Harnessing Multiple Correlated Networks for Exact Community Recovery
Authors: Miklos Z. Racz
Jifan Zhang
Venue: NeurIPS 2024 poster (2024)
Keywords: Stochastic block model
community recovery
graph matching
correlated random graphs
information-theoretic limits
Primary area: learning_theory
TL;DR: We derive the precise information-theoretic threshold for exact community recovery from any constant number of correlated stochastic block models, showcasing the power of integrative data analysis and quantifying the value of each additional graph.

Abstract: We study the problem of learning latent community structure from multiple correlated networks, focusing on edge-correlated stochastic block models with two balanced communities. Recent work of Gaudio, Rácz, and Sridhar (COLT 2022) determined the precise information-theoretic threshold for exact community recovery using two correlated graphs; in particular, this showcased the subtle interplay between community recovery and graph matching. Here we study the natural setting of more than two graphs. The main challenge lies in understanding how to aggregate information across several graphs when none of the pairwise latent vertex correspondences can be exactly recovered. Our main result derives the precise information-theoretic threshold for exact community recovery using any constant number of correlated graphs, answering a question of Gaudio, Rácz, and Sridhar (COLT 2022). In particular, for every $K \geq 3$ we uncover and characterize a region of the parameter space where exact community recovery is possible using $K$ correlated graphs, even though (1) this is information-theoretically impossible using any $K-1$ of them and (2) none of the latent matchings can be exactly recovered.

Harnessing Multiple Correlated Networks for Exact Community Recovery
Miklós Z. Ráez
Jifan Zhang
Northwestern University Evanston, 60208 miklos.racz @ wow.wwesttttttteteeeeeed.
Northwestern University Evanston, IL 60208
jifanzhang20266 haw.aaannnn2noo..oooooooeeeeeee u.oreeeeeoootwhesee.eddd
Abstract
We study the problem of learning latent community structure from multiple cor- related networks, focusing edge-correlated stochastic block models with balanced communities. Recent work of Gaudio, Rácz, and Sridhar (COLT 2022)
determined the precise cnormatiion-thooreeee threshold for exact community recov- ery using two correlated graphs; in particular, this showcased the subtle interplay between community recovery and graph matching. Here we study the natural setting of more than two graphs. The main challenge lies in understanding how to aggregate information across several graphs when none the pairwise latent vertex correspondences can be exactly recovered. Our main result derives the precise unformation-thhoree threshold for exact community recovery using any constant number of correlated graphs, answering question of Gaudio, Rácz, and Sridhar (COLT 2022). In particular, for every 2 3 we uncover and characterize region of the parameter space where exact community recovery is possible using K corre- lated graphs, even though (I) this is information information-thhoretiaalll theoretically impossible using any K of them and (2) none of the latent matchings can be exactly recovered.
Introduction
Finding communities in networks that is, groups of nodes that are similar is one of the fundamen- tal problems in machine learning. This task crucially important for understanding the underlying structure and function of networks across diverse applications, including sociology and biology [23]. The increasing availability of network data sets offers the intriguing possibility of improving com- munity recovery algorithms by synthesizing information across correlated networks. However, in many settings the graphs are not aligned- whhcch may happen for a variety of reasons, including anonymization, missing erroneous data, simply the alignment being unknown-whin a challenge. Thus graph matching-tte task of recovering the latent vertex alignment between networks. Our work follows an exciting
presents
trophs-plays a central role in efforts to integrate data recent line of work at the intersection of community recovery and graph matching.
Recently, Rácz and Sridhar [41] initiated the study of community recovery in correlated stochastic block models (SBMs), focusing on the simplest setting of two correlated graphs with two balanced communities. They determined the information-theonr has applications for community recovery. In particular, they uncovered a region of the parameter space where exact community recovery is possible using two correlated graphs even though it is inmnaationnhhoorticallll
limits for exact graph matching, which
information- theoretically impossible to do so using just a single graph. Subsequently. Gaudio, Rácz, and Sridhar [22] determined the information-theorrtie limits for exact community recovery from two correlated SBMs. This required going beyond exact graph matching and understanding the subtle interplay between community recovery and graph matching.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Gaudio, Rácz, and Sridhar [22] posed the question of understanding what happens in the case of more than two graphs. which arises naturally all the motivating examples. For instance. people participate in numerous overlapping yet complementary social networks, and only by combining these can we fully understand and make inferences about society. Similarly, synthesizing information across peeeinnprotein interaction networks from several related species can aid in inferring protein functions [44]. The main challenge lies in understanding how to optimally pass information across three or more graphs.
Our main contribution fully answers this open question by Gaudio, Rácz, and Sridhar [22]. Specifi- cally, we precisely characterize the information -hheoretic threshold for exact community recovery given K correlated SBMs, for any constant K. This result highlights an intricate phase diagram and quantifies the value of each additional correlated graph for the task of community recovery. In particular, for every K 2 we uncover and characterize region of the parameter space where exact community recovery is possible using K correlated graphs, even though (1) this is impossible using any K of them and (2) none of the latent matchings can be exactly recovered. See Section 3 and Theorems I and 2 for details.
