,)
such that root r(T) is mapped to V V(K)). Define X as the random variable representing the number of in-community edges in the tree T under random T. Then,
(Nn 2n :) x X
H x
(N, 22 -b. 2
Proof. There are N edges on the tree T. For any (u, v) E V(K.) x V(K)), we denote X ((u.v) as the indicator random variable of event & =(u and v are from the same community). ). Since 2 - < PE | H) I 1 + 2n - X ((u,v) H stochastically dominates Bernoulii{ 2n and is stochastically dominated by Bernoulli( + 2n- Thus, the summation over all edges on tree T, X | H. E(un)ER(T X(a.) H. stochastically dominates Binom(n, 1 2n 1) and is stochastically dominated by Binom(n, 1 2n 2
Remark 3. N = =(ná), Lemma 9 implies that with some non-negative integer N1 < N, P([x = N1) n H) = (1 + o(1) (N) (H)) Because P([X- N1) n H) = (1 - o Ni H), it
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suffices to show both the upper and lower bound on P(X = N1 HH).
P(X = Ni
H) = P(X
N₁ H) P(X 2 N₁ +1 H)
Σ
1
+ 2m 1,,0)
-nn
-2 Σ
N
( 2n 2 -,,1
+ 2n N
2
2
12N++1
$ (1+)(11)
(N)
1
+
E
)M (,,,,,).
N1
12 - M(M++1)
where f(n, N,t) 41+4n 1----y(11110 4n 11 11 4n 4n The first inequality holds because of the stochastic dominance in Lemma 9. The second inequality holds because (N-)) (m, N, N - t) = (7) f(n, N,t) and thus cancels out every term in the summation indexed from f N1 + -
(N1 + 1). For 2 (N N1) V (N1 + 1), we know that (?) (N)
Also, f(n, S (1+ 4n à 3 (1 4n 3 Onn AN) of +t) from our assumption on
N. Therefore, summing over t 2 (N - -N) A (N1 + 1), we have
N
P(X No HH) S (1+0(11)
N₂
From the other direction of stochastic dominance, we have P(X = N1 H) P(X 2 Ni H.) - P 2 N1 + 1 70) 2 (1 - (x)
Lemma 10. Let Kn be the complete graph of stochastic block model G on vertices. Let o, (t) (1) E (-1,+1) drawn independently and uniformly at random as the vector of community labels. Let E [mi] be arbitrary node of G. Let (T); be a sequence of arbitrary rooted tree with N vertices other than the root. Consider a uniformly random injective function T: VT) + (K,). such that root r(Ti) of the first tree is mapped to in ((K,). Also, the root of i-th tree is mapped to T(r(T))) Tu)) for fixed vertex previous trees E VT.) or fixed vertex on Kn, Define X as the random variable representing the number of in-community edges in all the trees under random Then,
(N 1 1- -22n 2
- <
X H <
(in(N,
2 + 2n 22-+))
Proof. Define X; as the random variable for number in-community edges on Tree Ti with random embedding The following holds immediately from Lemma 9,
1 Bin( LinNN,, 2n 1) = X; H & Bin(Nr,
2 + 2n -))
Since X H Σ, H.
Bin 2n - x X
H x Bin(N 2n =)).
To present the following Lemma 11, we need to introduce a few more definitions: decorated union graph and decorated edges, which will be explained in more details in the context of chandelier in Section I.
Decorated union graph. Let S1, S2 Ti, Tr be four rooted graphs. The union graph is defined as U : St U S2 T. TT, We define the decorated union graph as two-tuple Ú := (U, Drr), which associating with a decoration set. For each edge,
The subset of (Si, S2, Ti,Ta) where coccurs, if e E E(U), [0,
Dule)=
otherwise.
We call an edge e E(U) is t-decorated if [D) for 10, 4}. Decorated union graph Ú has one-to-one correspondence with (S1, S2, Ti,TT) as we can uniquely determine ú given (St, S2, Tı, T2) and uniquely recover (81, Tí, Th) given Ú.
25

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Lemma 11 (Asympiotic independence of the counts of i-decorated in in-community edges). Let Kn be the complete graph of a stochastic block model G on n vertices. Consider connected decorated union graph Up with d1 1-decorated edges, d2 2-decorated edges, d3 3-decorated edges, and da 4-decorated edges, rooted at on the complete graph Kn. Consider a uniformly random injective function т V(T) - v(K.) such that root r(T) is mapped to u in V(Kn). Define X(9 as the random variable representing the number of i-decorated in-community edges in Up. Assume that [V(T)) = O(log n) and Up has excess k. If = -1, then,
(( = MI, x = M2, = MM, x (4) = -M4 H) = MM
(x)) -M2 H)P((X Ms H)P(X( Ma 12)
d1 (dz
ds d4
1
=(1+ o1))
Mi M22
(M1) M3
2dy+da
Proof.
From the assumption k -1 we have Up is a tree. We can decompose Up to a sequence of trees as follows: Traverse Up in BFS order and include each maximal connected component with all edges i-decorated as subtree.
In addition, we break this sequence of tree into four sequences based on the decoration number: 15- ( { be mapped to either non-root vertex on the other tree. Let xx" be the random variable for the number of in-community edges for the j-th tree of i-decoration. x,() H satisfies the following stochastic dominance:
Jÿ=1, (T) (r/""//=-
In the random mapping, the root of each tree should
Bin (V " )1 2
1
2n #) x x H & Bin
)1 2
1 + 2n 4))
Thus, their summation satisfies the following stochastic dominance:
Bin (4. 2
2n ビ) x x H ক Bin(d;, 2 + 22 -)- == 1,2,3,4.
As each of the series of trees occupy O n2) vertices, every ( conditioned on other X(1), i' + still satisfies the stochastic dominance:
Bin(d:: 2 - 2n bn=+)) x x( ,, x Bin(d;:
2 + +2n -)
Specifically, following Remark 3, P(X(1) M4 H) = (1+ (2) 2dd. P(X(3) = Ms
M4, H) (1+ (()) de म... P(X (2) MM/x((3) = x (4) =M4, H)) (1+ (1)
and (1 = x = M2, (3) = X(4) = M,, = (1 o(1) dy + Collectively.
these form the asymptotic independence of X();
x() P(X(1) = =( (2) = M2, x = Ms, (+) = -Ma | H
d1
d2
ds
da
1
= =(1+ o(1))
M1
M2
MMs
M4
2d dg +d
Lemma 11 discusses the case of ls have an auxiliary result as in the following Corollary 1.
-1. We do not expect the same property holds for k 2 0 but we
Corollary 1. (If 2 O, then ÜP contains eycles.) By definition, Üp can be decomposed into a tree Tp with up vertices other than the root and additional ie + distinct edges Ex+1 = [[ug, u)) "))) k+1 fixed. By edges' decorations, Ex+1 = E E)2) u E E(A)) Let x() still be the rundom variable representing the number of i-decorated in-community edges on ÜN. Ifk 2 0, let (iaa) be the counts of i-decorated in-community edges on TN and (t.b) be the counts of i-decorated in-community edges on { From construction, we know ( = x(t.a) + ((i.)) Correspondingly, there are A, i-decorated edges on tree, B; = =IE,0,, and d; = A; + BB which are all fixed from Up.
k+1
Then, ((.(1))) a1, (2,a) 0g. X(3.a) az (4.a) aq, x(2M) by, X(1.) b, (3.6) by, (1.)) ba H) S (1 + o o(1) (dg)
As) (A))
as las) クク
The proof follows straightforwardly from the proof of Lemma 11.
26

---

r(722)
r(T]
(713)
Figure 5: Decomposition of decorated tree into three sequences of trees. Edges that are 2, 3, 4-decorated are painted as red, green, and blue color correspondingly. Roots of each subtree is marked by larger node and annotated as r(T) where is the decoration counts and is the order in its sequence.
D.5 Inequalities concerning automorphisms
In this subsection, we study how adding edges can change the number of rooted automorphisms of rooted tree. With a slight abuse of notation, we denote by aut(T") the number of rooted automorphisms (i.e., automorphisms which fix the root) of rooted tree T. We start by quantifying the effect of adding an additional edge.
Lemma 12. Let T = (V, be a rooted tree with root r. Let == UV E ((u, )) be the rooted tree obtained from T by adding vertex and the edge (u, v), where E V. The following bounds hold:
aut (T) x |V S aut(T") S aut(T) x [V..
(16)
where [ is the number of vertices in T.
Proof. Since T is rooted tree, there is natural notion of a parent vertex for every vertex other than the root. Namely, the parent vertex e A r is the neighbor of u which closest to the root r. We also define the subtree rooted at to be the subtree of T induced by all vertices whose shortest path to goes through v, rooted at v.
We partition V into equivalence classes according to the following rule: and are in the same equivalence class if and only if they share the same parent and the subtrees rooted at t1 and d2 are isomorphic. We denote the resulting partition (Vihez and refer to the equivalence classes as orbits. Note that the root r is always in a singlc-element orbit and thus the size of each orbit is at most VV| 1.
Observe that a permutation of the vertices is rooted automorphism precisely when it maps each vertex to a vertex in its orbit. Thus we have that
aut aut(T) (T)
II M SEZ
(17)
Now consider 7', which adds a new vertex I' with an edge connecting a vertex in By (17), in order to understand aT') we need to understand how the orbits and their sizes change due to the addition of the new vertex and edge. The new vertex u will either join an existing orbit form its own one. The parent of may change orbits, so might the parent of its parent, etc. In other words, vertices on the path from u to the root might change their orbit, but vertices not on this path will not. In the following, we argue iteratively based on the depth of u in 7' (i.e., its distance from the root). When considering the upper bound, we will ignore the possible size decrease of orbits. When considering the lower bound, we will ignore the possible size increase of orbits.
27

---

Assume first that the new vertex u is attached to the root r. If there are no leaves except for connecting to r, then u forms new orbit Viz whose size is 1, which does not change the number of rooted automorphisms. Otherwise, without loss of generality, assume that orbit V1 is the set that contains all leaves connected to r. Then u will join this orbit, so the set of orbits of T is given by !2] Vi U {u} and (v Thus, we have that aut(T") aut(T) + 1), so in particular
(T) I (T") = aut(T) + 1) S aut(T) x [V].
Now suppose that u has depth 2 in T", and let v(1) denote the parent of u in T", Without loss of generality, assume that v( E Vi. There are again two cases depending on whether or not there are leaves attached to w()) in T. Suppose first that there are not any leaves attached to v(1) T. Then u has its own orbit (of size 1) in T". The orbit of y(1) changes from V1 T to either new orbit or some existing orbit Vú, in T" (for some 2 S J1 I [Z]. In the former case we have that aut(T") = (T)) while in the latter case we have that
11,++1 +1
aut(T") = (T) X
[V]
The desired inequalities thus follow since each orbit has size at most |V| - 1. Now suppose that there are leaves attached to e(1) in T and let V2 denote the equivalence class of these vertices. Then, u joins the orbit 1/2 T". The orbit of v(1) again changes from V/2 T to either a new orbit or some existing orbit V,, in T" (for some 3 I j1 ΣZ). In the former case have that aT'( = (T) x (22 + 1)/[V/], while in the latter case we have that
aut(T") = aut(T) x (112) + 1)(,, + 1)
[Vi
The lower bound follows since |V S |V| ---1. 1. For the upper bound in the latter case, note that by the definition of orbits, in T" there are [V + nodes depth 1 who each have [V2 + 1 children that are leaves. This implies that T has at least ([V/ll + 1)(V,, non-root vertices, so ([V/ + 1)((V, + 1) S ([V)
1= IV.
The general case when has depth l in T" is analogous. Let vlf-1) denote the parent of in T", let u denote the parent of vi(f-1), etc. Without loss of generality, let V, denote the orbit of v(() in T, for E [e- - 1]. Suppose that (f-1) has children that are leaves in T (the other case, when it does not, similar and simpler), and let V/2 denote the equivalence class of these vertices. Then, using similar observations above, we obtain the following upper and lower bounds on aut(T"):
aul (T)
S aut((T)) S (T) (VV) + +1) II (V)++))
V
Here, for every E [ - 1], either ji [E + 1, [Z] (which corresponds to v(() changing from Vi
in T to some existing orbit V5, in I') or VV, 0 (which corresponds to ((()) changing from Vi
in T to a new orbit in T"). To conclude the lower bound, observe (using the definition of orbits)
that T contains a subtree consisting of the root and [[= [V] additional vertices, so [T [V| <
|V| - 1. For the upper bound, observe similarly that T' contains subtree consisting of the root and
([V] + 1) x I= f-1 (IV, 1 additional vertices, so ([Vi + 1) x I (IV,, + 1) & [V].
Lemma 12 is tight. Let T be star rooted at its center (i.c., a tree where all vertices except the root are connected to the root). Then all permutations of the vertices that fix the root rooted automorphisms, so aut(T)= = (T)) 1)!. Now let T" be a tree obtained from T by adding an additional child to the root (see Figure tree in the middle). Then au'( = (IV (T)))
aut(T') x (T)). This shows that the upper bound in Lemma 12 is tight. Now let T be a tree obtained from T by adding a child to one of the leaves of T (see Figure 6, tree the right). The rooted automorphisms of T are precisely the permutations of the vertices that permute the neighbors of the root which are leaves, so aTt( ([V(T) 2)! aut (T)/((V(T)) 1). This shows that the lower bound in Lemma 12 is tight.
From Lemma 12, derive the following corollary by an iterative argument.
28

---

Figure 6: Left: The simplest example of two rooted trees 7j and T2 satisfying v aut (Ta) > aut(Ti J T2); here Ti is induced by the blue and green edges, T2 is induced by the blue and red edges, and both trees are rooted at the vertex at the top. Middle: Example that shows that the upper bound in Lemma 12 is tight, where blue lines and black vertices represent T, and the red vertex is additionally added with an edge attaching it to the root. Right: Example that shows that the lower bound in Lemma 12 is tight.
Corollary 2. Let Ti - (V,, and T. = (V2, E2, r) be two trees rooted at the same vertex T. Define the union Ti UTT := (V UV, E E2, by taking the union of the two vertex sets (both which contain r) and the umion of the two edge sets, with multiple edges ignored (i.e.. if an edge appears in both Ej and Eg, then it appears E1 J E2 exactly once). Suppose that Ti U UT2 is also free. Let d = A denote the size of the symmetric difference of the edge sets. Then
V (Tat(Ti)anu (T))(()))) S aut(Ti UT2 x (2 max (V21).
(18)
Proof. By the assumptions on Ti and T2, the union Ti U T- has (V + [/21 d)/2 edges. There are two natural ways that we can think of Ti UT, First, we can start from T' and add ([V2 - -Vi + d2/2 new vertices-those that are in 1/2 but not in Vi-one a time, logether with new edge for each new added vertex, connecting it to an existing vertex, to obtain Ti U T2. With this perspective, applying the lower bound in (16) from Lemma 12 across each of the (|V/) - |V1 d)/2 steps, we obtain that
([V|+|V2|+ 22)22-2
aut(Ti) S aut (T; UT2)
II
k.
(19)
K=
On the other hand, we can equally well start from 72 and add (IV) - [V2 + d)/2 new edges and vertices to obtain Ti U T2. Thus, analogously, (19) also holds with au) on the left hand side replaced with aut (T2), and [V] 1 on the right hand side (the minimum value of & in the product)
replaced with [V/ 1.
Combining these two inequalities, we obtain that
[IVI+VV/l+ V/l+d/22
v Vaut(T:)aut(()) & (T))))) x
II
k.
k=min [V..IV11111
Since d < [[max{[VV, V2]} - 1). it follows that ([V1) + [V,] + d)/2 S max [1 [/2/. so all factors in the product above are most max{[V]] [2]]. Note also that d 2 max{ V1. 1/2
(i. V2[}. so the number of factors in the product above is Imaxf/V; V/13 min [11 Vol d)/25 5 d. Putting these observations together we see that (18) holds.
Remark 4. The leftmost example in Figure 6 provides an example showing that the right hand side of (18) would no longer be an upper bound without the factor (2 ma) [V]]})d.
D.6 Auxiliary result: Bounds on the cross-moments
In this section, we summarize the upper bound on cross-moments. In Lemma 13, we work on the regime sD_ (a, b) > 1 where we can recover the community label exactly first. In Lemma 14, we work on the regime s sD+ (a, b) < where we we have inverse polynomial fraction of vertices being labeled incorrectly.
29