Along the way, we also precisely characterize the information-theorete threshold for exact graph matching given K correlated SBMs, for any constant K. In particular, we uncover and characterize a region of the parameter space where the latent matching between two correlated SBMs cannot be exactly recovered given just the two graphs. but it can be exactly recovered given K > 3 correlated SBMs. See Section 3 and Theorems and for details.
To prove our results, we study the so-called k-core matching between all pairs of graphs. Recent works have shown the k-core matching to be flexible and successful tool in variety of settings for two correlated graphs [13, 22, 433. Our main technical contribution is to extend this analysis to more than two graphs. The main difficulty lies in understanding the size of intersections of "bad sets" for k-core matchings for different pairs of graphs. We refer to Section 4 for details.
Models and questions
The stochastic block model (SBM). The SBM is the most common probabilistic generative model for networks with latent community structure. First introduced by Holland, Laskey, and Leinhardt [25], it has garnered considerable attention and research. In particular, it can be employed as natural testbed for evaluating and assessing clustering algorithms on average-case networks [1]. The SBM notably displays sharp information-theorrtiic phase transitions for various inference tasks, offering detailed understanding of when community information can be extracted from network data. The phase transition thresholds were conjectured by Decelle et al. [15] and were proved rigorously in several papers [33, 34, 35, 36, 2, 3]. We refer to the survey [1] for a detailed overview of the SBM.
In this paper, we focus on the simplest setting, SBM with two symmetric communities. Let be a positive integer and let p. [0, 1] be parameters representing probabilities. We construct a graph G ~ SBM(nn p. 9) as follows. The graph G has n vertices, labeled by [m] == {1, 2, 3, n). Each vertex is assigned a community label o² (1) from the set {+1,-11 -1]; these are drawn i.i.d. uniformly at random across i E [n]. Let o+ := fo" (i)) denote the community label vector. The vertices are thus categorized into two communities: V+ == fi E [n] o'(i) = +1} and V := {i E [n] (i) 1}. Given the community labels o", the edges of G are drawn independently between pairs of distinct vertices. If of o² (j), then the edge (i,j) is in G with probability pt otherwise, it G with probability q-
Community recovery. In the community recovery task, an algorithm takes as input the graph G (without knowing σ²) and outputs an estimated community labeling ō. Define the overlap between the estimated labeling and the ground truth as follows:
n
ov(a a .)
==
(i)((1)
n
The overlap measures how well the true community labels and the estimated labels of the algorithm match. Note that ov(a*, a) E [0, 1]. where the larger the value is, the better performance the algorithm has. In particular, the algorithm succeeds in exactly recovering the partition into two communities (i.e., a* or o* -ō) if and only if ov(ar* .)) = 1. Our focus in this paper achieving this goal, known as exact community recovery.

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SBM(n,
10
2(4)
8(1)
7(8)
1(10)
3130)
4:20
8.0
5(9)
10(6)
5(3
G1
G2 (2)
G (G)
M1
4, 10,
8,8,7
Tix cm (1,,2.445,,,3333,,,
Figure 1: Schematic showing the construction of multiple correlated SBMs (see text for details).
It is well-known that exact community recovery is most challenging and interesting in the logarithmic average degree regime [1]. Accordingly, we focus on this regime: in most of the paper we assume that logn and b log for some constants a, b > 0. In this regime there is a sharp information- theoretic threshold for exact community recovery [2, 35, 3]. Let D (a, = (vā vb)² /2 denote the so-called Heellngf divergence. Then the information-theoree threshold is given by (2.1) In other words, if D+ (a, b) > 1, then exact recovery is possible (and, in fact, efficiently). That is, there is a (polynomial time) algorithm which outputs an estimator with the guarantee that limm-+oo PPov((o", 6) 1) 1. On the other hand, if (a, b) < then exact recovery is 1)= O.
(a 1.
impossible: for any estimator or, we have that 1im P(ov(o"
Correlated SBMs. The objective of our work is to understand how the sharp threshold for exact community recovery varies when the input data involves multiple correlated graphs. To do so, we first define a natural model of multiple correlated SBMs [29, 39, 28] (and see further discussion in Section 6 about alternative models).
We construct (GI....,Cee Gk) ~ CSBM(n, P. s) as follows, where the additional parameter E [0, 1] reflects the degree of correlation between the graphs (and the number of graphs is dropped from the notation for ease of readability). First, generate a parent graph Go ~ SBM(n, p. with community labels a* Subsequently, given Go. construct G₁, G2 ........ Gxx by independent subsampling. Specifically, each edge of Gp is included in G², with probability of everything else, and non-edges of Go remain non- non-edges in G¹,- The graphs G inherit both the vertex labels and the community labels from the parent graph Go. Finally, let i2 "ik be i.i.d. uniformly random permutations of [n] and let "r" == (a x Define G, := G¹ and, for all i E [2, K) J. define G, ni. (G)). In other words, for every > 1 and j E [n]. vertex j in G* relabeled to (i) in G.. This last relabeling step mirrors the real-world observation that vertex labels are often unaligned across graphs. This construction is shown in Figure 1.
independently
An important property of the model is that marginally each graph G2 is an SBM. Since the subsampling probability is s, we have that G3 SBM(n, ps, gs). Thus, it follows from (2.1) that, in the logarithmic average degree regime where
b logn the communities can be exactly recovered from G alone precisely when sD+ (a, (b,b) = D.. (sa, sb) > 1.
log and
The key question in our work is how to improve the threshold by incorporating more information as K. the number of correlated SBMs, increases. This question was initiated by Ráez and Sridhar [41] and then solved by Gaudio, Rácz, and Sridhar [22] when K- =
An essential observation is that, go beyond the threshold, one needs to combine information from the K graphs G1.... GK through graph matching. Then one can exactly recover the community labels using the combined information, even in regimes where it is information-theorettaaall impossible to exactly recover o* given up to K graphs.
To be more specific, if π were known, then one can reconstruct G", from Gj and then combine the graphs Gi...., G¹x to obtain the union graph H* defined as follows: the edge (i.j) is included in
3

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H' if and only if (2.5) is included in at least one of ......, Gkk- Note that H' is also an SBM; specifically, H ~ SBM (n. (1- (1 (1-s) s)*) p. (1 (1 s) g). so (2.1) directly implies that the communities can be exactly recovered from the union graph I1 if (1 (1 - 8)*) D_ (a, b) > 1.
Graph matching. In real applications, the permutations π = (Tfz:..... xiix) are often not known. The arguments above highlight the importance of intermediate task, known as graph matching: how can one recover the latent permutations π given the graphs (G1, G2...... Gx) While here we regard graph matching as an important intermediate step, it is of great significance in its own right, with applications in social network privacy [40], machine learning [10], and more. For two correlated SBMs, this problem was resolved by Rácz and Sridhar [41], who proved that the information- theoretic threshold for (pairwise) exact graph matching is sa + 6)/2 = 1. Note that this is also the connectivity threshold for the intersection graph of G1 and G²2 (see [41]. Denote
Te(a, b) a b
(2.2)
2
With this notation, the (pairwise) exact graph matching threshold is given by TT b) = 1. This directly implies (by union bound) that if TTTa(a, b) > 1, then π can be exactly recovered given (G,, Gz...., Gx) ~ CSBM(n. logn 6 n .), for any constant K. By the discussion above, this also gives a sufficient condition for exact community recovery given (G1, G...., Gk) ~ CSBM(n a lognn , log 8):
TT > and
(1) (+ (a(a.)
(2.3)
We will generalize the exact graph matching result of Rácz and Sridhar [41] and show (see Theorem 3 below) that π can be exactly recovered given (G1, G2,, ...,,........................... Gk) CSBM(n, a loga b logn if
s (1- (1- - s) ")) Tc(a, > 1,
(2.4)
which (for K > 2) is weaker than the condition sT b) > 1 implied by [41]. (Moreover, in Theorem we show that the condition in (2.4) is tight for exact recovery of πτ") Thus, by the discussion above, this gives a sufficient condition for exact community recovery given (G1, G2...., . Gk) CSBM(n, logn Jogn s):
8 (1-(1-s) (a b) =1 and
(1-(1-))*) D+ (a, b) 1.
(2.5)
The interplay between community recovery and graph matching. The condition (2.5) is, however, not tight. To attain the sharp threshold for exact community recovery given K graphs, we need to answer the following question: does there exist a parameter regime where exact community recovery is possible for K graphs, even though (1) exact graph matching is impossible, and (2) exact community recovery is impossible using only K- graphs?
For K = 2 graphs, Gaudio, Rácz, and Sridhar [22] proved that the sharp threshold for exact community recovery given K correlated SBMs is given by
8 TT, b) + s(1 - (a) b) >
and
(1- D+ 1.
(2.6)
The condition (1 (1- (1 =)²)
) D+ b) > necessary due to the work [41]. The first condition in (2.6) demonstrates the interplay between community recovery and graph matching. To be more specific, the first term s²T (a. b) is the threshold for exact graph matching given (G1, G2), while the second term s(1 - s)D b) comes from community recovery.
Our main contribution generalizes this result, determining the exact community recovery threshold for K 2 3 graphs. If (1 (1 - {) D_ (a, b) > 1, then the sharp threshold is given by ()
(2.7) The condition (2.7) also clearly exhibits the interplay between community recovery and graph matching. The first term comes from graph matching, while the second term comes from community recovery, as in the case of K 2. We refer to Section 3 and Theorems I and 2 for details.
s (1-(1-9) (a,
> 1.
Despite the apparent similarity in results, when K 2 the situation differs significantly from that of two graphs. The primary challenge lies in the existence of multiple methods for matching K 2 3 graphs. When K = 2, there is only a single matching that needs to be recovered from G and G2. In contrast, with three or more graphs, the graphs can be matched pairwise, or to some anchor graph, or potentially in many other ways. Integrating information across different matchings requires substantial additional effort. We present the formal results in the next section.

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3
Results
Our main contributions are to determine the precise information-thhoretti thresholds for exact community recovery and for exact graph matching given K correlated SBMs.
3.1 Threshold for exact community recovery
We first describe the precise dnformatiin-theoretti threshold for exact community recovery, starting with the positive direction.