---

a no 4 b Denote A, B as the centralized adjacency matrices of G. and G2 correspondingly Let = (u. [n|] [n] be an arbitrary edge. Define the edge type indicator c(e) = + if is an in-community edge and cle) = if a cross-communtty edge under a.. The following holds for 0 S e, m S 2,2 - l m S and any o..
Lemma 13. Let G2) ~ (....(....q..))
(f, 2,,0,,0,2)
(1 + O logn )p
(f,m) - (1, (C, m)) - (2. 12,1),(1 (1,))
((e-m) (c.m(e) == a
-((-m)) E[A Be
cle)
Ve(png) s(pAgg
(L) = (2.
Proof. Conditioning on a specific true community label vector that satisfies the balanced community event H. we can apply Lemma 5 from [42] for each case of c(e) = + and c(e) - Then, the upper bound are different in terms of p, q and P+: p-. When m)) = (1, 1), Br.m S max 1--1
(1 + O (og"))n When (f, (2, or (1,2), if c(e) +. then we have Be.m S Vsp else
we have ft.m= I Van"
Therefore, 3r,m I Vs(png)
The same argument holds for the case when
(e, m) (2,2).
blogn Denote A A _ as the approximately centralized adjacency matrices of G1 and G2 correspondingly Let (u, v) [n] x [n] be an arbitrary edge. Define the edge type indicator cle) = + if is an in-community edge and c(e)
clog Lemma 14. Let G2) ~ CSBM (.
"
if is cross-community edge under o.. Define A sp sql. The following holds for 0 S l, m S 2, s + m < 4. and any o., Ō = OB))
((VA) 1 O log
(e, m) = (0, 1), (1,0)
(e, m) (2,0), (0,2)
1t (e)
σ (1+m)
A BB c(e) E[A
1o.,ol
p(1 + O log
(f.m) = (1,1)
+ O lug
(f,m) = =(2, (1,2)
Vs(pag)
(1 + O
logn
(f, m) = (2,
s(png)
n
Proof. Denote the two vertices as u and We denote p' == sp and g' := sq.
(a) l + Without loss of generality, we consider and m
a..l 6 o,, =
= +D
[0,
= (0))))))))(((0)0()) ()) +
T.
When F(A
A..l
o., a is non-zero, d(e) F a..l
0, a @(VA).
(b) l + m 2. We first consider the case of f or m 2.
We explicitly calculate the expectation on the following two cases. If a. (u) =
then E[A A.2 o. p/1 (11pp)) = od ( then E(A 2112
ox, 0x, =
p' - 2p'd' 2 oeee) 22
In summary, we have
x..2 E[A
6..6]=E[1
=E[B
0, o +1² = -0(ee (1 + O
log n
))
n
We next consider the case of l = If of (v) = ( then ElA F Be ..
a., o) = o(te) If (u)o, + ((4)@(v), then E(A
o A.1 Be 0a.1
o.. θ Pe(c) 0² (le) + A
In summary, we have
a E[A B
-n. lo. 6] s O2 + po (1 0
log ))
n
30

---

(c) C f+m m = 3. With loss of generality, we assume that this edge connects two vertices from different communities. Conditioned on correct centralization, we can compute
E[A, 1.2 Be Be o., = [[ o.] (1 - (1) 2qs) - P- p- 24/) V 9')
< Vq
which is the same as in Lemma 13, the centralization incorrect for both graphs, the expectation changes to the following:
2 n B
o.,
2 (//
From observation, we see that the dominant part in both cases are the same, which is q's = 0 l) Then, we can show that the difference between two quantities are minor:
Be
a 0 E(A
o-]
(p'
- -2p'' 20" + + 20's) O(v)).
q'(l (11-9) -1''))
(20)
The above (20) also holds for the case when is classified correctly in A or only.
By also considering the other case, that is, this edge is connects two vertices from the same community but wrongly centralized according o we conclude with
E E.A
B a.
o., S o2
1
(1 ((s(pA =2(e) Vs(p A
+ O
log
V ((p
n
(d) f 4. With loss of generality, assume that this edge connects two vertices from different communities. For the correct centralization, the moment stays the same as in Lemma 13,
B[ o.) - 4g 00 + 24 39" -3q" (2- + (1- 2q') s 0² 1
If the centralization is incorrect for both graphs, we have
E(A A.. (.. = q's Ap'g's + 2 - - + 11/
and we can show that the error has the following order
AA.2 2
B
A.2 o E
B
B.
o.. O((
log (ogg O(o
(21)
When the centralization is incorrect for only graph, (21) still holds.
By also considering the other case, that is, this edge connects two vertices from the same community but wrongly centralized according to o, we conclude with
A. 5n.2 2
a. S ((ra(e)) + ddte) s(p A 4)
S dle) ((p
1 A
(1 + +
log ))-
n
E Proof of Theorem 3: Almost exact graph matching
Fix constants a * > 0, e [0, 1]. Throughout the paper, we refer to sD+ b) > 1 as Regime I
+o and sD4 (a, b) 1 as Regime II. We define μ : |T p o where o² = 2 Depending on the different parameter regimes, the analysis different. We first present the first and second moment bounds for both regimes and then prove Theorem 3.
E.1 Regime Exact community recovery is possible for a single graph
Proposition (Mean calculation, sD (a, b) 2 1). Given (G1, G2) CSBM(n, the similarity score satisfies,
blog
n
(f) (1+))))))
F F 1M]
1,
ff) $j.
31

---

Proposition ( (Variance calculation -rrue pairs, sD=(o, b) 1). Suppose that j = T. (i), that (D+ b) L, and that
14L2
22R 1)(113)2(K+M))
1
2
m
2
AR (11 4K+1M
1 + 2T
1
(22)
ns(p AopAg Q)
2
ns(p ^g) 4
Then, for any i [n], we have
Var ky 1x
L ppns(p A 4)
L
(1) 1x
(2(K=
M)
Proposition 3 (Variance calculation-Fake pairs, sD+(a,b)
1). Suppose that # TT),,, that
sD+(a,)) > 1, and that
4(K+M)) 4R (11,8
4 2 (113)KKM (2N 1)3
1
(23)
2
ns(p A 2
Then, for any i [n], we have
Var 1x]
1
O
(T
E sn. (0) 1x]2
E.2
Regime II: Exact community recovery is impossible for single graph
Before getting into the details proof, let us first give more intuition on the approximately centralized adjacency matrix.
In this regime, for each element the adjacency matrix, there are four cases: (1) Correct central-
ization for in-community edges; (2) Incorrect centralization for in-community edges; (3) Correct centralization for cross-communt cros-ccomunntyy edges; (4) Incorrect centralization for cross-community edges. Figure 7 gives example of how the graphon for balanced 2-community SBM changes under community label misclassifications.
3.300,0.3.0011 Craphon
Labels
5BM|jn003300.1 Incorrect Labels
8,4
0.0
2.0
0.0
0.300
0.25
0275
0.2
0.2
0.250
0.20
0.200
0.6
0.175
0.5
0.175
0150
0.100
0.8
(a) Regime I: exact recovery is possible.
(b) Regime II: exact recovery is impossible.
Figure Illustration on the impact of incorrect community labels on adjacency matrix centralization
with G SBM(n, 0.3, 0.1). Centralized adjacency matrix A A [[[A), where matrix F[A] takes the value of discretized graphon as above.
Proposition (Mean calculation, =D+(a,)) < 1). Let (G1, G2) ~ CSBM(n, a log blog 8).
Given (K, L,M, D) D)Chhnlleeiiirr class T. Assume that D ol Tog log
log
For all i [n] with (1), we have
E (1) 1x] o(1))
32

---

For all i E [n] with j # (1), we have
E E 1H] = =(1))
Proposition 5 (Variance calculation-True pairs, sD+ (a. 1). Suppose that bD+ < 1, = onn). and that for some > 0,
T. (T).
AR dr (15,3)
#
1
+ (1(15)))
1
ns(p ^
2
2
&D+ (D+ b)
(log log n)
2TT(4TM)) & log" n.
(24)
D
log
Then, for any i [n], we have
Var 0 1x
O (
L Pnslp A 4)
L p2(K-M133
E
(x) 1x
Proposition 6 (Variance calculation-Fake pairs, sD+(a, < 1). Suppose that # (T)
ba, < 1, N @(log n), D ol log n and that
log
AR (153)2 M
1
sD+(a,))
(log l)) log
ns(p 4)
2'
D
(153))(K++0)00 30 (4n + 1)2²
(23) ( (AL)! 1
(25)
n
Then, for any i E [n], we have
Varl 1H] El T sn, (1)
1
1j
0(
(T
1xl
E.3 Putting things together: Proof of Theorem 3
Proof of Proposition 3. Firstly, the following conditions imply (22), (23), (24), and (25):
log e²
L L C1 log n log logn
A c6 Revnst ns(p A g),
C
M
<
KL >
C3 log n
log(ns(p A))
K
log p
log p²
log n
K + M S c4 logn,
R ==
D DS 07 (log log logn)?
(26)
for some absolute constants c1, Cz,..., e7 > O.
Furthermore, with growing ns(p/g), we have for j = (1), Varls, Lu]
E[0.r. 1x] = o(1) when sD b) >
1 and Varl0;;1xl o(1) when sD_ (a, < 1. For any & 0 there exists ex > such that
$² 2 a +E + p² 2 a + E'. We have for j # n. (i), E(P 1 = o( when (D+ b) > 1
[ar
Var and E[0 Axl = ol //) when sD+ b) < 1.
Next, we claim that almost exact recovery is achievable by counting chandeliers. Let = cjs, for arbitrary E (0, 1), where y F [P... 1x]- By Chebyshev's inequality, the probability that the similarity score of fake pair (j A n. (t)) of vertices exceeding 7, is upper bounded
Var Dr 1x]
P 2 T) S P (10 1x El 1x] 2 cE 1x) S
e ; 1x 2
o( 1 n
0
T p2N
Next, by the definition of H, we have
P(D,, 2 2) S
Sc) +c)
of
n?
33

---

Applying union bound over all i + j E [nl, we have P[3i j. 2 2 o(1). Thus, for all possible pairs of vertices (i.j) [n] [n]. Pi < with high probability.
We next study the probability that the similarity score of a true pair of vertices falling below the threshold r. We even consider a larger containing F {i E [n]] (15.(1) 77).
Var i 3.. (1) (1)
i( pl > c)p) -
(1 c- ix (x) (6) 1H
L
L
) =: =:7 =
=0
p²(k+ F
p2ns(p A
Therefore, E[[F]] S my. Denote I [n] the vertex set this algorithm matches. For every vertex not in the set F, it is guaranteed to be matched up correctly, so
E[|r]] 2 F|] n(1 - 7)-
By Markov's inequality,
P([F) >= //)) S
7 V
o())
Therefore, Algorithm matches |7| 2 (1 / vertices correctly with high probability.
In Regime II, with Proposition 4, Proposition 5, and Proposition 6, the same argument follows. These two regimes together complete the proof.
F Proof of Theorem 4: Efficient algorithm for almost exact graph matching
F1 Color-coding- similarity score estimation
In this section, re-state and modify the efficient graph matching algorithm for correlated Erdós- Rényi graphs discussed in Section 5 of [41], ]. and then analyze for correlated SBMs.
Let (G1, G2) be pair of correlated SBMs with adjacency matrices A and B. Let H be rooted connected graph with N vertices. We now want to approximately count the signed subgraphs rooted at E [ri] on the centralized adjacency matrices. In general, we do not have access to the centralized adjacency matrices A and 73, we would use community label estimates ō and ē from Algorithm 3 to approximately centralize the adjacency matrices as A A and BT"
First, we generate a random coloring μ [n] + [N + 1]. which assigning every node on A one of N + 1-the same as the number of vertices on a chandelier colors independently and uniformly random. For any vertex set V C [n], we define X₂ 1 (Va.ee We call the
(N+1) vertex set V being colorful if X (V) 1. We denote r (V) = (N+1)N41-
We define
the approximate signed rooted subgraph count as
X1. 3a :p) = Σ ((
(27)
S0DEH
Observe that E[X,H((A' (A [)] = (A where Wi, (A the ground truth of signed
FA ,n)/r an unbiased estimator of counts as defined Section C. another word, (A
Wa. (A
Let /. : [1/r], we repeat the random coloring for times and then average over the estimates before taking the inner product of two signed counts vectors. Formally, we generate 2/ independent colorings independently and uniformly at random, denoted as (Na Pa=z and {vp W For vertex i in G₁ and vertex j in G2, we define the approximate similarity score as follows
( Σ Xin (0))))
1 2
Σ
Hut(H)
Σ X (A (ma)
34

---

We have E 2 deterministic function on We summarize the algorithm Algorithm 4. There are two additional steps than Algorithm in [41]: First, in the construction of the chandelier class this work, we need to filter out instances with maximum degree greater than threshold D, which takes O(K) time for each tree; Second, we need to obtain the community label estimates before estimating the signed subgraph counts.
A A "]= " 2 and also 1
B o. 2 1y also holds as 14 is
Algorithm Efficient Almost Exact Graph Matching Algorithm
Input: Adjacency matrices A and B on vertices for correlated stochastic block models (G1, G2).
Step 1 Construct the chandelier class
(Rooted tree generation [6]) List all non-isomorphic rooted trees with K edges.
2: (Automorphism constraint [12]) Compute (H) for each rooted tree using the automorphism
algorithm for trees.
3: (Maximum degree constraint) Compute the maximum degree of vertices.
4: (Chandelier class) Return J the subset of rooted trees whose number of automorphisms is at most R and maximum degree is at most D. Construct (K, L, M, R. D)-hhandelier class T.
Step 2 Estimation of the similarity score
(Random Coloring) Generate i.i.d. uniformly random colorings {Na) and { each maps from [n] to [N ++1 1]]
(Community recovery) Obtaining o and o for A and B independently by Algorithm for all (i.j) E [nx] x [n]] do
for all H T do
(Signed counts estimation [42]) Compute (Xi,tA" [aa))a and {X,,I(B²) ; _))]== end for end for
Output: The approximate similarity scores (.
When we can obtain the correct centralized adjacency matrices A and B, we replace A a Bda) with A (resp. B), and everything defined above is still valid. In that case, we denote the approximate similarity score as & rather than の
(resp.
F22 Analysis
The analysis is similar to that in Section 5.1 of [41], while we split the analysis into two regimes.
Under Regime I (s b) 1), we can recover the community labels on G1 and G2 exactly correct with high probability. We define Pif as an upper bound of Var[Dy) v) as follows:
Var[D.;11] <
Σ (H)(H))
Σ
E [E[As, Bs, Ar, Br, 1x]
H.JET
=: Ti (28)
0. then there exists Sr, S2. Tı, and T2, and so [E[As, Bs, Ar: B,, 1x]1 the cases where every edge occurs at least two times among these four chandeliers. If 1 ( #T, (VV())))))))
If 1 LSS.ATTCSSSU.SS,AATTCCCC...
edge occurring only once among 0. Therefore, we only look at
= 0, then S, = S2. Ti = T2, and S has no common ver- tex with Ti except for the root. In this case, S and Ti, and Sn and T2 have no common edges, so the covariance between As, Br, and As, Br, is always zero.
For Regime II (sD+ () < 1), where cannot recover the correct centralized adjacency matrices with high probability. Alternatively, we define T as:
T2 = Σ (ut(H)))
Σ
E EAA
3 B T A=1B00 A7, B 1x).
(29)
HIIET
S1.8 M.T.
From the definition of variance,
Varl 1x] IT
35

---

Lemma 15. Fix constants a + b> 0 [0,1]. Let (G1, G2) ~ CSBM(n2, log bogg variance of estimation error has the following upper bound:
s). The
Varl( Di [ S i5
and
Varl( 1x) 33
where it is defined (28) and : defined in (29).
Proof. Regime I: sD (a, b) > 1. Define
Yij (I, == Σ
(A
HET
where μr, v are two (N co-clriin of the vertices in [n]. Then, we can represent approximate similarity score
Yi (Mc.
(30)
2212
c=ld=1
For any Pes Va, 1 S c, d S t, (Me) /22 is an unbiased estimator of Dig given A, B as
[[Yis va) 1 A, B, o.. = Σ HET
aut(H)W,,n ( = 2 1x
Note that (A, and Hj.H(B, p) are independent conditioned on A, B. Moreover, note that (Yg (Ne; va))iso,ast are identically distributed. Hence, we have
+ 1x] = E
Yos La)
E(P,, 1x].
222
c1d=11
Next, we bound the variance of P.j. In particular, we can get
Var(( - (ii)11l = Var(E[( 2; 1 E[[ar ( B.
B, o.]) + [Var((Q Piy) A, o.)]
because Dij and 1 are fixed conditional on A, o. Furthermore, conditioned on A and B, for any e, d, e, S i, Yiy(Me, va) are independent with Yıy(ues uy) if and only if + e and d + f.
Var(( - 1x) r444
E(Cov va), Yig (Me, u1) 1x)
Applying Lemma 16, have
Var((0. 1z))
5 11
( (20) - T'
d=1 1=1
(t2 + I (t2p2 + 2
222
< 31'
where the last inequality holds because t = tr
Regime II: (a, < 1. With community label estimates ōA and ōg. we define
(M,)) = Σ (A A , B 2 .))
(31)
HET
where μt, v are two (N + 1)-coloring of the vertices in [n]. The proof for Regime still holds in this case by replacing A, B, Yij, Pij, T.g and T with A B ya
to .y:
20 and 1 correspondingly.
36