Theorem 1 (Exact community recovery from K correlated SBMs), Fix constants a, b > 0 and 8 E [0. and let (G,, G2 G2....,Grr Gk) ~ CSBM(nn, a logn log 8). Suppose that the following two conditions both hold:
(1 - (1 - (1-s))) (+,bb)
(3.1)
and
s(1 s)* D 1. Then exact community recovery is possible. That is, there is an estimator ē (Gi, G....., Gk) such that lim P (ov (ō,
(1- 1) Te (a
(3.2)
=1.
Combined with Theorem 2 below (which shows that Theorem 1 is tight), this result precisely answers an open problem of Gaudio, Rácz, and Sridhar [22]. The condition (3.1) is required for exact community recovery for K graphs by [41]. We now focus on the condition (3.2). In the prior work [22], proved that the threshold for exact community recovery for two graphs is given by (2.6). The primary contribution of Theorem is to go beyond this threshold as the number of graphs K increases. In particular, this showcases that there exists a regime where (1) it is impossible to exactly recover o* from (G,, G2..... GK-1) alone and (2) any exact graph matching is impossible, yet one can perform exact recovery of o given (G1, G2...., GR). This requires developing novel algorithms that integrate information from (G1, G2,..., Gx) delicately and incorporate multiple graph matchings carefully.
Here we first provide detailed discussion of the algorithms for three graphs (G1. G2, G3) CSBM(n, Jogn 6 Jogn s), which is the simplest case with intriguing phenomena and challenges as mentioned. This avoids complicated notations (which arise for general K) for easier understanding. The new techniques used for combining multiple matchings and integrating information with three graphs are subsequently generalized to K > 3 correlated SBMs. The high level idea of the algorithm for exact community recovery when K = 3 consists of five steps (in the following discussion assume a >> b; when a < b, change majority to minority everywhere):
n
1. Obtain an almost exact community labeling of Gi.
2. Obtain three pairwise partial almost exact graph matchings P12, P13, and fizs between graph pairs (G1, G2), (G1, G3), and (G2, G3), respectively.
3. For vertices in G that are part of at least two matchings, refine the almost exact community labeling in Step 1 via majority vote in the (union) graph consisting of edges that appear at least once in (G1, G2, G3).
4. For vertices in G1 that are part of only P12 (resp. P13), label them via majority vote of their neighbors' labels in the graph consisting of edges that appear only in G1 and not in G2 (resp. only in G1 and not in G3).
5. For vertices in G1 that are not part of any of the three matchings or only part of figs, label them via majority vote of their neighbors' labels in G1.
Each step in the algorithm involves abundant technical details. See Section for a detailed overview of the algorithms and proofs. Note that the threshold (3.2) captures the interplay between community recovery and graph matching, which we now discuss more detail.
The first term in (3.2), which is s (1 - (1 (1- Ta(a, b) for K = 3, comes from graph matching. In [22] it is shown that for one matching, say P12. the best possible almost exact graph matching makes n² (1=a"T.(a.))++
errors. Here, we show that is possible to obtain almost exact matchings #1 and #13 (namely, these will be k-core matchings; see Section 4 for details) such that the size
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of the intersection of the two error sets is n - n. This quantifies how synthesizing information across graph matchings can reduce errors and this exponent is precisely what shows up in the first term in (3.2). This observation is important and relevant for Steps and in the algorithm.
which is smaller power of
On the other hand, the second term in (3.2), which is s(1 - s) (a, (a, b) for K = 3, comes from community recovery. In fact, this term arises from the majority votes in Step where we use only edges in G₁. Note that the nodes that are unmatched by 2u12 and Ji13 are, roughly speaking, the isolated nodes in the intersection graphs of G1 and G2, and G1 and G3, respectively. Thus, while we use all edges in G1 in this step, the relevant edges are not present in G2 nor in G3. giving the "effective" " factor of s(1 s)². By (2.1), the exact community recovery threshold for SBM(n, s(1 - 3)²a log
n s(1 - 5)²6 log is s(1 - 3)* (a, b) = 1, giving the second term in (3.2).
The following impossibility result shows the tightness of Theorem 1.
Theorem 2 (Impossibility of exact community recovery). Fix constants a, b > 0 and s [0, 1]. and let (G1, (e.....GK) CSBM(n, log logn s). Suppose that either
(1 8)*) (a, <1
(3.3)
or
s11 (1 () -TT + s(1 - - ba, b) < <1
(3.4)
Then exact community recovery impossible. That is. for any estimator ā T G2. .... Gk), we have that lim TP (ov (ō, o²)) 1) 1)
n-x
Impossibility of exact community recovery given K graphs under the condition (3.3) is proved in [41]. Hence, Theorem 2 focuses on proving impossibility for exact community recovery given K graphs under the condition (3.4). In particular, condition (3.4) reveals parameter regime where exact community recovery from (G1, Gg...., Gk) is impossible, yet, if πr were known, then exact community recovery would be possible based on the (correctly matched) union graph.
Theorems I and 2 combined give the tight threshold for exact community recovery for general K correlated SBMs, see (2.7). Fig. exhibits phase diagrams illustrating the results for three graphs.