---

Lemma 16 (Extension of Lemma 12 in [41] to correlated SBMs), Fix constants a # b > 0,8 e
[0. 1]. Let (G1, G2) ~ CSBM(n log.n logn s). Fix any 1 S c, e, S t, and i, E [n]. If
(D+(a, b) L, then
E[Cov(Yis (Per va), ( uy)l o.) (1) S (12t)(01)) (ote) M
(32)
(TD+(a,,) < 1, then
E[Cov(Yi; Va), Yig (Mes vy) BB" oB. .o.)1ul S (ozt - r
(33)
Proof. We first prove (33) and then explain why it implies (32).
For two independent (N 1) coolorins μ and of the vertices in [n]. From the definition of yi (u, (31), E[X. (H, v)) A Cov(Y (Mes va), YiJ (Me: V3 us)lA² のΛ B ...) = El (Pe, va) YiJ (Me, Us)l 3Λ RB .0.1 - -0)))
,B
TB o.. = Then,
(34)
From the definition of v (1, v) in (31) again,
E[Kiç(Me; Ma) Va) // A ,B80 o.)
E (n AA .Pa) B vu)
НET
au R1 (1o)Xi,11B " の0 ,v))[A ah BBe o...
From (27) and the independence of colorings.
E| (Me va) Yig us) A B .o.l
=E Σ (H)(H))
Σ
Σ
Xuc (V(S 3A
HIIET
x Xva (V (V(2)) 3g x Xa. (V x Xu, (v, 3g 31 ,B
A
Σ aut(H))ut(()
Σ
Σ
F(xm. ((8))))x..(( (V(Ti)))
HIIET
(1))) (1)
Eix.. (V(])))) B' AT 0B
(35)
From (34), (35), and (10),
(Ys va), (is AA an Ba BB,,,,,,,, 1.,1
Σ aut(
Σ
Σ
E L.As, B Ar, B1 1x]
HIIET
x
(V - 70)
(V(T2)))
Observe that E[xpe (V(S))))xx (V(T:)) 5 (lt(() and [[xu, (V(S2))x,, (V(T2)) s plt4e
E[Cov(Yes(He, va), (Me, A 2 B o.,1x]
Σ aau(H))
Σ
E[As, AT, BTg 1x 2 - 12)
3. Bãs ē
I.IET
= (r +1eze) - ,;
37

---

and this completes the proof for Regime II.
In Regime I, L[As, Ar, Br, + 0 only if { 4T C U2T C CS happens. In addition, if Si = S2, Ti T2, and ((Si) (ITi) {i}. then Lx (V(S))) (Vii))
(v)2)) (V(E))) = r², because S1 (resp. S2) shares no common edges with Tj (resp. T2) Therefore.
E[Cov(Y va), (io vj) (A) A,,)),)
Σ (nt(H))
Σ
Σ
E(As, Bs, Ar, Br, 1 12 -
H.IET
(1)
-E (n)(H))
Σ
Σ
E(As, Bs, Ar, BT, BT,1H1((22 22t r4)
HIIET
(r -ry(22t1 r²),,
where the first line follows from the proof in Regime II and the last line applies the definition of
Tú (28).
Now, we are ready to prove Theorem 4.
Proof of Theorem 4: For the first part of this proof. our goal is to show that the estimated score
preserves the asymptotic upper bound on Var(#;, 1x] for Regime I and Var[*[1x] for Regime II. (1)
F[ 17.(0) 1x]
For Regime I (s b) 1), we have
Var P.g 1x]
Var P 1x]
E 4 im. (1) 1x]
E( sT, (1) 1x
1
2 [x] + Var[(e. - 11z)1x] + +2 Cov((e.) - 1ii)) L p,y1H))
E (1) 1x
where the first equality is from the fact that o 1u is an unbiased estimator of 4; 1n conditioned on A, B3 and a.. Further,
Var 2 1x]
El (i) 12 Fe 31 2 2.y) I
mm. (1) 1x]2
E E 1x
where the first inequality holds from (28), Lemma 15, and E( 1 0, and the second inequality holds because E[((0,; P,y 1x) E[E[(
The proof of Proposition 2 and Proposition are based on analyzing F and thus the upper bounds Therefore, we conclude that for all i E [n], if (22) holds,
on Var 1xl all hold for E0
(6)
then
Var E ix. (8) 1xl
(
1 o (pns(png) +
L
mm (1) 1x]
and that for all ú, E [m]. + T (1), if (23) holds, then
Var T., 1x]
=0
-(1TTOx)
E E E (1) 1x)
For Regime (sD+(a,)) < 1), from (29) and Lemma 15.
Var T 1x]
T
+ 31 + 2
1x11 5
E @ 15
1x
E T
1 H
(m)()
sm.
38

---

The proof of Proposition 5 and Proposition 6 are based on analyzing H' and thus the upper bounds Var 2 1sl
on Fl (.. 6x]
all hold for
F/
Therefore, we conclude that for all i E [n], if (24) holds,
F
i0 "x)"
then
Var T (T)
14]
L ppns(png) +
L
=0
E 1 (1)
1x]
p2(K+N013
and that for all ú, E [m]. # n (1), if (25) holds, then
Var t 1xd
=0 =0 ((Tibe))
F 0o fm (1) 1x]
For the second part of this proof. we are going to show the time complexity of Algorithm
First, we know that step 1-1 costs time O(3*) ) by the algorithm in [6]. step 1-2 takes time O(K) by the algorithm in [12], and step 1-3 takes time O by enumerating through all edges. In summary, he total time complexity to generate J is O(K a Afterwards, it takes O(T)) time to complete step 1-4, the generation of chandelier class.
The signed counts estimation step takes ((7) N3 n?)) times as shown the proof of Proposition in [41]. Then, the total time complexity is
O OK + [[[L+NN(88 m2))) = ((7 N (32
= =0 (0(Kkt (3e) =0 ((3e3)
Under condition (3) and for large enough log(ns(p A A)) 2 log(2), we have
C" C²
N = (K + My C logo (1 log(ns(pAg))
C"
log
log m1
for some constants C' and C Hence, there exists some constant C' depending only on e such that the total time complexity of Algorithm 4 is Onn)).
G Proof of Theorem 5: Exact graph matching by seeded graph matching
For a pair of correlated SBMs (G1, G2) ~ CSBM(n, p. g.s,, where p = e lng and = bog Algorithm 4 efficiently matches (1 - o(1 (l vertices correctly with high probability. Our next step is to finish the matching on those remaining vertices.
n
To do this, use the seeded graph matching algorithm? (Algorithm 2): Starting with an initial partial matching on at least (1 1- =/16)n vertices which is correct on whatever matches, we form new matches between vertex in Gi and vertex in G2 if the common neighbors of those two vertices under the current partial matching sufficiently large. We then update the partial matching and repeat this rule until get complete matching, which will be shown happens with high probability. Define h(x)) = x log x - x + 1. The sufficiently large common neighbors threshold is set as 7 "² +.2 s2(n + 2n where 7 E (1, oo) such that h(y) (n=2
3 log
Denote N₂ (i, as the number of common neighbors of and j under correspondence m. In another word, N₂ (i, is the number of vertex v E I such that v is neighbor of in A and T(v) is neighbor j B. If π. = T. we also write it as (i..)).
The following lemma for SBM is an analogy to Lemma 13 in [41], which studied the property of Erdös Rényi graph.
2 Lemma 17. Fix & > and a, b > O such that a+b 2 1 & Let = logn . = b log and
G ~ SBM(n, p. q). Let I C. [n] be a subset of vertices, 19) denotes the number of edges
between vertices in I and vertices in Ie = [m] \ I. With probability 1 n- à, for any such
Relevant seeded graph matching algorithmss also occur in 5, 40J.
39

---

that 11 1 16 n, lG( 2 7/1 A where lj is the unique solution in (0, 1) such that log
1((1) [x+0)( n-/4 n) In particular,
++ E 821 (1+ f)
Proof. If eG 1/441191 ptt holds trivially. Let be an non-empty set with S | To.. n. We denote kt as 141. The number of edges between I and I is the summation of k(n k) independent Bernoulli trails. For arbitrary vertex i E 7 and vertex j E Ie if they have the same community label, then the Bernoulli trail between them has mean p. Otherwise, he Bernoulli trail has mean q. We assume that there are Nj trails with mean p and N2 trails with mean q, where N₁ + N22 = k(n k). We can write out the distribution ea(1, 19)
Binomm(N: p) + Binom (N2, q).
Under the balanced community event H. 2 2//4 < |V < + n//14 Assume that there are 4+4 vertices in I with label +1 and the remaining ki_ vertices with fabel -1
N1 = 1 (k+ k k
(1[+) k+) + k- (IV -
n
3/4
k + k2
3/4 E n,
z n
2
n1
2
h
2
16
(36)
N₂ 1 (K-(VV-- k4) k4 - ) 2
2k
3/4
N 2
n
3/4
n. (37)
k
32
We are interested in the probability of ea [ Ie) being less than 10 p A q:
P(eG(I, 1") S ())(g)))pgg) S P(ee(II 19) S N(Nip+NN2g) S exp(-(Nip+N2g))to
where the first inequality holds because N1 N2 k(n k) and p Ag q S p,q and the second inequality holds because of the multiplicative Chernoff bound (Lemma 4).
Let h(v)) (ptt(( 2344 tenj.
(1+ log
we can show that
A(1 + 3 logn
(1+ logn
1+ Su
e) F) 1
Nup + N2g
(p + 2 n3/4 16 (1 (1+e)(1 - 2n 1/4
8
(38)
where the first inequality holds because of (36) and (37), the second inequality holds because -. > 1 ++ s and the third inequality holds for sufficiently large n. Since l())) E (0, 1). there is an
unique solution of (0, 1) due to the monotocity (decreasing) of the function lt(-).
Applying the fact that (N1p + N24)h(()) > (1 + logn (38) and using an union bound over all
subsets C. [n]] with size 1 < [ S 16. we have
P(3I C [n] s.t. 1 15
|T| S
co 19) s (n [7)p^g)
16"
17
ntP(ee (1, 19) s nk(n -a k))
k- -(1+++))) = =O(n
k=1
Because ((z)= 2 (x-1)² 2
for E (0, 1), we have 2 1- / > 1 - v
1+f (I+ex|I<))
Now, we are ready prove Theorem 5.
Proof of Theorem 5. Firstly, we study the size of common neighbors. For an arbitrary vertex [n]] in G1 and v E [n] in G2 such that v is not the true correspondence of u, we study the number of common neighbors N(u, e) in the intersection graph corresponding to the true permutation.
Case These two vertices come from different communities. If v * n. (u), o (v) -1. then we have N(u,v) v) ~ Binom(n - 2, pqs²). By the multiplicative Chernoff bound (Lemma 4) for Binomial distributions, for 7 (1, (1), we have
p² q² s2 (n- + 2n P(N(u, v) 2 7(n 2)pqs²) S c( ((n-2)pgg?h(())
PNN(u, v) 2 2 2
40

---

where h(7)
3 1n (n-2)
> 1,7 (1, The first inequality holds because e' (n + 2n
(n 2)pq.
By union bound over all u, such that # (u),,, (()) = =1, we have
P(3c + (u),,, (1) -1, s.t. N(u, 2 7 p² q² n2n(1 2n
2n = ) -( 11
Case b: These two vertices come from the same community. If v x n. (m), (w) 1, then we 2²) because there are |V vertices left from the same community as u, v and IV -=()) vertices from the different community. Denote n as [V and n2 as IV |V-o(o) ni + n2
have N(u, u) ~ [inom[[Vaa()) 2. p²)²) + (anoo((V-a(")))
2.
p² + snn (n + 2n 9 }} < ((N(u, v) 2 (n + n29²)s²) ((N(u,,) 27 2
S (n + n2 >s² >>>(())))))) < n
2 (n 2n n npp² + n2 The last inequality holds because for sufficiently large n.
The first inequality holds because
log n
3log n
3log nip"
(n-2)) 21pg
2 n11 2
Then, with union bound over all u, such that + T.(u), ( (u))((u) 1, we have
P[Je + m. (u) (u)) = 1, s.t. N(u, v) 2 p² +
8'n(1 + 2n 2n- = 0
8
2
Combining the above two cases, we have
p² 4 8n + 2n = = o(1)
P( + T. (u), N(u, 2
(39)
2
Secondly, we show that the algorithm is working properly. From (39), we assume that for all # T. (w), N(u, v) < <y = (2 ((++2n2) We want to show मे = T.l, in every step of Algorithm by induction. This is true for the initialization o से as a base case, from our assumption for Theorem 5. Suppose that this is true for t-th round of the algorithm, then at the (t + 1) round, we have that for all e # (u),
N (u, v) S N(u, u) < 1²
nn + 2n
2
Therefore, as the (L + 1)-th round, the algorithm will still not add an fake correspondence. Either the algorithm terminates, new vertex i is added to J such that (i) i, which preserves the property =.JJ.
Next. we show that Algorithm 2 always ends with J = [n] by contradiction. Assume that the algorithm terminates with [J] < n, then Je + V. Then, by definition, for al E Js, N=(1,,,
+4² (n + )n Therefore,
12 + q² 2 + 2n
eGI.r. (JJ9) [N S 119h7
(40)
2
$EJe
On the other side, Gj An. G2 ~ SBM(n, s²p, s² q). (in view of Jr as J), with probability of - O(n ))
If 2
> (1
+ 8)
logn
then from Lemma 17
2
eG Go (J,J9) d|ll|ll 2 +
n
E
(41)
16
To reach a contradiction between (40) and (41), the remaining is to prove that
1
)nss A
<
(42)
(n + 2n $)
2
41

---

We observe that h(7) for sufficiently large n.
3 log (n- 2)pps
(1), while 2(1 - to )aa(pg)) 0 log Therefore (42) holds
(n+2n
Finally, we analyze the time complexity of Algorithm 2. For each E [n] to be added into the seeded set I, we update the number of common neighbors (t, for i,j I. This step
takes Oddega, (u) degg, (u)), where dega (u) denotes the degree of u in graph G. With probability 1 ol ). dega, (u) On1(p q)). By summing up for all possible to be added, the time complexity of Algorithm 2 is O (p- 2)
H Proof of Proposition
In this section, calculate the expectation of similarity score for correlated SBM. Recall that have defined of == sp(1 sp), o² :- =sg(1 sq) and that o :=
to throughout this paper.
Proof of Proposition 1. By definition of the similarity score,
E E/ 1x]
Σ
aut(H H)
Σ Σ
E [ss Bs)) 1m]
HET
S(OEN 8U)EN
The expectation is zero if S(1) = S(j), which implies considering (1) suffices:
E 11) (1 o(1))
Σ
(H)
Σ
E [Aso (s) HH)
HET
SO@H
Define X1 as the random variable for the number of in-community edges of S(i). which takes possible value from 0 to N. Define [{S(i) S(i) H}] the total number of S rooted i and isomorphic to H, we have
iT m. (1) [m] (1 (1+
Σ
aut (H)(H))))() s()) = WHE E (s( 1x,, H|]
HET
N
$1+
Σ n HET
P(X = Nı H)p N Σ N1=
o 2N o 2N-
The second inequality holds by (S((): = (N)
aut(ff (1 + o(1)) aut(H) since (N) N!
the number of possible vertex embedding of the chandelier onto the random graph and also
N1} n H] = p p N- No o o_
2N1 2(N-N1) = (1 +1+ o(1)) o 2N, 2 2(N-NN)
Then, according Lemma and the binomial theorem, have P(Xi N1 H) (1 (N) 22) Therefore.
N
1x 1x] = (1 + (1))) N
Σ
2N1
2N-
N₂
2N
0
0
N1=0
02 + o
(1 (1+
N
(1
o()))]
N PMO2N
I Proof of Proposition 2
L.1 Decorated union graph and union graph partition
The analysis of the first moment involves two chandeliers, while the second moment analysis requires four chandeliers. Before delving into the analysis, we introduce the notation for the decorated union graph and establish a rule for union graph partition. We adopt some notations and definitions from [41] and further introduce concepts that are particularly useful for correlated stochastic block models.
42

---

(Decorated graph) For any graph G, define G as a decorated graph G := (G, De), where De is decoration mapping from the edge set to a decoration set. Define the edge set of decorated graph ((G) == E and the vertex set of decorated graph V(G) == (G)). (Decorated union graph) For any pair of chandeliers H, 1 E T, let S (f), S2()) = H and Ti (i), T2(5) = I. The union graph is defined as = Si U U. UT; UT2. Now, let us define the proper decoration for U:
The subset of (2, T2, where occurs,
if E E(U). if $ E(U).
(u(e)=
(43)
0,
We call an edge e E(U) I-decorated if - 111E 10,1,
(Decorated set operation) Assume that U = (U, Dv) and U = (U', Dv') are two decorated graphs. We define the union, intersection, and difference operations, from which the complement and symmetric difference naturally follow.
= (U UU, ,U"" : (e) = Du(e) (e)((e)) Union. O'Ú"=
Intersection. Ú (U Dy" Do- Du n Der(e)). Difference. U/
(U U Du": Du"(ee Du(e)) \ (c)).
(Union graph partition) Assume that is the root of union graph. Consider the graph U with edge (i, a) removed for all neighbors a of Let C(i, be the connected component therein that contains a. Let Ú(i, a) be the graph union Č(i, a) and the edge (i, a).
Then, we divide the set of root neighbors Nii) into the following sets depending on whether Ú(i, a) is a tree: Nr- = {a: : (i, a) E E(U), ((U))(iia a) is tree), = {a (i, a) E E(U)} \WG. Furthermore, we breakdown. N into two sets depending on whether there are at least M edges 3-decorated: Na1- {a E Nr. Iefe e E(ğ(i, a)): (Du(e)) 2 3H 2 M).
Nr. = NT \WM..
Next, we define the decomposition of decorated union graph Ú: G(i,a), aENN
ÜL = U G(i,a),
ÚM = Ú(i,a, ÜN == (Ü UÙM).
(44)
aENL.
If any of these become an empty set, we define it as the graph consisting of the single vertex i. To provide an intuitive understanding: UN contains those chandelier branches that form cycles in the union graph, while Ü2 and ÜM are the collection of those chandelier branches that do not tangle with other branches from bottom and remain a part of tree rooted at (or i) in the union graph. As a characteristic of trees, the decorations on edges within Üa and UN are monotonically decreasing, meaning that the decoration set of an edge at depth d 2 1 is always subset of the decoration set of the preceding edge at depth d 1 that connects to it. For each node E ÜM that is connected to the root, the connected component g(i, a) should have at list M vertices that are at least 3-decorated. UL is the union of all connected components remained that are trees. The definition of Ün implies that the branches cannot be fully 3-decorated, otherwise this branch would fall in ÜM-
For every union graph, Specifically, we want to keep track of how many times each vertex appears on the chandeliers that rooted at i and the chandeliers that rooted We present the definition as follows.
decompose the decorated union graph based on its decoration sets.
Definition (Decorated union graph decomposition by decorations).
Kin = (V(U), fe E(U): (CeS)) + (eET)) k [(ee))) + (eeTa))))
(45)
Definition 4 (Edge counts on and 4-decorated edges). Based on the definition of union graph partition and Kim. we further define
er 2 (e((K22 U K21) (1)) + U12))
(46)
CM 2
[e((K2 U Ken) n M + e((k U K12) n UM)),
(47)
CN = 2
je((Kz U K21) n nU)) + e (K2 [U))],
(48)
43

---

where 2el (resp. 20M, 2eN) is the counts of 3-decorated edges and two times the 4-decorated edges on Ur. (resp. UM. U).
We use (VL, E) to denote the collection of all possible Üz with er vertices except for and j. e7 counts of special edges as defined, and edges in K11 n Uz. Ua (UM, (UM,,M))) denotes the family of all possible UM with UM vertices except for i and and counts of special edges as defined. Uv (uw, eN, k) denotes the family of all possible ÜN with uN vertices except for and j. ex counts of special edges as defined, and excess k.
We further define the weights for of decorated graphs Ut. UM, and UN-
Definition 5. Let G be an arbitrary UL Uns, or ÜN, we define the weight of G with regard to chandelier S as
ws(G))
II
aut(B)².
(49)
BCC
We set (G(()) = 1f G contains no bulbs in K(S). We define the weight of G as the multiplication of its weights over S1, S2 T2
w w(G) == : (C)ws,(G))r; (G)wr;(G))rg (G)wT, (G).
(50)
We have the following observation with regard to the concepts introduced above:
(Counting edges) 2(1z + Var + UN + k
((21)) + 2(ez em + eN) = 4N. This
holds because 2( + uM + uN +++ + 1 (ixz)) counts all the edges on union graph twice and 2(e,, + eM + eN) makes up for additional count for 3-decorated edges and two counts for 4-decorated edges.
(Automorphism decorated graph weights) From the definition of aut(-) and chandelier,
(51)
(Trivial upper bound on the decorated graph weights) Since aut(B) (B) is upper bounded by Rt for arbitrary bulb B, this inequality follows from the definition of decorated union graph weights
w(ú)) s (VR Wel
(V R2WNE,
(52)
The same holds for ÜM and ÜN.
Specific to the moment calculation for correlated stochastic block models, present the following two definitions.
Definition 6 (g. o-)). Fix chandeliers (S1, S2, T2) on the complete graph, conditioned on the ground-truth labels of correlated SBMs, we define a function as follows:
96 00(0+.0))==00 ( ((S)))
where h(-)) is the counts of edges connecting two points from the same community, T((-) the counts of edges connecting two points from different communities on the complete graph K
Definition (Extension of ozr)) We define 7/2 = -0--t0² fo and 7/1 = (02+02
o Thus,
(02 107
72 and (09 = (1) (o 7..
= =(1
Remark 5. We observe that 7/2 < 2 and 7/1 S 72- The first holds simply because 0f + 01 2 20 By Cauchy-Schwarz inequality, for an arbitrary random variable X, E[X"][[[22] <
[[X311 Let X takes 02 and o² uniformly at random, then 1[ W EX Y1 72 is implied.
E[X31
EIX and thus
44

---

1.2 Proof of the Proposition
(a. b) 1, we can recover the correct community label with high probability [2, 47. 1]. Therefore we can effectively work on the correct centralized adjacency matrices. Recall that H
In regime sD+(a,
{ - n |+ |V S + n as defined in Section D.
Proof of Proposition 2. From the definition of variance,
Varl Var[D151x1
Σ
aut(H)aat(()
Σ
Cov Bs, 1n Ar, Br, 1x).
HJETT
81 S.STTTTT
S S2, - T2, and V(SI) (V()) {i), the covariance becomes zero. Also, we know from Proposition that [[As, Bs, [ Br,)] either 0 (1 + (p)))(pa)t)) )2N Thus,
Varl#; 1x] s
Σ (at(H)a
Σ
E[As, Bs, BT, 1x].
HJIET
S.S.
E[Ag, Bs, Ar, Br,] is non-zero if and only if each edge E U is least 2-decorated. Let Wij
denote the collection of decorated union graph D. where (i), S2(1) H, (1). T2(j) = I for some H1 1 T, such that each edge is least 2-decorated and satisfies + S2 or Ti # T2 or V(S)) + {i}.
Var 1,y11] S
Σ DeWS
Bs, As, B 1n]
(53)
Conditioned on the ground-truth labeling d..
go(0+.0-)-
II
II
[6+m) c(a,w)
25F+ (u.v)EKem
where c(u, v) = (u) (v) and c(u, v) if (u) # (v).
Conditioned on a.. H. is determined.
E[As, Bs, Ar, BT, 1x] = E [E[As, Bs, Ar, BTd [0]11]]
(54)
Conditioned on a., 96 (o_ o-) is fixed, so as 9ú (a+, o-).
00(0+,00)--E(AA Bsg Ar, Brg
o.] II
II
o((u,)) A B o.]-
((+m))
25t+m54 (u,u)EKrm
We define BBm (e) as σ c(u,v)
(L+m)
E B au oo1, for - (u, E Ke, Thus,
96 0 Bs, AT, BT, [..] = - II
II
(em
25E+m54 (u,))E
Then, we apply Lemma 13 to bound each 3rm (e). Since the upper bound of 3Bm (e) only depends on and m. we have the following:
(sp A sq) <((Ka))
- Bs, Ar, BTd o.] S maxx p_xx(pt,,_)1c(8..1 (((Kk.)
(sp A sq) (KKa)t((Kaa))
=(1 + =11)
(K()) (sp sq) -2N+o(()
(55)
where the last equality holds because 2 [ e K21 t [(22) (2 2) 2)e (2n) (2 2)e(Ko2) +(3 2)e(K12) + (3 - 2){(K()) + (4 - (22)) 4N e(U'). From (54) and (55), and the fact that 7 happens with high probability, we have
E[As, Bs, Ar, BT, 1x] (1 ((1))p (Kn)
(sp / sg)
-2N 22 Eg (a_ ( H). (56)
45

---

It remains to compute E [gg (0+, H/. Here, U is fixed and we assume that there are d2 2- decorated edges, d3 3-decorated edges, and d4 4-decorated edges without loss of generality. Let x x and ( be the number of in-community edges for 2,3, and 4-decorated edges on the complete graph respectively, In Ü is tree, we can apply Lemma 10 to determine the distribution of X X (3), and (4)
Generally, Ú contains cycles. Ü can be decomposed into a tree with an additional set of edges connecting vertices on the tree. We assume that there are A; i-decorated edges on the tree and B; i-decorated edges in the additional edge set of size e(U) - v(Ú) + 1, for i E (,,3, 4}. We apply the Corollary 1 and have that (P
0g (3,a) 0z X(4.) a1 X( b2 X (3,6)
by, ( by H) S (1 + o(1)) oz
A2) As) (Aa) ado+dgta, where X(aaa) is the number of i- decorated in-community edges on the tree-part of U and ((.))) is the number of i-decorated in- community edges among the additional edge set.
ag
A; and B, are fixed but summed up to d, for each Ü. ag (be) takes possible values from 0 to A, (B.), for E [4]. The number of i-decorated in-community edges is a; + bi, and the number of i-decorated cross-community edges is d; O₂ b; = (A: a.) (B; bz).
4 Ag
B >ΣΣΣΣX
A
B4
B
96 (04::00)
=
Σ
2()z
0 2(d2-a2 a2- bb) 3(as 3(ds- 0+ 0=
02
03
94
b22
his
ba
Martbss)
Ald: -04 -o4
A2
A3
A1
x
+
0=
12+
a22
03
04
2 + +
A2
+ 0²
0
+
Aa
2
2
B2
14
2b 2(Bz-ba) b2)
3b3
3BBs
by)
4b4
4(B4
0
b==0
b==
b,=0
<
0²
+
2
d2
3 + 3
dg
o + 0
d4
2
(57)
2
2
where the last inequality holds because we can upper bound by multiplying binomial coefficient (2
B3) Ba) and that B₃ + B3 + B1 ls + 1. By the definition of 7/2 and 71,
by)
E o H1 < of 2d,+ id1, 1 (2gk+t S def 72
Weettertexx)
(58)
The last inequality holds because 71 < 72 and dz + +1 = <((K2) + e(K21) + el e(K22) S 2(e1 +
eM + ex).
Denote (1y k) as the subset of Wiy that contains all the elements that have exactly vertices except for and j and excess k: By applying the definition of union graph partitions, decorated graph weights, and (53),
VarlD S (1 + o()) ΣΣ
(sp A sq) -2N -2++++++1
Σ
p (o(Ku) ((U2))((Us)))())))))
(EWi)(o,))
AN 2(ertterttr))gtt)
oeff 12
(59)
For the ÜN.UM. UN partition, we define
PL(UL, el, e) 1=
Σ
w((t),
(60)
PM]( ==
Σ w(ÜM),
(61)
RN(uN,eN,K) = Σ w((n).
(62)
(NEUN(eN.eNN)
46

---

Then, we can write out the upper bound as 2N+
Var(D. 1u] 2
Σ ΣEO sp A sq)
Σ
Σ
AN 2(er, tenttede) eff '2
VLAUM 20
PE e,, (x) k).
Applying Proposition Lemma 18, Lemma 19, and Lemma 20, we can upper bound the variance by: Σ Σ
Var[ 1x] x
Σ
Σ
(sp sq) 2N+
(1138m))M R aM Ses 4K(AM)
&
c,20c,600
AN teur
2n" 2n"'(1+ 2 fr, (K+ 1 t2LNN))
off 7/2
n (118) UN (22R'(NN + +)) 8++ (UNS2(++M((+0)
4K+1M we have Ever 20 (113) & 2(118) 4K4ah
Note that aM & eM
Also, we know that
M
UN 2N, UN 2(K
M) 1). Then, E UN 20 (113) S (111)) Thus,
Var 1x] 2 Σ
(sp A sq) 2N+o+441
Σ
nnn " R (113)
4
k.
2N-
(22R*
((
1)
+2 212)
4N H(ectertee) off 12
Σ
VL 202120
Regarding to the Lemma 4 in [41],
121" (13)
S 1
then
2'
Var( 1x] x
Σ (sp A sq) 2N++0++41
Σ
2'n" (R (113)
awteM
Un untuu +e,, tel #cN=2N tearte))
22 + 1)(18))(( My s+1
2(
(1 Mn)&t)
222)
def 12
121.2
8p2
2cl 1 10
( 1 lc,=0 IT?
12 Note that ([K+M0(]]121) large enough n.
I is guaranteed by the first condition (22)
p2Ck ( S for
In the following step, divide Varl 1x1 by E(Pin. (n) and use the fact that eN 2N (u k ++++e + teM):
22R (2N + +1)(133)((KK0)) 6+1
Var sg 1xd
4N I (1+ o
Σ
E o
(1)
1x]
#2-1
RM (11,3) 12
4K+4M Mr
ns(p A ②)
eM
1 21
ez
12I²
72
ns(p Aq)
20l (er>00
p2(K+M) 1 (en=0)
ez.20
2N
2N (ottt++tta)
n
sp A
20
In view of 7/ < 2, the last three conditions in (22) guarantee that
22R (2N + 1)(1L8)2(K+M))
k+1
Σ
< 2,
n
R (113)
3K+M nss(p ^ 9)
1 2T,2 a
) 22
Σ
1/2
22
Σ
Y2
ns(p ^ q)
ex
e 20
47

---

2/
Also, for sufficiently large n, ns(pAg]
2 such that
2N k-1
ns(p A g)
72
2NN
(14
22
In conclusion,
Var($5 1x] E[ (1) 1x]
(2 + 4T.3)
-0
12T.
1,2 (onsspp Ag) (00
I,2
S
0
ppns(p Aq)
- p2(K+M)]J)
p2(K+Mi]J
I.3 Proof of auxiliary Lemmas
To complete the proof of Proposition 2, it remains to prove Lemma 18, Lemma 19, and Lemma 20. In this case, those upper bounds have been shown in [41]. We briefly re-state the proof idea. Lemma 18 (Upper bound of (L EL. () (True pairs: j = T (i), sD (a, b) > 1)).
Pu(uL,eL E) s (1 (1+ ((1))nn"' 2 (k to
Proof of Lemma 18. Firstly, we define unlabeled union graph class Ü(v,, ez, E) and set of labeled union graphs isomorphic to ÜL Ü(UL,eL, e) (Ú))).
CL, (),
Σ w((L)=
Σ w((L)H((LL.
ÜLEMU(VL.EL,))
The number of ways to label Ü2 is H(LL) (:5) sut((t)) S =ut(())' ("")vL! S n² In addition,
n
w(Ur) au)) This is true because every bulbs exactly 2-decorated so for the union graph, the automorphism number coming from the orbits in this bulb is the same as the weights of this bulb. However, there can be other orbits outside the bulbs, for example, some bulbs isomorphic the vertices on the wires can be in the same orbit, thereby increasing the automorphism number.
From Lemma 7 in [41], know that
Uu (Uz, ez, ()] < 2(1 + 222) (K (votors2NY,
where fi. counts the possible structures of chandeliers before merging edges to form union graph and is depending on l.
Combining these pieces together, we derived the upper bound: PL(UL, () S nn (1 2 ( tK
te,
See Lemma 3 in [41] for more detailed discussion on 1, and Ji,--
Lemma 19 (Upper bound of PM eM) (True pairs: j = k. (i), sD+ (a, b) 1)).
(M eM) < (11,8) R " 22x 72 eM fowr Sem 4K 14M
off
Proof. We have WM M IM 2e because each connected component Ú(i, a) contains at least M edges that are 3 or 4-decorated and Nas is the number of neighbors in this set. This gives the constraint on UMs S A (2K 2.M) and from the definition of w(+), w((m) < R22 < "M Define Uss(UM ess) as the unlabeled union graph class and HŪMM) as the set of labeled ÜM for UM
PM((MMEE)=
Σ
w((U) R M
Σ
[((UM)
(63)
Un (vMM..)))
We have [H(Um)] (m))a. There most goar rooted unlabeled undercoated trees, where [T-o(aja [55] and at most 1M decorations for each tree. Thus, (MM,UM,eMM) $ (118)aM. Finally, we obtain that
PM(UM,eN S (116n) vMr 3
for S (2KK+22) 2 2004
48