3.2 Threshold for exact graph matching
The techniques that we develop in order to prove Theorems and 2 also allow us to solve the question of exact graph matching, that is, exactly recovering x" (zi....., iix) from (G1, ....... Gx) ). In the context of correlated SBMs and community recovery, exact graph matching can be thought of intermediate step towards exact community recovery. However, more generally, graph matching is a fundamental inference problem in its own right; see Section 5 for discussion of related work. We start with the positive direction in the following theorem.
Theorem 3 (Exact graph matching from K correlated SBMs). Fix constants a, b > 0 and & E [o, 1].
and let (G1, Gz,..., Gk) ~ CSBM(n, a logn log
s). Suppose that (1- - -1) To(a, b) >
(3.5)
Then exact graph matching is possible. That is, there exists an estimator मे = #G Gz...., ... ,,K) (fi2...., Rik) such that lim TP (# G2....,
"22....,G]]] ,Gk) T²) =
Since the condition in (3.5) is weaker than TT (a, b) > (which is the threshold for exact shown in [41]), Theorem 3 implies that there exists parameter
graph matching for K = regime where 112 cannot be exactly recovered from (G1,G2), but मे can be exactly recovered from (G1, G2,.... Gk). In other words, it is necessary to combine information across all graphs in order to recover न (and even just to recover 712).
The estimator मे in Theorem 3 is based pairwise k-core matchings (see Sec. 4 for further de- tails). Roughly speaking, for each pairwise k-core matching the number of unmatched vertices is 11 TT (o.)) however, we shall show that the number of vertices which cannot be matched through some combination of pairwise k-core matchings is n (1=1(1_(--)) +)))) which is of smaller order. So, when 8 (1
-1) Te(a, b) > 1, then exact graph matching is possible. The following impossibility result shows the tightness of Theorem 3.

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(a) Fixed 0.15.
(b) Fixed 0.25.
Figure 2: Phase diagram for exact community recovery for three graphs with fixed 8, and E 40),
E [0, 40] on the axes. Green region: exact community recovery is possible from G1 alone; Cyan region: exact community recovery is impossible from C₁ alone, but exact graph matching of G₁ and G2 is possible, and subsequently exact community recovery is possible from (G), G2); Dark Blue region: exact community recovery is impossible from G2 alone, exact graph matching is also impossible from (G1, G2), yet exact community recovery is possible from (G1, G2); : Pink region: exact community recovery impossible from (G1. G22) (even though it would be possible if 412 were known), yet exact community recovery is possible from (G,, G2, G3): Violet region: exact community recovery is impossible from (G1, G2, G3) (even though it would be possible from (Gi, G2) if 412 were known); Light Green region: exact community recovery is impossible from (Gi, G2), but exact graph matching of graph pairs is possible, and subsequently exact community recovery is possible from (G1, G2, G3); Grey region: exact community recovery is impossible from (G1, G2). exact graph matching also impossible from (G1, G2). but exact graph matching possible from (G1, G3), and subsequently exact community recovery is possible from (G1, G2, Ga); Yellow region: exact community recovery is impossible from (G1, G2), exact graph matching impossible from (G1, G2, Gs). yet exact community recovery is possible from (G1, G2, Ga): Orange region: exact community recovery
impossible from (G1, G2, G3) (even though it would be possible from (G,, G2. G3) if π were known); Red region: exact community recovery is impossible from (G1, G2, G3) (even if π known). The principal finding of this paper the characterization the Pink, Violet, Orange, Yellow, Grey, and Light Green regions.
Theorem 4 (Impossibility of exact graph matching from K correlated SBMs). Fix constants a, b > 0 and 8 E [0, 1), and let (G1, G2...., GG) CSBM(n, alog n blog 8). Suppose that
- (1- - s)--ss ()
(3.6)
Then exact graph matching impossible. That is, for any estimator मे #G1, G2,..., ) we have that lim P P( (G1, G.....,.Kk)) Gk) xx)== = Tt' 0.
Gr)
Theorems and 4 combined give the tight threshold for exact graph matching for general K correlated SBMs, see (2.4). We note that, in independent and concurrent work [5], Ameen and Hajek derived the threshold for exact graph matching from K correlated Erdōs-Rény random graphs; in other words, they proved Theorems 3 and 4 in the special case of a b.
Comparing Theorems and 4 with Theorems and 2, note that there exists parameter regime where exact community recovery is possible even though exact graph matching is impossible.
4 Overview of algorithms and proofs
In this section we elaborate on the technical details of the community recovery algorithm, for which high-level ideas were presented in Section 3. We focus our discussion on the setting of K graphs, which already captures the main technical challenges; we highlight these and explain how
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we overcome them. We subsequently explain the generalization from 3 graphs to K graphs. The overview of the impossibility proof is discussed as well.
k-core matching. We now define a k-core matching [13, 22, 43], which is used for almost exact graph matching in Step 2. Given a pair of graphs (G, H) with vertex set [n], for any permutation x, we have the corresponding intersection graph G Ag H. where (i) is an edge in the intersection graph if and only if (i is an edge in G and ((])),((j) is an edge in H. The k-core estimator explores all possible permutations π of [r]] to seek a permutation : that maximizes the size of the k-core of the intersection graph G ATr H; recall that the k-core of a graph is the maximal induced subgraph for which all vertices have degree at least k. The output of the k-core estimator is then a partial matching fi. which the restriction of the vertex set of the k-core in ^g H.