---

Lemma 20 (Upper bound of Pv JuN,eN, (True pairs: j = (1), (D+(a,b) > 1)).
PN(UN, k) S nMM " (I1 (1IH2 + 1)3) fuN (UN
Proof. From Lemma 2 in [41], we know that [Wvl S 2k and thus w (Un)) I R!NM R4(K+1), Briefly, this is because the excess is k and whenever two branches tangle with each other, the excess of this graph increases by one. To maximally involving branches this k times of branch tangles, we never re-tangle a branch after it has tangled with the other branch. In this way, we see that there are most 2(k + 1) branches in UN. This immediately gives the condition of un S (2k 2) + K). Define UN (NNNN,N eN, k) the unlabeled union graph class and H HÜNx) the set of labeled ÜN for ÜN E (N(UNNEN,H)
eN,
E
w((N)s R((++1)
Σ
H(UN)]
UNEUx(eN.eN..)
ÜN (NEUN(VN..N
Ur consists of a tree with uv vertices and additional k + edges connecting vertices on the tree.
What's more, each edge can be associated with at most 11 possible decoration
(1x++)) k+1)
ÜN(eN. CN. k)l BUN
gUN
11
< (113))M (11(UN + 1)2)(++1),
Therefore,
PN(UN,eN,N) S nnd (113)) + 1)))tt1 L ((K
J Proof of Proposition
In this section, show the Proposition complete the analysis on variance of similarity score. In this case, as j o n. (i). the decorated union graph structure is different from the case when j = .. because the roots of and S2 are different on the complete graph.
J.1
Graph partition
The definitions in Section I still apply, except that we need to re-define the union graph partition.
(Union graph partition) We first decompose Ú into three edge-disjoint subgraphs. Specifically, for any neighbor of in Ú, consider the graph U with the edge (i, a) removed and let C(i, a) be the connected component that contains a. Denote Gi,, a) the union of C(i, a) and the edge (i, a). Let
(1 {a: (1) {a (o((i, n 0) (1).
0, a) is a tree not containing j):
Symmetrically, let
Nz = {a Du (() 0, a) is a tree not containing i),
Nx {a: Du Du ((. a)) n (S2,T2) + (J)
Next, we further decompose (r)) into two edge-disjoint subtrees and similarly for ÚT.
In particular, define
(i) = E Nr He B()(i,))) DDal 2 331 M).
Ni = Nr(ii (M)
Then, we decompose i according to the node set N into two tree-pane parts on root (resp. f):
Úz())= U G(i,a), Úm())= U ((i,a). OENL()
(ENu((
49

---

and the non-tree part
ÜN = U ÜN= U102000000000000000.
(64)
For convenience, we denote
Un = U00-(()))0)), ÚM UM()) UtM
(65)
We again assign weights to each of these decorated union graph partitions. w((n)) = ws, (Uz.1)) (1)( (Ur()) w(ÜM)= = wss (UM (1),, (iss (U)
(66)
(67)
w(UN) = ws, (Ú \ (r()))rr (Ü (T( (Ú Wr( (Ú (r()).
(68)
J.2 Proof of the Proposition
Proof of Proposition 3. In the case of * (i), the excess of the union graph starts from 22 This is because the minimum number of edges of decorated union graph for j * TT) with vertices except for and j is v, when Si, Ti and S2, T2 form two disjoint trees rooted at and j
respectively. Therefore, & 2 0 - =(( -2
The proof of Proposition 2 in Section holds for j # n., until (59), which holds with placing + with k 2. This is because the decorated union graph can be viewed as two disjoint trees with v + vertices plus k + 2 edges connecting vertices on these two trees. We have,
2N-
VarlD, 1x] & (1+ (1))
Σ
Σ
(sp A sq) 2N+
Σ (eW,s(v.k)
pelKu) aut H (au((I)))
e(K (Ka))
x off 12
We define Pi)(u,k)== = Edew aut (1)
2N 2N-k-2
Varl 1x] $
Σ
(sp A 8q) 2N+ 4N 2 (1
ell
Case a: Ao = 22 We first consider the special case when k 22 When kt = -2, there are two disjoint trees in the decorated union graph. Then, it must be the case of S Ti, S2 T2. 2N, and = T. Therefore, el (21)) = e(K2)) = e( (22) = 0. We have,
P2,(2N, -2) S Σ 2 S ((() S2 = H 1
HET
Under this parameterization, -2N + + k + is also zero, we have
2N-k-2
[
Varl2 1ul s (1+0(1))
AN off
Σ
(sp A sq) 2N 2N o 2N Pay (iy(o,))) (o,
turns out that the summation over k 2 would have the same order as the first special case.
Case b: & > 2. We enumerate through three parts of the union graph. Recall that el ((12) +
e(K21) + el <(K22) < 2(er, + eM + ex) 4N - 2(v + k 2), therefore
Po(u,k) < Σ Σ Σ
eM VL.UM (EW().A)
72
2
Σ PL(ve))P((eMm
(69)
es
where Pa (UM, eM) ) is defined as before, while we let PL(v)) and PN (uN, k) have no constraints on the number of 3 and 4-decorated edges for Un and ÜN.
50

---

We can plug those upper bounds from Lemma 21, Lemma 22, and Lemma 23 into (69);
(r)(u,)) - Σ
n"1 2 (0N-2NN(11R42))
E
((1))))(((113)) 22))8+2 0 12
4N (N-2((++42)
x
x fom SX(K
Note that UM, UN (2N + 1))((+2)) Applying eM 2N (v+k+2), UN 4 (k)(k and UM S 2(K + M) cu we can get the following upper bound:
Sus 2N - 2N + as k -1. Therefore uM(UN 2)2(k+2) S
Var D. 1x]
S (1+)
TTp2 (1+ o(1))
4 (22
(1) 1x
TTD2N
(113)a(K+M02218 + 1)33
n
2N 2--
40
2N 2N-(0+R+2)
RT (11
X
ns A 4)
From the first condition in (23), because 72 have
2N k-
R (113)
2N 2N-(v+k+2)
Σ
22
ns(p A 4)
From the second condition in (23), we have (2
Σ therefore,
(118) 1)
A+2
(113) M) 22R (2N + 1)3
Σ 12-1
<
From the second condition in (23), we know that
44 22L (63) (63 (119)4(K+M)2221)) (63)) 4M
+1)3)
1
S
n
In summary,
Var D 1H]
11 1 (T|p2
01
E| 1 in. (1) 1x]
J.3 Proof auxiliary Lemmas
The following Lemma 21, Lemma 22, and Lemma 23, have been shown by Mao et al. [41]. We briefly re-state those results for completeness.
Lemma 21 (Upper bound of (V( (Fake pairs: j + n. (1), (a, 1)).
(v.(vr) "
Proof. Firstly, we introduce unlabeled union graph sets (vr, en.) and the set of labeled isomorphic members as H (Ú)), for Ür El(ul, (). From definition (60),
PL(vL) = Σ w(())=
Σ ((ÚL))H((LL)
ÚLEMc(eL)
As we have repeatedly seen, (H(U)) S a From the Lemma and Claim 5-(v) in [41] 2)-2; (6))))++))
we have the following takeaways: (1) (11)) I (68)4(( (63
w(z) <
(i)(Uz())aut(())))) aut(Uz). Putting pieces together, complete the proof.
51

---

Lemma 22 (Upper bound of PM (uM eM) (Fake pairs: * n (i))). PM(UM,EM) S uM
2os n (113)"M 1 (2M (PM (2Ms2(K+M) "M
Proof. Define Uy EM) as the collection of unlabeled decorated union graphs and [ (UM))) Ml be
the set of labeled isomorphic members for each ÜM UN (Var, ear).
In UM- there are 2 edges that are 3 or 4-decorated. Since for each branch connecting to the root, there are at least .M edges on it being at least 3-decorated, [W) + [Wa s 2 Thus, this directly gives w w(üM) I Ra x2([Nx(01++ (a) < R and PMr(UM,EM)
Σ Ú (VM.MM
w(UM) S R M
Σ
H(
US (vs
Since (1) and (j) are two vertex-disjoint trees with UMr edges. There are at most UM 3"M unlabeled non-decorated because each way the number of rooted unlabeled trees are bounded by grM There are at most 1M ways of decorating each edge. Therefore, [M (Mu,,MMe ex)] I UM (113)uw, In addition, the number of labeled isomorphic members (Uxss)) S nM and the number of 3-decorated edges plus two times of 4-decorated edges, eas, is upper bounded by 4N 2(v 2). Putting things together,
allocate uMr edges 2 trees in at most UM ways and under
(MM eM) S UM 3 PM (118)"M1(s S2M
Lastly, because each connected component Č(i, contains at most 2(K + MM edges, for all E (i) (same holds for E NT)()). This because there are at most four wires, each from S1, Tı. T2 go through vertex and every edge is at least 2-decorated. Therefore, UM S
-
(K + M) 20M
Lemma 23 (Upper bound of Py(un k) (Fake pairs: j T., (i)).
(N(NN, k) S nM" n""""""L"""" ("133(1)" (LIK* (uN + + (oNS((++M)(+2))+
Proof. From the Lemma 9 in [41] we know that (1x (N(() S 4(k + 2 The intuition behind this bound that ÜN be viewed as bunch of branches (wires plus bulbs) coming from two different roots j.j tangled together. Whenever two branches intersect with each other, the excess grow by one. Since the excess of decorated union graph is k and the starting point is separate trees (with excess 2),, there must be k + 2 times of intersection between different branches. Each time of intersection involves at most four branches.
Thus w(Üx)= = RIN (W)+ < (2()+)) As usual, we define Ün (UN, k) as the collection of unlabeled decorated union graphs and (H ) be the set of labeled isomorphic members for each Ü, (x k).
PN(uN,k)
Σ w(Ün) S R1(++2)
Σ
(((L)))
UN ExEMN(uNs
UxEldv(ex.))
By (H(UN) S (1)
uN! [N) I nN we have:
(N k) & (k++2) ÜN (UN,
In this part, the total number of unlabeled non-decorated graphs ÜN with uN + vertices and excess
k is bounded by +1 ((xx)))) < + + 2))(k))) This is because for a unlabeled connected
141
graph with un + 2 vertices, we construct spanning tree with uw + 1 vertices first and than add additional + 1 edges connecting some of the uy + vertices. We can also bound the number of
unlabeled UN as S (2N,+2)
k+2 which constructing two trees rooted i and j. with total of Un ways of vertex number allocation, and then adding an additional + 2 edges. The latter bound is looser. Also, there are at most 11 1 ways of decoration. Thus,
[uN(NN k3]l < 3ux+1 (uN + 2) 11 +x+2 < gex+ + 2)2(k+2) 11 Lastly, UN is upper bounded by (K + ())) + (Wv())) by Claim 4 in [41].
52

---

K Proof of Proposition
In this section, prove Proposition 4, which extends the Proposition I to the case when we can only perform almost exact community recovery with a single graph. We denote ō = (ā A, ā B), which is the combination of the community label estimates for both graphs.
Proof of Proposition 4. By definition of the similarity score and H.
[[
Σ
aut(H)
E
Σ
[[As
a
BT 1x]
HET
S(1 T()EH
+1+
Σ
aut(H)
E
E
E
E(EAAS Br |o,
HET
S(j TG
Define C. as the edge collection of the intersection graph of S and T: C : E(S) n E(T).
E 1 1f
1x]
Σ HH)
Σ
Σ
B
II
A
II
Bed o.. 1.. 1
E
HET
S(G) TG)EH
EEC
e P'E(S)CC
e"EB(TNE
Σ HH)
Σ
Σ
E(Ag 3 B
3B
Ox, σ)
II
E[A 04
10,,
HET
S()
GEC
e'EB(S)(C
II
E[B B
o., 1
C'EE(T))C
We perform case studies for E 1x] based on the structure of S, T union graph and then later sum each case up.
(a) S that is, all edges appear in pairs (this case is only possible for i j). There (1+)())))n) /aut(H) labeled union graphs of Si and Ti satisfying this condition. In this case, every edge is 2-decorated and it longer matters whether ø gives correct output not, as we can show the following upper bound.
Σ aut(H) nN
E| E[ 1j (1) (1 o(1)
E
E[A a "B= o., Ō|1
au)(H)
#ET
LEEC
$ (1 + o(1))TTT)
Σ P(S N+) Σ IP[ee = Not: Me- No-1C N4)
2 (p= (p-o2) No- (p+07 N + A22) A22 10 + N
N
+1+001)
Σ
2N (po + 2200~~(p0= -2 N-
N
1
N
N+=0
=1+ (po + 22)
where ( is the number of in in-community edges out of the N edges in S, Ne+ is the number of in- community edges that are centralized incorrectly, 7e- the number of cross-one coss-community edges that are centralized incorrectly, and A := Ip gl. (i) The first equality holds by definition and counting cases. (ii) The second inequality holds because there are No+ (N,-)
edges centralized correctly, each of which contributes the same L[A.B o..] (1 ((+o))))++o3 o²)
N4 Not (N N4 N.-) of them has E(A 7 B
(Lemma 13). For the remaining edges,
o., o (1 + (L)) + E[A" Z
o., 1o, " o., 6] s (1 + ((L) + A2). (iii)
The third inequality holds because A > 0. Lemma 9 gives the distribution of Ç, and p (1
( log )> p= = (1 O log n pp_ (iv) The last equality holds from the binomial theorem.
Observe that A = (1 + e log (p 2.1) Assume that N = O(log n).
1x (A) (1 + +o()) (por.t)
53

---

Figure 8: Left: One possible labeling such that all edges are centralized incorrectly, with the colored vertices indicating those that are labeled incorrectly and the black vertices indicating those that are labeled correctly. Right: Another possible labeling such that all edges centralized incorrectly.
(b) S and 7 have no common edges, that is, n ((T) = We can use the trivial hound on
the labeled S and T as au)
E [ 1x] (B)
Σ H
E
Σ :C)
E II
E[A 0
o ō]
II
E 3B o..
HET
SOJH
e'EB(8)
e"EE(T)
n 2N
$ IT|
P(E)A2N,
(70)
Hu)(H)
where E denotes the event (ā: E(E) E[A "IIT TEEEE E[ ((Aa())+(T)) n H, that is, every edge is centralized correctly and H happens. This inequality is true because conditioned on H and ō being correct, E[A a 0| o(n -D DD (a.b..))) the upper bound of the probability that one vertex on this edge being labeled incorrectly.
We further denote E' as (ē IeE "] ()) nH. is obvious that P(E) < P(F).. Then, need to upper bound the probability of & a giving incorrect centralization for all edges in T.
We observe that there are only two situations, that is, no vertices in S that has neighbor with the same label correctness as itself. See Figure for an illustration. We denote those two possible outcomes constraint on S UT ōJ and Ø2. PPEE)) = P = H) P((@A ō2) n H).
By the label correctness, we separate V(S) into two disjoint sets: V(S) for the correctly labeled vertices and V(S) for the incorrectly labeled vertices. The deterministic relationship between 6 and 0₂ is: V(S) ã V(S).2, without loss of generality, we assume V(S) (V(S) Observe that event ( = ai (resp. ( )2}) is equivalent as saying the set of vertices on odd (resp. even) levels are labeled incorrectly (falling in the bad vertex set as defined in Section D).
Denote Pa,b.s.e.6.V == n Vx Dx + +n =6
We can apply Lemma 8,
P( = 3 n H) S (1 - -P a. "a.......d (p aas.e..dvvi P( )2) ( H) < (1 - Po.ha) (P (absssdii(s))))))
P(E") 2(1 P (a.b,..r.& (U
Consider each chandelier has L branches and each has a M-wire (assume that M O log Log logn
as in condition (3)). Even if we don't know the structure of bulbs, we have a coarse lower bound log n Q log log n because least half of the vertices wires should be labeled incorrectly.
(V(S)
For some constant C₁ > 0 such that IV (S) 2
log n log log
P(E) < ((EE) S O(p (...))) & O(n (a.b,,,),)
54

---

By substituting P(E) and A into (70), we have
1 1=] (B) I 0
T
aut(
n
n
ID (a,b,s,c)) loglog lo (p )2N (2)))
log
Recall that = - (po )a,
Denote u²
A (po 201)
(= (1+ + o(1)) platb)) For some constant
22 such that po S C2 logn
E| E
1 (B
(B) 0
1cg uugiaN V2N
logn ) ynn N Nn D+ s),) Tog
1)
=0 uey
n D+ (e, lag
log
For some constant C3 >> 0 such that N = c3 log n, as in assumption (3),
= (p( n (c2R² log ni log log ne (4(abac))
lox lox lok
=(u/n²).
(c) S and T have some common edges, that is, E(S) n E(T) 0. There are most [n2N n (n - Jautt/ = o(n (a) cases in the enumeration of S and T.
N-1
E [n (O) S
Σ
aut H)
Σ
Σ
(po +
P(E"),
HET
M=1
MEL C
where we denote by E" the event
To II
E(A
= (1A)) n H. C"EB(T)C
II
e'EE(S)CC
Let Xa denote the number of different edges between and 7 under the true permutation. The structures of S and T such that (Xa = M) happens, Es(yen Σrgr# (X,=M), is upper
x bounded by aut(H) aut(Hr changing M edges from S to T allows changing most M vertices for tree. Then, we have
nM
[ 1x] (C)
S 1
(po A3) M A (MP((")
N-1
2(a -)a- b) 2M P(E") p(a b))
S 7 N
(po N-M M (po 2M
N-1
(1 + ((1))
(mpo 2 M ,2M M22MP(EE")
eff
(71)
M=1
We observe the following:
P(E") n (a..a
This inequality holds because there are 2M edges centralized incorrectly (required from event E") in S and and each incorrect labeling vertex can lead to incorrect centralization on at most D edges. Therefore, at least 2M/2DD vertices should be wrongly labeled for E" to happen.
Substitute P(E") into (71). Assuming that D = ol
log n log Jog
(3),
-
nppo 2 off 12
N-1
Scc logn )M = o(1).
E 5j
1x (C)
Si
M S
Σ
D_
(.,8.
nD (ab.e.s)
M=1
n
M=1
In summary,
[ [ 1x] (A)
+ E 1x (B)
+
E[
1x (0) = + +1
if = n. (1),
E 1x]
[[
(B)
+ E[ E[ o(v)
if j # n. (i).
55

---

Remark 6. (Denser regime) we are not restricting the sparse regime a log and blg 12 interested in what general p. conditions are for Lemma 4 hold. Assume that pVq (()))))) mp u²
we
then the geometric series 2 n (..b. converges if and only if c(n) > 1 (+(a,b)),e)
L Proof of Proposition 5
L.1 Proof of the Proposition
In this section, analyze the second moment of similarity score. We expect the variance of Φig to be infinitely small in comparison with the squared expectation of true pair's similarity score.
Proof of Proposition 5. Recall that St, Ti are rooted on and S2, T2 are rooted on j.
Var( 1x]
Σ aut(H)aut(I)
E
Cov (As B 1 Hi 1HAAT,
30 Be 1H)
H,IET
SL.
= Σ (H)
Σ
E[As, S B9, Ar: Br, 1x1
a Fy
(72)
HJIET
Vi
Σ (at(H))
Σ S1.5;-H.Ti Tee
EAAs B 12
2a 1x Lx][[AA
Brz
HJJET
(a) Analyzing V22 We first give the upper bound of the latter part. When analyzing with correct centralization, we ignore this part as it is non-negative in the correct centralization case. However, in this case, it is possible to be negative because there can be odd number of edges occurring once and also being incorrectly centralized.
After factorizing Vi2
V/2
Σ HH) HET
Σ
Bsg nn E[As,
(Eau()) aut(T)
Σ TTIT,
3* F[AT, BTg 174]
STS,
IET
can see that V2 and thus we have Varl Vi for (1).
(b) Analyzing VnI The main challenge here is that we not have the condition that every edge occurs at least twice in the union graph as in the analysis of Regime I. However, we can put union graphs into two categories based on whether every edge is at least 2-decorated or not. We keep the notation Way as the collection of decorated union graphs Ú that are at least 2-decorated and we decompose (72) as follows
V = Σ (H)(a))
Σ
Φ E(As,
B
Ay.
G BT,
1:]
S.
H.IET
S.S.HHH.1.T33 T
Σ dEwB
Jant
B 2
Ar
B 1n1 +
Vi 2
(aut(a))a )aut (aut(())) E(A,, B Ar; BTZ
F Fy 1x
Uewy
V12
We first show that V/1/2²² -0 (
56

---

The incorrect centralization affects the upper bound of moments as the following:
E [As,
S2
Ti
T2
1H]
=E
90 (0+,0-)
II
II
-(++m)
E (Au
B Ban. o 11
25t+m54 (a,0))KKm
=
90 90(0+-0-)
3e(Kem)
1
25ttmga
log n2
I 0 (o(k))
sp - +N(+ oeff (1 + O
e)(())
log n 4N
I <pekkn (sp (q) -2N+e(U) -
AN off
(1 + O
(73)
n
The first two equalities follow from definitions. The third inequality holds because of the upper bounds of Be(K ge(Kem) from Lemma 14 holds for all 6. m
Then, the structures of the union graph, alone with the assigned weights bounded the same in V11 (0
Section I. This implies that
S SO
L2
1a
under the same condition
(-ns(ppg)
as (22).
(c) Analyzing V12- To conclude Varlo 1x]
I
Vi++VaaVV 0 (pns(ppg)) + pack+mnpp
La
((()n.(1)) 1x
(() 12
remains to show V12 = o (na(ppng) 7 [
1.2
Lemma 24 gives a even stronger result,
because as assumed in Proposition
La o(log n)), giving pine(pra) + pacttnn11 > -5 for all
> 0.
L
Lemma 24. Under the same conditions as Proposition 5, for some e' > 0,
V12
o(n o(n
kx. 1x
Proof.
For the expectation inside V/2 part, it can be separated as eight sets Kem
2 3a As Bs, Ay; B1 1+] =
9096 E[
II
II Ae an Ba
10,
where 9ú the abbreviation for 9ú (0+,0
Conditioned on and ã, approximately centered edges are still independent with each other,
[As, B Ari BTz 1x = E 9096
II
II
E(A 3 2
me]2,,1 of Kem.
Lemma 14 gives the upper bound over Me (4+m) enumerating through the products, have
E(A
# B a.. @]
mmmm
and thus by
c(e)
90
II
II E.A.
B a.m.
0
66(2) (22
Kmn
(1 + θ
log ni ) 4N po(Ki)) (q)
-dkKa)))kk)))( P())a=))))))
(1 + (Kn)) - PNN)))112NP((((((( 1
u+
aAb
57

---

where & := (Every I-decorated edges are centralized incorrectly] } n H. and & is the number of 1- decorated edges, namely, == e(Km) + 10)) The second inequality holds because (e()( + el {(22))) - el (22) = (v + l 1) - 2N - z/2.
According to Lemma and condition (24), have that for any e > O,
((E) S n- -+(ab)))))
Putting things together, E[Ag, B 8, AT, BT, 1x] S (1+o(1))n (a,,,,,)) ( (spAss)utkt1 2N (gg HH..
It remains to calculate (o+,00+ H].
Similar as in the calculation in Section I, where we only consider U being at least 2-decorated. Here, we need to generalize it into the case when Ü has 1-decorated edges. U can be decomposed into a tree with an additional set of edges connecting vertices on the tree. We assume that there are A; i-decorated edges on the tree and B; i-decorated edges in the additional edge set of size e(t²) - x()) + 1, for i E [4]. We apply the Corollary and have that P(X = (2,a)
02: (3,a) 03 ((.a) (14, (1.b)) b,, (2.b) b2, (3,)) b3, ( by H) S
(1) (49) (1+)
2TT where (1.0) is the number of i-decorated in-community edges on the tree-part of U and ((,)) is the number of i-decorated in-community edges among the additional edge set.
(4g) (44)
A; and B; are fixed but summed up to d, for each Ú. (b:) takes possible values from 0 to A: (B,), for E [4]. The number of i-decorated in-community edges a, + by, and the number of i-decorated cross-community edges is d, a, b, (Ac a.) (B, - b.).
E (100
H
A
Ag
44
B
B2
BA
S (1 + (())
(b1) 2b22 2(d2 (id2-o2-bb) b2) 3(a 3(ds 0g bs)
0+
0+
01
b2
by
by
4(ds hs)
04 h4)
A
A2
Ag
A4
x 0
0
A2
Q1
02
04
=(1+0(1))
0+ -
A1
0
+
A2
+
+
As
04
+
o²
Aa
2
2
2
B
B2
B3
B.
bn (Br b1)
2b2
2
3bg 0
3(B
by)
Abs
1(B-bb1)
x
b1=0
b==0
b==0
b==0
$ (1 +
(1)) o(1))
0+ -o_
0 + t
0 +
0²
ds
0
+
ot di
2
2
2
where the last inequality holds because the upper bound holds with multiplying a binomial coefficient
(B:) B2) BB)
(Bd) and that Σ; B, 1o 1. By the definition of 7/2 and 71.
by)
E(go(a+,0_) | H < (1 + o(1))
(5+
0+ + o_
di
2ds 4ds Ads, do 24
71
Define 7/o : 2
)/ off Gef and we can see that 70 < 1. So, we can upper bound (z )ds as off Because 711 7/2 and d3 + d1 = 1(K))) e(K21) + e(K2)).
(
E [lgg(o,,00))
H| < (1 + o(1))
1N c( (((Ka))= (Kn)))) 72
In summary,
F EsAs, Bs, Ar, By, S (1 + )()))) (abbsss)) -- à (sp
sq) u 2N p((1))
x off 72
((K12)) ((k)))))((
(74)
58

---

After plugging the upper bound on cross- moments to the ratio, it remains to bound the number of different union graph structures and their corresponding weights. In parallel Waj. we define Siy as the collection of decorated union graphs that have at least one 1-decorated edge. (1, k) denotes those with + 1 + jzz, vertices and excess k:
V12
E [ it. (1) 1x!
E 20th+1=0 Evesh(er)) aut(H)ant(D)))((k)) A sq) (ww.akkk)eeeBaaaeeaaaaa + ([]))n2N p2N 1 D
72
We define the (UZ Un) partition of decorated union graph (44) in Section I. We define Ur (VL, 2, f) the collection of Un that has uz vertices, edges belonging set c(K1), and no more than 1-decorated edges. We define Uar (M)(wM) as the collection of ÜM that has uM vertices. We define (UN, k) the collection of ÜN that has un vertices and excess k. We also keep the notation of as the corresponding unlabeled decorated union graph sets. In addition.
Σ
w(ú)),
(75)
(OL.E))
(Mr( = Σ w(UMs)
(76)
ÜLEML(UN)
PN(uN,k) =
Σ w(Üx).
(77)
From the above partition,
V12
E 0 iT.. (2) 1x,2
ENN Es sq) +Q) Ee Evr wM P.
(1+0(1)) 2N D (a.b,,,)) 2(04h+2)-1N
We show upper bounds for PL (L 2, Lemma 25, which is
2N Σ p PL(u,, E=
= (4N) n ( p
The upper bounds for PM and Pv trivially follows from Lemma 19 and Lemma 20, with a replace- ment of 11 to 15 as the possible decorations of each vertex increase by 4 for 1-decoration and different maximum value of eas, parameterized by v, k,
PM(UM) < R 2 PM (K+ 2 1 Sew
PN(UN, k) S NN (15 (15R"(uN + (
Putting all the pieces together, we have
(2N ()3(113))( R+1
V12
(x)
1
n
12-1
4N --1
R (15,3 KTM
2N 2N-
x
ns(p A 9)
=m N
AN Σ
K+M RA (1533 M
2
(a.b.e..)
pn
59

---

From condition (24), 2
I log' n. Also, since with (24) and 12 < 2,
R (15 A5
7/ 12R ()(
1
ns(p A q)
n
For 2N - k - 1, we have
2N-R-1
R R (15,)) K+M
2N-v-R-1
Σ
2
ns(p A Aq)
N
For the first summation, we always have
30N + ()23
h+1
2.
n
For the last summation with those additional terms, from condition (24), we have
4N
R (1533) KLM
2 o(n o(n
(a.b,))))) pm
for some E' > 0 because (abbss)) the only term being polynomial.
If v > 2N - k 1, then we know that > 2(v + k +++1 2N). The summation over and together upper bounded by
1+++1-2N
kN-k-1 Σ
R (153)2 KIM
(4 (15,3) K-M
22
2NL(4LM) = o(n o()
np A
D. (a.b.x.e)
2=
Vp
because, again n (a.b,,))) is the only term being polynomial.
In summary, we have
Vi2 E in 1x = o(n -') for some s' > 0, which completes the proof.
L.2 Proof of auxiliary Lemmas
Lemma 25. For true pairs,
2N
Σ
PE(U,,2,,) (4N) L(LMM))) p-
Proof. We define the unlabeled union graph sets corresponding to LVL, 2) as ((1, 2, From definition (60),
From the definition (75) and Claim 1,
2N
2N Σe 8=
nh w()) aut((
2N
p PL(ULL2, 2, <
Σ
<
Σ n (1LL (78) 1=
(V
Recall that ez. = [e(K12 U K22 n Ú2) + e(K21 U K22 (z)). The total number of edges on S1. S2 and T2 involved in Üg is (2(u) + EL) -2) 2) and thus the total number of bulbs on S1, Ti, S2 and T2 involved in Üz is b = -01
K+M < 4L. Those b bulbs can be partly or fully overlapped
(namely, tangled) with another stay on their own. From the definition of U. (44), is impossible to have three more bulbs tangling with each other. If two bulbs are tangling with each other, we put them into pair. If bunch of bulbs are all not tangling with any other bulbs, we pair them up arbitrarily. We denote & the number of pairs of bulbs that have decorations being subset of (S,, or (Ti, }2}. For all 2-decorated edges among these pairs, they are in K Since Ü, has at most 1-decorated edges, we have
8 22 tik 2/2.
(79)
60

---

Next, we introduce three types of bulbs. The first type is called effective non non-isomorphic bulbs, which a selection of bulbs that are not isomorphic each other and always pair with a bulb that are not of the same type. The selection is not unique and we take the largest possible set of bulbs satisfying those rules as the set of effective non-isomorphic bulbs. Fixed the effective bulb set, for bulbs that are isomorphic to those effective non-isomorphic bulbs, we name them as shadow effective bulbs. For the remaining bulbs, we name them as non-effective bulbs. We have the following Claim 2:
saas+ 4 2
(80)
2
From definition, there is at most one non-effective bulb and at most one effective non-isomorphic bulb in each pair of bulbs, while two shadow bulbs can pair up.
We call those effective non-isomorphic bulbs as effective because when enumerating through the chandelier structures, we let them having the priority of taking any possible structure from J and serving the base of that pair. Shadow effective bulbs are named so because they mirror the structure of effective non-isomorphic bulbs and thus will not increase the union graph richness too much. For non-effective bulbs, we let them take any possible structures with the constrain that there are at most z 1-decorated edges in ÜL.
We denote the number of effective non-isomorphic bulbs as For all pairs, we upper bound the number of different non-isomorphic tangled bulbs as following combinatorial factor
(1))
2N 2/2 (4N)
where (1)) comes from the structure of a effective non-isomorphic bulbs, (-) is the upper bound of choosing a effective non-isomorphic bulbs from b bulbs, (2/) is the upper bound on the selection of which edges on effective non-isomorphic bulbs and shadow effective bulbs are overlapped as there are most 2K (< 2NN) 2-decorated edges on bulbs if all bulbs are perfectly overlapping with one another, and (4N) 2 bounds the placement of those remaining 1-decorated edges from the non-effective bulbs as each of them has most 4N possible vertices attach to on the union graph.
Since wires cannot tangle with bulbs (otherwise, it is not a tree), we bound the possible structures separately. There can be at most 4 wires tangling with each other, from top bottom. We apply very loose bound even without using this fact, which is (b- - 1)!M--1 This is because there are at most b wires and we determine the structure of wires the union graph one by one. When the f-th wire comes in, it can determine which of the t wires to tangle with and the length of overlap, from 0 to M.
Putting together (79) and (80) with the combinatorial observations, we have
21 4L
b
(VL, 2,11
1)!M
(os
2a
24
25
4L
b)
< ITI44))
21J p
(b
11)!
a
h==0
b=
2L
< (T
Σ
21
4==0
< IT
(81)
In the second inequality, we loose the upper bound of t, from a 2L and change the or- der of summation. In the third inequality, we bound the summation over a and b. Lastly, (HE < L([]]p²K) L
Plugging (81) back to (78), after summation over we complete the proof.
Claim 1. Assume that j = (i). For any arbitrary Ün E Uu 2, f), we have
w((r) (2t((
61