One significant advantage of using k-core matchings is a certain optimality property in terms of performance. Specifically, if (G1, G2) ~ CSBM(n, a logn log n s), then the k-core estimator between G1 and G2 fails to match at most n Taa) +o(l) vertices, which the same order as the number of singletons of G₁ Axi, G2 and any graph matching algorithm would fail to match these singletons [11, 41, 22]. Another significant benefit of utilizing k-core matchings is the correctness of the k-core estimator for correlated SBMs, as discussed in [22]. The k-core estimator might not be able to match all vertices under the parameter regime that we are interested in; however, every vertex that it does match is matched correctly with high probability.
Community recovery subroutines. The high-level summary of the algorithm is follows. Since exact community recovery might be impossible in G1 alone, we first obtain an initial estimate which gives almost exact community recovery in G1, as described in Step 1. In Step 2, we use pairwise partial k-core matchings with k= = 13 to obtain j == {u12, 313, P23} } (see Fig. 3b). which we will use to combine information across (G1, G3) to recover communities. Note that each partial matching ūij only matches a subset of the vertices, denoted as Miji we denote the set of vertices not matched by P.g by F.y := [n]] My,. Subsequently. we split the vertices into two categories: "good" vertices and "bad" vertices, where "good" vertices are part of at least two matchings and "bad" vertices are part of at most one matching (see Fig. 3a). We conduct several majority votes among "good"" and "bad"' vertices to do the clean-up after the graph matching phase, where each subroutine is meticulously executed to disentangle the intricate dependencies among (G1, G2, Ga) and ft.
Exact community recovery for the "good" vertices. The major distinction between being "good" and being "bad" is that "good" vertices can combine information from all three graphs via their union graph (which is denser), whereas "bad" vertices cannot. Suppose that vertex is part of Ju and ăi (i.e.,& E 22 M13). We can then identify the union graph Gi Vin. G2 V Vž13 G3. which consists of edges (i such that (i,j) is an edge in G1 or (P12(t))P11(j))
(i1(()) is an edge in G2 or (P1s(1), Pis(j)) an edge in G3. and i is part of this union graph. Similarly, i E M12 n M23, then is part of the union graph G1 2 G2 V Rm G3 that also integrates information from all three graphs. On the union graph, we can refine the almost exact community labeling by reclassifying "good" vertices based on majority vote among the labels of their neighbors that are also "good", and this reclassification will be correct as long as the condition (3.1) holds.
There are many underlying technical challenges and roadblocks in the theoretical analysis. The key difficulty arises from the structure of the union graph. It is statistically guaranteed that in G GiVe G2 Vxt, G3. all vertices have a community label which is the same as the majority community among their neighbors [35]. However, whether this is also the case for G1 VĀ.a G2 Vžn, G₃ is unclear, since the latter graph is only defined on the "good" vertices M1 n M13. One would like to demonstrate that the removal of "bad" vertices does not significantly affect the majority community among neighbors of "good" vertices. Prior work [22] addressed a similar problem for two graphs by employing a technique known as łuczak expansion [27] to F12 to ensure that the vertices inside the expanded set Fi2 are only weakly connected the vertices outside of the expanded set [n] \ F12- Unfortunately, this method is no longer applicable for correlated SBMs with three or more graphs. Even though the size of the expanded set F12 is orderwise equal to the size of F12. the size of the intersection of the expanded sets F12 n Fi3 might not be orderwise equal to the size of F12 n Fis. which directly leads to the failure of the algorithm working down to the information-theome threshold. To overcome this challenge, we consider the graph [[[m] \ b} to decouple the dependence of being connected to vertex w and w being part of the k-core. Applying the Luezak expansion on such a graph for any given U, and through a union bound, we prove that unmatched vertices are contained in the set of vertices whose degree is smaller than constant, with high probability. This
8

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allows us to quantify the size of F12 n Fis and meanwhile directly ensure that "good" vertices within the k-core are only weakly connected with "bad" vertices.
Another hurdle needed to overcome, as stated in [22], concerns the almost exact community recovery in Step 1 which is subsequently used for majority votes. Therefore, it is of great importance to guarantee that the mncorretly-classiffi
vertices are not well-connected and do not have a great impact on majority votes. Consequently, we utilize an algorithm originally developed by Mossel, Neeman, and Sly [35] which allows us to manage the geometry of the misclassified vertices and demonstrate that the vertices classified incorrectly are indeed only weakly connected.
Exact community recovery for the "bad" vertices. The remaining step is to label the "bad" vertices. The "bad" vertices can be further classified into three categories (see Fig. 3a): vertices in F12 Fis. which are only matched by f223 or are not matched by any of the three matchings; vertices in Fis O F23 \ F12, which are only matched by F12; and vertices in F12 n F23 \ F13, which are only matched by 7113
Consider the vertices in F12 n Fia (the other cases are similar). First of all, as discussed above, we show that Fign = (1==((1(1-=))TTaaa))
with high probability. Consider the graph G1 \az G2 'ais Gs. which consists of the edges (i.j) in G1 such that (f12(t), (i)( and (i)a (1s not edges in G2 and G3, respectively. Due to the approximate independence of F12 n Fis and G1 \Fi G2 \Fs G3. for a vertex E F12 0 Fis we can calculate the probability of the failure of the majority vote in the graph G1 \012 G2 \Fia G3 in relatively straightforward manner, giving n -===(==)) D+ (a.b)))(
The factor s(1 - 8)² arises from the fact that the edges in this graph are subsampled in G1 and are not subsampled in G2 and G3. Now since a vertex in F12 O Fi3 can have at most 12 edges outside of F12 in G1 Auz G2, and also at most 12 edges outside of Fia in G1 Apps G3. the majority vote for i E F12 n F13 essentially does not change whether it is performed in G1 or in G2 Am G3. Putting all this together, the probability that the majority vote fails is at most:
G1 \e,
P(exists a vertex i E F12 n Fi3 such that the majority vote fails)
= =IF12 x Pmmaoority vote fails for a vertex) =
Thus, if with K = 3 holds, then majority vote will correctly classify all vertices in F12 n Fin-
Generalization to K graphs. For K graphs, we have (K) pairwise matchings to consider (see 3b). We again categorize the vertices as "good" and "bad". The "good" vertices can integrate in- formation across all K graphs through the pairwise partial k-core matchings (7. : 1,j E [K], í A j}, while "bad" vertices cannot. To illustrate this concept more vividly, for any vertex consider new "metagraph" MG, on K nodes, defined as follows: there is an edge between and j in MG, if and only if v can be matched through Paj (see Fig. 4). If the metagraph MG. is connected, then there exists a path that can connect all of its K nodes. Equivalently, there exists a set of matchings that allows to combine information across all K graphs. Subsequently, quantify the number of "bad" vertices to be n² n=1x(1-(1-s)= "yT, (r..y....))))) (o()) The remaining analysis for K graphs can be derived by generalizing the analysis for three graphs.
Impossibility proof. As discussed in Section 3, focus on the proof of (3.4) for impossibility. We compute the maximum a posterior (MAP) estimator for the communities in G.. We show that, even with significant additional information provided, including all the correct community labels in G2, the true matchings Th for i,j E (2,3, ..., K), and most of the true matching কं2 except for singletons in the graph G1 Axis (G2 V... V Gk), the MAP estimator fails to exactly recovery communities with probability bounded away from 0 if (3.4) holds. The proof is adapted from the MAP analysis in [22]. The difference is that here we are considering K graphs G1, G2...., Gx with different additional information provided for the MAP estimator. Given that the MAP estimator is ineffective under this regime, all other estimators also fail.
The proof of the exact graph matching threshold is implicitly present in the proof of the exact community recovery threshold. Essentially, since we show that the number of "bad" vertices is n (=<(1-(1=>))* (o.b(),)) the condition (3.5) implies that there are no "bad" vertices with high probability. Since all vertices are "good", and "good" vertices can integrate information across all K graphs, the latent matchings can be recovered exactly. For the impossibility result we analyze the MAP estimator and show that, even with significant additional information, including the true matchings {Ti i, j E ( ... ..,,KK }}}, it fails if (3.6) holds.
Exact graph matching threshold.
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5
Related work
Our work generalizes- and solves an open question raised by-the work of Gaudio, Ráez, and Sridhar [22]. Just as [22], our work lies at the interface of the literatures on community recovery and graph matching'-tve two fundamental learning problems whichh which we briefly summarize here.
Community recovery in SBMs. A huge research literature exists on learning latent community structures in networks, and this topic is especially well understood for the SBM [25, 15, 34, 35, 33, 36, 2, 3, 7, I]. Specifically, we highlight the work of [2. 35], which identify the precise threshold for exact community recovery for SBMs with two balanced communities. Our algorithm builds upon their analysis, taking particular care about dealing with the dependencies arising from the multiple inexact partial matchings between K correlated graphs.
correlated Erdös-Rényi random graphs. The past decade has seen a plethora of research on average-case graph matching, focusing on correlated Erdös-Rényi Rényi random graphs [40]. The information thhooretic thresholds for recovering the latent vertex correspondence π have been established for exact recovery [11, 46, 12], almost exact recovery [13], and weak recovery [20, 21, 24, 46, 16]. In parallel, a line of work has focused on algorithmic advances [37, 6, 18, 19, 30, 32], culminating in recent breakthroughs that developed efficient graph matching algorithms in the constant noise setting [31, 32). We particularly highlight the work of Cullina, Kiyavash, Mittal, and Poor [13]. who introduced k-core matchings and showed their utility for partial matching of correlated Erdös-Rényi random graphs. Subsequent work has shown the power of k-core matchings as flexible and successful tool for graph matching [22, 43, 4]. Our work both significantly builds upon these works, as well as further develops this machinery. which may be of independent interest. We also note the independent and concurrent work of Ameen and Hajek which determined the exact graph matching threshold for K correlated Erdós Rényi random graphs [5].