---

Proof. Denote all bulbs contained in UL as B1, Bz..... B (1) Some of them can be fully overlapped form 2-decorated bulbs in the union graph. (2) Some of them can be partly overlapped. (3) And the remaining of them are fully 1-decorated. There cannot be three or more bulbs overlapping with each other thanks to the definition of Ü2 (65).
Each bulb B, contributes to the w((() ) by (B)( independently from definition (66) and (50). For any vertex on the bulbs, it will not be at the same orbit as any vertex the wire, studying the automorphism of the overlapped bulbs gives a lower bound on the automorphism decorated graph. Since each bulb occurs in at most overlapped bulb, to prove the claim, it suffices to examine the relationship between weights and automorphism for each of the three cases aforementioned.
the whole
For i,j E [w),, if bulbs B, is partly overlapping with By. From Corollary 2, (a, =
(B; J (2K) (BBB)))BBBB))))))) If B; is fully overlapping with By, then v /aut(Ba (B,,0B,,ttaui Jaut aut(B, U By). If B, is fully 1-decorated, then (a, S aut(B,).
Denote /, and 12 the collections of index pairs that the corresponding bulbs fall in case (1) or (2).
Denote 13 the collection of indices corresponding to the bulbs falling in case (3). Therefore,
w(úL)s II aut(B, U (2) (k.))h
II aut(B, U BB) II aut(B,)
(KJ)
iETs
< (is.E)
(82)
The union graph has at least 2 x ((.e) EBB.)) 1-decorated edges and we know that UL has at most 1-decorated edges. Therefore, E(s)er (E()))) S 2. Substituting this into (82) completes the proof.
Claim 2. Let be the number of effective non-isomorphic bulbs and t1 be the number of pairs of tangled bulbs that are decorated by a subset of either {2) or (Tr, T2)- We have
1 < a
4 2
< L
(83)
2
Proof. For an arbitrary bulb, it occurs at most one time in S1, S2, Ti. T2 each and they are paired up into sets, a 2 3. Assume that there t' non-isomorphic bulbs among the t pairs, they can all be assigned as effective non-isomorphic bulbs. Then, without loss of generality, S1 have at most L -- bulbs unspecified. By assumption, they cannot pair up with each other, every one bulb from S1, S2 remaining will pair up with another bulb from Tı, Tg. Therefore, the remaining bulbs have at t' effective non-isomorphic bulbs.
most 2
Together, we have s (L ()) s L 2. as t' t
M Proof of Proposition 6
M.1 Proof of the Proposition
Proof of Proposition 6. Recall that S and Ti are rooted on S2 and T2 are rooted on j. The minimum value of
when the union graph consists two disconnected trees, S₁ T2 and S2 TT. We use the same notation for different parts of the variance as in Section L.
62

---

Var 1 1x =
Σ
aut(H)autip
Σ
Cov(Ag, Bs 1 Ar; BT
81 3A
3 の 1x)
H.IET
Σ
(T2)
E[As
B2
AT,
#
BB,BBT #B
BT₂ 1x
VeW;
V11
(aut(S))aut(S,) (auttttS))aaaaaaaaaaa (att (aut((T)))))))) (Tb))
[As. Bs
34 Ari
BTz 1xl
W
Via
Σ (at(H))
Σ S1 ,,
El A
B 1x
[AT,, BIg 1x]
(84)
H.IET
V/2
Since V2 O, it suffices bound the first two summations. The same argument in Section I. to bound the Vn for true pairs works for this case with an additional fluctuation coming from using Lemma 14 bound the cross moments rather than Lemma 13. We have
log n J Var D 1x] (a.))))) = O |T
Vn/p²² < (1 +
n
22
under conditions (23). The additional fluctuation comes from the cross moment bounds.
Lemma 26 shows that V/2/1² = ol ( ). In summary, we have Var(# EpPrr
O (
Lemma 26. Under the same conditions as Proposition
V12 o
n²2
TT 1T
2. Proof. First, we apply the upper bound on E[As; B B A7; By, 1x] as derived (74), with replac- ing k ++1 to le + everywhere as from definition is the number of vertices except for i and j. When j # (1), the minimum value of j starts from 2 when the union graph consists two disjoint trees.
3.
a GB a 1x] S (1 + o(1))n (4.bb.ss)) (sp sq) u4)2+ (((K))
E[AS; Bs AT,
4N e(K (((KK)))((K)))(((Xa))))) x off
(85)
After plugging the upper bound on cross-moments to the ratio, remains to bound the number of different union graph structures and their corresponding weights.
V12
E (ii 11)
Σ AN
20 Eves, aut(H) aut sq) ((K)))))2
(1 + o(1))n2N (211))22N22NNH)) p2N (T De
(a) Case -2 We first consider the special case when k 22 When 2, there are two disjoint trees in the decorated union graph and all edges are decorated by subset of either (S1, T or (S2, }.}. Then, el (Kn)) eeK 12) e(K2)) el (K22)) Also 2N,
Vi2
AN eff
Σ
aut(H))uut()n" a01la
20 Fij.
(182))22N)
where
Fgg Σ
Σ
aut H
Σ
xD (a.h...)
HeT aeTaan(())auut()
(ES,(022N,H,1)
63

---

and (1 2 2N, I) is the collection of decorated union graphs that have at least 2N + 2 vertices, excess -2, least 1 edge 1-decorated, and that S S2 H, Ti ..... T2
For specific decorated graph U, denote 11 (resp. 1/2) the number of different edges vetween Si and T2 (resp. S2 and Tz). There are 2 = 2(t1 + t2) 1-decorated edges and the remaining edges in set K0 K We write Fig under the summation over i and l2.
N
N
Fig = E
E
Σ
b..6...))
n
HET S1 an H BIET anti ant(())) mn (1.Ts T T to
0
Σ
S(v 2 2N, H. f2 (u.bu. Σ
41 h WET
where S(v 2 2N, H, 12) collects all the possible decorated union graph that have St, S2 = H, T2 = / for some T such that aut(/) < aut(H), and S₁ (resp. S2) differ in i (resp. 22) edges with Ti (resp. T2).
Next, we bound [S(v 2 2N, H. t2)) by the following way: First, enumerate through all S1, S2 ~ 2 Il on the complete graph with all possible structure, this gives aut(Hyy Then, choose which edges that are overlapped with 7i on S₁ (resp. overlapped with T2 on S2). This is most (c)) (w) I
N After this, we draw t + 22 new vertices for T1 and T3 and allow them arbitrarily connecting edges among those N vertices its chandelier, which is upper bounded by (2) 41 (ignoring the constraint that Ti = T2 = I for some chandelier / having less automorphism number than H). Altogether,
n
N3
t1
1S 2 2N, H,I)
aut (H)2
22
Therefore,
N3 n2D (a,bss,))/
litt
Fú = 2N 1T Σ
As assumed in Proposition 6, N = O(log n) and D = ol
loga Tog logn!
Therefore,
V12/1² ol (T2NN))
(b) Case k. > 22 In general, we define Ur. UM. and ÜN partition the same as (64) and (65). We also define the weights of each part the same as (66), (67), and (68). We define (ut, Eveu (vz..) w(')). The definition of Pas (Uss) and Px (UN, k) follow. All union graph class should have more than z 1-decorated edges, but specifically we only need this constraint for Üz.
Note that el 12) + e(K2i) + e(K22) & 4N 2(v + A 11+
4N AN Σ
4N Vielas-2 S Jeff
AN ((= 2 A sq)" 2N 12
VEN k+2 z=1
72 ( (DDtlabeee))
Σ PN((M))
X
PN
We show the upper bound for Un part in Lemma 27. The upper bound for Par and Py trivially follows from Lemma 22 and Lemma 23 they do not use any assumption on edges are all at least 2-decorated, except for the number 11, the possible ways decoration. So, we change 11 to 15 and then every thing follows. When z * O, eas as defined before an arbitrary Ü*s has maximum value 2N (v + k 2) 2/2 (same holds for eN, eg but we do not need to use them in our bound). In
64

---

summary,
(1( S n 2 [ (LMM)) (4L) 4( (ANV)
Pas(uM) - R f TM (15 2
nn' (153) -
2N 2N- [1 (15(1 (158) K*M
(N(uN,)) k) S NN ((15 ((15,) (15R + 2)33ttt2
nn
15R" (4N + 1)3)
The last inequality holds because UN 1 - 4N - (k: + 2) and h 2 -. Putting every pieces together,
V12la>-2 2
(15,3)((K M03KK + 1)²
R+2 4N244-1 1224---2 R-2
12 (15 R+M
R 2N-v-k-2
Σ
1
T1
ns(p Aq)
12-1
V=N
AN
12 (4(4 (15,3)
K+M
B BRTM
x
(a.b.re.))
== v(2N 2-
84K -
(2LM)
IT|p2N
From the second condition in (25), we first look at the summation of from N to 2N -- 1 2,
AN
22(4N) R (153)
KM g
=
From the first condition in (25).
2N k-2
K KIM R (2
2N-v-k-2
Σ
2
ns(p ^ 9)
When v > 2N- 2, the power 2N r 2 < Observe that = 2 2 + 2) 4N
((112) e(221)
2e(K22), we have the product of two summations upper bounded by
4N-k-2
ns(p Ag) q)
72 72 (2N) R (153) KLM BKİM
0+4+2-2N
Σ
x
1= k-1 (153) 2/ R
This is clearly o(1) because N = Ologgn and from the first and second condition (25). D+ is the only term being log=(1), n.
From the third condition in (25),
(K (4n + 1)2) k+2
(153))7k++00300 1)9) ( < 1.
<
n
Therefore, Viz/p2 ol (TTTeN)
M.2 Proof of auxiliary Lemmas
Lemma 27. For + T1).).
PL(UL, =) S n (LL"" ((N))))) (ANN)3xttu))
65

---

Proof. We define the unlabeled union graph sets corresponding to (((1,,2)) as U(V,, 2). From the definition (75) and Claim 3.
n² w(UL) au)(UL)
(
Σ
S (2K)*n""\u4 (u z)]-
(86)
Üg (eL,,))
Recall that Un consists of two disjoint trees, one rooted at i and the other one rooted at j. We consider the branches of chandeliers without overlapping with each other.
Then, we specify two categories of branches. A branch is called an invader if it is rooted at i (resp. j) but in (i) (resp. (i)).. A branch that is not an invader is called residence. We observe that there are at most I, + k + effective non-isomorphic bulbs among residents (defined in Section L, the proof of Proposition 5) because of the followings: I) If branches are perfectly matched and overlapped, there are at most L pairs of them rooted at and another L pairs rooted at j. We define effective non-isomorphic bulbs the same as in Lemma 25. Here we have L effective KKM is the maximum number of fully I-decorated branches in allowed Ür. and 3) There are at most 4 invading branches, each which can at most fully overlapping with one resident bulb, due to the fact that there cannot be two branches on the same chandelier passing through the same vertex.
non-isomorphic bulbs because S1 _ S2. T = T2. 2)
The remaining is to bound (VL, 2)|. We observe that there are at most 4(K + M - 1) edges from invading branches, which might be attaching to at most 4(K + M 1) resident branches. This is because an invader rooted j may only have its bulb overlapping with (t). In this case, one invader can overlap with multiple resident branches in (1).
(1+
IJ
(3P))
(N
(UL(vL,)) N L
(4N) $
(AL)!,
(87)
LRtTJ + 4)
where (++ wctwee (+) is the structures of all resident branches, (2/2) is the upper bound of choosing
[J]
which edges to be not overlapped on bulbs assume starting from perfect overlapped bulbs, ( is the bound for placing the remaining z/2 1-decorated vertices, (4LM) is trivial bound on how resident branches have their wires tangling with each other, (8)4(K is the structure of invading bounds the different interactions between invading edges and the edges, and lastly understand the quantity 2 MM (4L)),, this comes from the fact that each resident branches. To invading edge connected to the root can choose one out of at most 4L resident branch to attach, and that the following invading edges can choose to stay overlapping with the current resident branch or leave.
Plugging (87) back into (86) with basic binomial bounds and [J] S 8K we complete the proof.
Claim 3. Assume that j # n. (i). For any Ün (VL.
w((L)) $ aut(U))((K)).
(88)
Proof. Denote all bulbs contained in (i) (resp.. .)) and are attached to wires rooted at i (resp., j) as B1, B2.... By Denote all bulbs contained in Üg (i) (resp., j) and are attached to wires rooted at j (resp., i) Ti, although they can have their bulbs in (i) (resp., ÚL(1), they contribute to the weight of non-tree part UN from definitions (66) and (68).
.mm For those branches rooted at (resp., j) but connect to j (resp., i),
For an arbitrary bulb B, in (x))) U ÚJ), we discuss the following three cases. Without loss of generality, we assume that B, is attached to a wire rooted at i.
Firstly, if there exists another bulb B, such that B, and B, be two bulbs with wires rooted both at and overlapping with each other. Then, from Corollary 2, we have that Vaut(BB, Jaut s (B, U (2K) F(8)) because each bulb has size K and the difference between two edge set is at most K. Secondly, if B, is full 1-decorated, then it contributes to / (B,) to w(Úú) and ABt( to au)( Thirdly, assume that there is another bulb B, attached to wire rooted at j partly
66

---

overlapping with B,, then / ( Jant I aut(B, (2K) [((B)) from Corollary 2. Note that in this case, full overlap is not possible because this invader branch spends least one edge connecting from j to i. The third case can be considered as a generalized version of the first case.
By product over all overlapping bulbs, we have w(UL) I of edges in the difference sets of overlapping bulbs are upper bounded by z, the number of 1-decorated edges, and the automorphism number of Uy. is greater than the product of automorphism number of all bulbs.
because the total number
67

---

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Reviews:
[{"review_id": "NRZe1iTjDg", "reviewer": "Reviewer_Nbs5", "summary": "The paper under revision addresses the problem of graph matching and community recovery in correlated stochastic block models (SBM) with two balanced communities. In particular, it studies the regime where the vertices of the parent graph have a logarithmic average degree, considering the within-community edge probability to be $a\\frac{\\log n}{n}$ and between-community edge probability to be $b\\frac{\\log n}{n}$. The edge correlation parameter is equal to $s$.\n\nRegarding graph matching, they proved in Theorem 1 the existence of a polynomial-time algorithm that can achieve both almost exact and exact matching. Almost exact matching is achieved when the square of the edge correlation is greater than Otter’s tree counting constant ($s^2 > \\alpha$). To obtain exact matching, an additional condition is necessary, namely $s^2(a+b)/2 > 1$. This result extends known results on almost exact and exact matching for correlated Erd\\H{o}s--Rényi graphs obtained in Mao et al. (2023). It corroborates the previous work by Racz & Sridhar (2021), which determined that exact matching in correlated SBM is only achieved when $s^2(a+b)/2 > 1$.\n\nIn the context of community detection, Theorem 2 of the manuscript states that exact community recovery in correlated SBM is achieved by a polynomial-time algorithm when \\( s^2 > \\alpha \\), and it satisfies the information-theoretic threshold for community recovery in correlated SBM established by Racz & Sridhar (2021). The authors provide a brief description of the polynomial-time algorithm in Section 5 and a more detailed explanation in Appendix C. In this way, one of the main contributions of this work is to propose an efficient algorithm for graph matching for correlated SBM.\n\nIn terms of the algorithm itself, it is structured by first applying a community detection algorithm proposed by Mossel et al. (2015) to each of the graphs to ensure exact recovery of the communities. Then, an adaptation of the algorithm proposed by Mao et al. (2023) for correlated Erdős–Rényi graphs, based on rooted trees known as chandeliers, is applied.\n\nThe manuscript combines established ideas from previous works, specifically integrating the concepts from Racz & Sridhar (2021) and Mao et al. (2023) to develop a polynomial-time algorithm for graph matching in correlated SBM. The manuscript is well-structured and provides detailed explanations of the algorithms, results and their connections to related works.\n\nReferences:\n\nMao, Cheng, Wu, Yihong, Xu, Jiaming, & Yu, Sophie H. 2023. Random graph matching\nat Otter’s threshold via counting chandeliers. Pages 1345–1356 of: Proceedings of the\n55th Annual ACM Symposium on Theory of Computing.\n\nMossel, Elchanan, Neeman, Joe, & Sly, Allan. 2015. Consistency thresholds for the planted\nbisection model. Pages 69–75 of: Proceedings of the forty-seventh annual ACM\nsymposium on Theory of computing.\n\nRacz, Miklos, & Sridhar, Anirudh. 2021. Correlated stochastic block models: Exact graph\nmatching with applications to recovering communities. Advances in Neural Information\nProcessing Systems, 34, 22259–22273.", "questions": "X", "limitations": "X", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "The manuscript combines established ideas from previous works, specifically integrating the concepts from Racz & Sridhar (2021) and Mao et al. (2023) to develop a polynomial-time algorithm for graph matching in correlated SBM. The manuscript is well-structured and provides detailed explanations of the algorithms, results, and their connections to related works.", "weaknesses": "X", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your review!", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "7t56AI1DN6", "reviewer": "Reviewer_K457", "summary": "The author(s) consider graph matching and community recovery on correlated stochastic block models. In the stochastic block model (SBM) the algorithm's input is a graph G generated by first partitioning vertices into two equal size clusters and then adding every intra-cluster edge with probability p and every inter-cluster edge with probability q, independently. In the correlated version onG_ f irst samples a graph G from the SBM, and then takes two independent edge samplings of G, referred to as G_1 and G_2 -- these are the input of the algorithm (with randomly permuted vertices).  Thus, the correlated SBM setting generalizes the more basic correlated Erdos-Renyi setting, in which there is only one community (or, one can think of p=q). In the latter setting the objective is graph matching. In the more general setting of correlated SBMs the objective function is community recovery, but graph matching is a natural sub-objective along the way, as it is quite reasonable to match G_1 to G_2 first, and then use the improved edge density for community detection.\n\nA strong recent result of Mao et al (STOC'23) established a tight threshold for reconstruction in the correlated Erdos-Renyi setting, and provided a polynomial time algorithm in the regime where  matching is possible. The present paper basically ports the techiques of Mao et al from the correlated Erdos-Renyi to the correlated SBM setting. Specifically, they develop technical tools for using Chandelier counts, which underly the result of Mao et al, in the SBM setting.", "questions": "None.", "limitations": "None.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The correlated SBM model is a very natural setting to study, and some of the author(s)' results are tight.", "weaknesses": "There is a number of lower level technical challenges that stem from the need of extending an Erdos-Renyi result to the SBM setting, but the most conceptual one appears to be the fact that when using partial matching results to perform community recovery on the union of the two graphs, one needs to handle errors that come from the non-exact matching. This, while definitely an interesting technical challenge, does not seem sufficiently compelling as a conceptual one.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your review and comments! \n\nWe agree with the reviewer that there are several technical challenges to overcome, and a significant part of our contributions is indeed showing how to overcome these technical challenges. \n\nHowever, we also believe that there are compelling conceptual challenges in our work, the main one being how to center the edge-indicator random variables for the centered subgraph counts. Let us elaborate on why this is so. \n\nFirst of all, centering the edge-indicator random variables is crucial for the statistic to work, as discussed in Mao, Wu, Xu, Yu (2023) [ref. 35]. In the Erdős--R\\'enyi setting it is trivial how to center the edge-indicator random variables, since it is a homogeneous model, where all edge probabilities are the same. However, for heterogeneous models, it is no longer clear how to do this. Moreover, for general models, this may not even be possible. Consider, for instance, the graph matching problem for correlated inhomogeneous random graphs, studied in the recent works [51] and [17]. In particular, the work [17], titled \"Efficiently matching random inhomogeneous graphs via degree profiles\" studied efficient algorithms for this task. However, they note that \n\n\"in the inhomogeneous case it is also very difficult to obtain good estimates on these edge probabilities (for the obvious reason that the number of parameters is proportional to the number of observed variables).\"\n\nIn other words, this challenge is made explicit in the work [17]. The authors get around the challenge by considering a much simpler statistic: degree profiles. However, this comes at a significant cost: their matching algorithm only works when $s = 1 - O(1/\\log^{2}(n))$. In other words, it only works for vanishing correlation; it cannot handle constant correlation. A major point of our work is that the matching algorithm works for constant correlation. \n\nMore generally, it is not at all clear for which classes of graphs a \"good enough\" centering of the edge-indicator random variables can be computed. We view this as both a conceptual and a technical challenge. A main point of our work is to show that SBMs indeed fall within this class.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lfyUGCpbjR", "reviewer": "Reviewer_xbYm", "summary": "The paper extends graph matching from correlated Erdos Reyni graphs to correlated SBM graphs. \nTo do so, they apply the recent breakthrough work of Mao, Wu, Xu, and Yu (STOC 2023) for ER graphs, which are based on counting a special kind of graph called chandeliers. A prerequisite for this method is to be able to center the graph. The primary challenge is that SBM graphs a graph cannot be easily centered (as the intra and inter-community edges have different means). To this end, they first apply a weak-community recovery approach to get a rough estimate of the communities in the different graphs. Then, they use this information to get a match. \n\nAs an application, they show efficient community recovery given multiple correlated SBM graphs in a regime where recovery is information-theoretically impossible given only one graph (This also shows that the approach is interesting, as they use a rough community recovery on individual graphs to obtain a matching, which in turn leads to a complete community recovery using the graphs collectively). This is the first efficient algorithm for this problem.", "questions": "It would be great if the authors could comment on the aforementioned weaknesses.", "limitations": "The authors have adequately addressed the limitations.", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "This technically involved paper extends work on graph matching for correlated ER graphs to (in my understanding, a restricted version of)  correlated SBM graphs. \n\nThe key technical difficulty seems to be that the results for the ER graphs require centralization of the adjacency matrix, which is not directly possible in SBM graphs. To solve this, the paper first obtains a rough estimate of the communities in the graph. Then, it shows that they are able to obtain a matching using the resultant centralization (which may not be completely correct).", "weaknesses": "1. The definition of correlated SBM seems a special case of the usual definition of correlated graphs in the ER model. Specifically, the authors assume that the non-edges of the first graph also remain non-edges in the second graph. \n\n2. The paper does not contain any simulation results. Implementing this algorithm and applying it to SBM graphs should not be hard.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your review and comments!\n\nComment #1: \n\nWe would like to kindly make some clarifications regarding these comments. \n\nFirstly, the correlated ER model is actually a special case of the correlated SBM model (not the other way around). Specifically, when the in-community edge density $p$ is equal to the cross-community edge density $q$, then the resulting SBM is in fact an ER graph.\n        \nSecondly, we do not assume that the non-edges of the first graph remain non-edges in the second graph. Since both graphs are subsampled from a mother graph, it is possible that some non-edges in the first graph are edges in the second graph. That said, it is true that the non-edges in the mother graph remain non-edges in both sub-sampled correlated graphs.\n\nComment #2:\n\nIn the literature on random graph matching, the quest for polynomial-time algorithms has been a major driving force behind the recent surge of papers on the topic. These culminated in the recent breakthrough works [33] and [35] on correlated Erdős--Rényi random graphs. Our work continues along this line of work, going beyond the correlated Erdős--Rényi model, and giving the first polynomial-time algorithm for graph matching on correlated SBMs. Unfortunately, the running time is a high order polynomial --- this is an issue that is present already in the prior work [35]. Thus, the implementation of the algorithm for simulations would only be possible on small graphs. As such, our contribution should be viewed as a conceptual, theoretical result.\n\nAt the same time, we do believe that fast and implementable algorithms do exist under some weakened condition, such as $s^2>\\alpha'$ for some $\\alpha' > \\alpha$. One recent empirical evidence towards this is the work of Muratori and Semerjian [42], who empirically studied a significantly accelerated version of the subtree counting method, using a restricted family of subtrees with a limited depth and where each node has at most $2$ or $3$ descendants. It is conjectured in [42] that for some $\\alpha'>\\alpha$, the proposed faster algorithm can achieve exact graph matching on correlated Erdős--Rényi graphs much more efficiently than the original algorithm. In future work, we plan to empirically study their algorithm on correlated SBMs. Furthermore, we plan to attempt to rigorously show the exact graph matching result on correlated Erdős--Rényi graphs and later extend it to correlated SBMs. This direction is exciting to explore, but it is beyond the scope of the current paper.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "EXyprEt2g7", "reviewer": "Reviewer_9uZn", "summary": "The text presents progress in solving learning conditions within correlated stochastic block models with two balanced communities. The main result is the creation of the first efficient algorithm for graph matching. This algorithm works well when the average degree is logarithmic in the number of vertices and can accurately match nearly all vertices with high probability if the edge correlation parameter $ s $ satisfies $ s^2 \\geq \\alpha + \\epsilon$, with $\\alpha $ being Otter's tree-counting constant. The algorithm is extended to achieve exact graph matching, solving an open problem posed by Rácz and Sridhar; 2021. This new algorithm generalizes recent work by Mao, Wu, Xu, and Yu (STOC 2023), which relies on centered subgraph counts of structures called chandeliers. A key challenge addressed is managing estimation errors since the latent community partition cannot be perfectly recovered from a single graph in certain parameter regimes. The findings also lead to an efficient algorithm for exact community recovery using multiple correlated graphs, even in cases where it is impossible with just a single graph.", "questions": "How differentiated the SBM model has to be from an ER model to effectively achieve exact and almost exact matching if $s^2>\\alpha$?", "limitations": "The discussion section indicates some of the limitations of the work including the lack of runtime analysis, and that the condition $s^2>\\alpha$ is believed to be necessary but that is not proven in this paper.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper has several interesting points. First, the paper deals with an important problem in graph mining, namely, graph matching in correlated SBM with two balanced communities. \n%\nThe proposed solution is based on probabilistic bounds based on well-established works applied to SBM with a logarithmic average degree.\n%\nThe paper also discusses an example of community recovery on CSBMs in polytime.", "weaknesses": "The paper is highly theoretical and, as such, most limitations are associated with this. For instance, it does not incorporate simulations as a sanity check of the proofs which could be useful to illustrate not only the usefulness of the theorems for graph matching but also the assumptions.\nA minor weakness in this regard is that although the introduction indicates the assumptions for which the framework is developed, e.g., the level of sparsity and logarithmic average degree, it could be useful to discuss the implications on highly dense graphs when the parameters are the complement of the SBM parameters p,q.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your review and comments!\n\nComment #1 (runtime analysis): \n\nIn the literature on random graph matching, the quest for polynomial-time algorithms has been a major driving force behind the recent surge of papers on the topic. These culminated in the recent breakthrough works [33] and [35] on correlated Erdős--Rényi random graphs. Our work continues along this line of work, going beyond the correlated Erdős--Rényi model, and giving the first polynomial-time algorithm for graph matching on correlated SBMs. Unfortunately, the running time is a high order polynomial --- this is an issue that is present already in the prior work [35]. Thus, the implementation of the algorithm for simulations would only be possible on small graphs. As such, our contribution should be viewed as a conceptual, theoretical result.\n\nAt the same time, we do believe that fast and implementable algorithms do exist under some weakened condition, such as $s^2>\\alpha'$ for some $\\alpha' > \\alpha$. One recent empirical evidence towards this is the work of Muratori and Semerjian [42], who empirically studied a significantly accelerated version of the subtree counting method, using a restricted family of subtrees with a limited depth and where each node has at most $2$ or $3$ descendants. It is conjectured in [42] that for some $\\alpha'>\\alpha$, the proposed faster algorithm can achieve exact graph matching on correlated Erdős--Rényi graphs much more efficiently than the original algorithm. In future work, we plan to empirically study their algorithm on correlated SBMs. Furthermore, we plan to attempt to rigorously show the exact graph matching result on correlated Erdős--Rényi graphs and later extend it to correlated SBMs. This direction is exciting to explore, but it is beyond the scope of the current paper.\n\nComment #2 (sparsity and other density regimes): \n\nIn Section 2 (from line $89$ to $100$ in our manuscript), we have argued that the logarithmic average degree is the most interesting parameter regime to study. The logarithmic average degree regime is also called the \"bottleneck regime\" of random graphs and it is the bare minimum for the graph to be connected. If we go sparser to constant average degree regime, then it is information-theoretically impossible to recover the communities exactly. If the vertices have polynomially growing degrees, then community recovery is easy as long as $\\liminf_{n \\to \\infty} |p_{n}/q_{n} - 1| > 0$ (see [39]).\n\nIn Section 7 (from line $363$ to $372$ in our manuscript), we have also discussed that our result can be extended to a denser regime (where the average degree grows as a (small) polynomial of $n$) too. In the even denser regime, whenever $\\liminf_{n \\to \\infty} |p_{n}/q_{n} - 1| > 0$ is satisfied, we can do community recovery (exactly and efficiently) by applying the work of Mossel, Neeman, and Sly [39], and then using the exact graph matching algorithm from Mao, Wu, Xu, and Yu [35] as a black box on the correlated Erdős--Rényi models under each community.\n\nComment #3 (on differentiating SBM from ER): \n\nTo apply our result in the logarithmic average degree regime where $p=a\\frac{\\log n}{n}$ and $q=b\\frac{\\log n}{n}$ (Theorem 1),  we only need that $a\\neq b$. If $a = b$, then the correlated SBM degenerates to the correlated ER model, where the result by Mao, Wu, Xu, and Yu [35] already applies. In other words, we do not need any nontrivial differentiation.\n\nComment #4 (on the necessity of the condition $s^2 > \\alpha$): \n\nIndeed, we conjecture that the condition $s^{2} > \\alpha$ is necessary for a polynomial-time algorithm for graph matching to exist, just as [35] conjectures the same for the correlated Erdős--Rényi model. Thus, this graph matching in these models is conjectured to exhibit an information-computation gap; that is, a region of the parameters where the problem is information-theoretically feasible, but it is not possible with polynomial-time algorithms. There are many interesting problems that are conjectured to exhibit such an information-computation gap, but as of yet there is no problem where this has been proven. Instead, people often resort to proving results on restricted classes of algorithms (such as within the low-degree polynomials framework), which can be seen as evidence of computational hardness (though not as a full proof). Related to our work, in the easier task of correlation detection in the case of correlated Erdős--Rényi random graphs, the recent work [16] proved a corresponding low-degree hardness result.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "WKpvex5LCc", "reviewer": "Reviewer_PrRf", "summary": "In this paper, the authors tackle two related problems: graph matching and community recovery of correlated balanced SBM under the logarithmic average degree regime. The authors extend proofs from the ER model for similar tasks and overcome the difficulties in the proofs raised by working with the SBM model.\nThe main two theorems are: (1) for correlation levels above a certain threshold, almost exact and exact graph matching are possible, and (2), again, for correlation levels above a certain threshold, community recovery is possible.", "questions": "There are some verbs missing in several sentences, and other grammatical or spelling errors (for instance in Erdos). I suggest a profound check on the writing.\nI think the other comment (about the suitability of this paper for a conference) is not something the authors could change in the rebuttal phase.", "limitations": "The authors claim to address the limitations in Section 7. However, this section is more of a \"future work\" section (as it is named). It's true that every possible line of future work is a limitation of the present paper, but it's not presented in this way, and no other limitations are discussed.", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "I liked the paper's presentation in general. The problem setting is well described, and the motivation is clear. \nThe major strenght is the theoretical result (Theorem 1, and Theorem 2 as an application of Thm1). It is important and is a nontrivial generalization of the ER case for 2-class SBMs.", "weaknesses": "The weakness I see is not of the paper itself but more of the paper-conference combination. The strengths of the paper, as stated above, are the two theoretical results. But the 9-page paper ONLY includes the statements of these results and some ideas about the proofs. I found it very very nice, well motivated, etc. But I cannot review a ~70-page paper in all detail like it deserves. Not for a conference. I can see this as a beatuiful journal paper, where a reviewer can spend some months to study the paper like it deserves.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your review and comments! \n\nWe concur with the reviewer that lengthy papers present challenges with the conference format and its restrictions. However, we argue that this is a widespread general issue, not specific to our paper -- indeed, there are numerous NeurIPS papers with lengthy appendices. Within the constraints, we certainly aimed to provide a good summary of the results and the proof ideas and techniques within the main text. \n\nWe will certainly do a careful check over the whole paper for grammatical and spelling errors. We will also ask a native English-speaking colleague of ours to proofread our paper, specifically focusing on spelling and correct phrasing of sentences. (We (the authors) are non-native English speakers, and we appreciate your patience as we improve the writing of the paper.)", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The problem of graph alignment -- ie. of matching the vertices of two random graphs with correlated edges -- has received significant attention in the literature. The present paper considers the case of correlated stochastic block models (cSBM) with two communities. The main result is to show how a certain estimator from previous work, which is based on on counting "signed chandeliers" in the graph, achieves good performance, and (most importantly) does so in polynomial time. The paper adds to recent information theoretic results for this model, and also to algorithmic results for the simpler Erd\H{o}s-Rényi case. At least in some cases, it reaches the natural information theoretic threshold below which alignment becomes (in some sense) impossible. 

Reviewers' scores were 5, 6, 6 and 7. They all seem to agree that this is a technically challenging paper, which solves an interesting problem. Differences in scores seem to reflect differing views of the relevance of the paper. My own opinion is that this paper contains important ideas required to deal with the challenge brought about by the fact that edge probabilities take two values. Moreover, the problem considered is of broad interest, albeit the contributions here are mostly theoretical. Overall, I believe the paper should appear in NeurIPS.
Decision: Accept (poster)