Graph matching:
Graph matching: beyond correlated Erdös-Rényi random graphs. Motivated by real-world
networks, a growing line of recent work studies graph matching beyond Erdös-Rényi graphs [8, 26, 9, 3939, 42, 49, 41, 22, 45, 43, 17, 48, 47], including for correlated SBMs [29, 39, 28, 41, 22, 48, 47). The works that are most relevant to ours are [41, 22], which have been discussed extensively above.
6 Discussion and Future Work
Our main contribution highlights the power of integrative data analysis for community recovery, yet many open questions still remain.
Efficient algorithms. Theorem characterizes when exact community recovery is information- theoretically possible from K correlated SBMs. Is this possible efficiently (i.e., in time polynomial in n)? The bottleneck in the algorithm that use prove Theorem I, which makes it inefficient, is the k-core matching step; the other steps are efficient. Recent breakthrough results have developed efficient graph matching algorithms for correlated Erdös-Rén random graphs [31, 32], which promisingly suggest that an efficient algorithm for exact community recovery may indeed exist in this regime. We refer to [22] for further discussion on this point.
General block models. We focused here on the simplest case of SBMs with two balanced communi- ties. It would be interesting extend these results to general block models with multiple communities. This is understood well in the single graph setting [1] and recent work has also characterized the threshold for exact graph matching for two correlated SBMs with k symmetric communities [47].
Alternative constructions of correlated graphs. An exciting research direction is to study different constructions of correlated graph models. For general K, there are many ways that K graphs can be correlated. In particular, the following is a natural alternative construction of multiple correlated SBMs. First, generate Go SBM(n, p, q). Then, independently generate H; ~ SBM(n, p', g') for i E [K] Construct G == Go V H,, and finally generate G3 through an independent random permutation of the vertex indices in G¹,. This construction is equivalent the one we studied in this paper for = 2 and it is different when K 2 3, and investigating it is interesting and valuable.
We note that graph matching has both positive and negative societal impacts. particular, well known that graph matching algorithmm can be used to de-anonymize social networks, showing that anonymity is different, in general, from privacy [38]. At the same time, studying fundamental limits can aid in determining precise conditions when anonymity indeed guarantee privacy, and when additional safeguards necessary.
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Acknowledgements
We thank Taha Ameen, Julia Gaudio, Elchanan Mossel, and Anirudh Sridhar for helpful discussions. We also thank anonymous reviewers for constructive feedback.
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A Organization
The rest of the paper is structured follows. First, we elaborate on the recovery algorithm for three graphs in Section C. Section D includes some useful preliminary propositions, including some nice properties of almost exact community recovery on G1. Section discusses the k-core estimator. After these preparations, we are ready to prove the main theorems in the paper.
Section F proves Theorem I for three graphs, where we first validate the accuracy of the community labels for "good" vertices and then classify the remaining "bad" vertices. Section G presents the proof of the impossibility result (Theorem 2) for three graphs. Section H discusses the recovery algorithm for K graphs and provides a general proof for K' graphs, with additional arguments on how to identify "good" and "bad" vertices. Section I discusses the proof of the impossibility result (Theorem 2) for K graphs. Section J contains the proof of the threshold for exact graph matching given K graphs, that is, the proofs of Theorems and
B
Notation
We introduce here some notation that will be used in the rest of the paper. In most of the paper we focus on the setting of K = 3 graphs: (G1, G2, G3) ~ CSBM(n, logn blogm, s), and this is the setting that we consider here as well.
Let = [m] = {1, 2, 3, .... n denote the vertex set of the parent graph Go. and let V+ == (i E [n] : (i) +1} and V == (i [n] (i) = -1} denote the sets of vertices in the two communities, Let ((r) == ((1,j) : i,j E [n], + j} denote the set of all unordered vertex pairs. Given community labeling a e [+1,- -13". we define the set of intra-community vertex pairs as 8 (a) == (i..)) j) E (7)) (1) = a(j}} and the set of inter-community vertex paits as E+( (ii.j) (()) : a(i) (j)). Note that (a) and E (a) form partition of (())
Let A, B, and C' denote the adjacency matrices of G1, G2, and G3, respectively. Let B' and C" denote the adjacency matrices of G22 and G² respectively. Note that, by construction, we have for all (i.j) E ((r)) that B', = (1 and C% = Cris (i)mm)))) Observe that we have the following probabilitiess for every (i.j) E (()). If afabbe b, {0, 13s and a b+ >> O, then
P ((Ang, Bij, C/3) (a, (,),,)) = [satbtal1 - 5) 333-abb-c
= (
gaatbtcl 8) s33-a-b-sg +
Furthermore, we have that
IP ((Ai, Biy, C22) = (0,0,)) = (1 -1 (1-))pp if)'()=='())
(1 q + (1 s)84 if''())
C The recovery algorithm for three graphs
Our recovery algorithm is based on discovering a matching between subsets of two graphs.
Definition C.1. Let G3 and Gj be two graphs in vertex set 'n] with adjacency matrix A, B, respec- tively. The pair (Mays psg) is a matching between G, and Gj if
* Mij [n].
* Maj Mij + [n].
*
Pij is injective.
Given a matching (Mry, H.j). here are some related notations. Define G. G. G, to be the union graph, whose vertex set is M, whose vertex index is the same as the vertex index of G, and whose edge set is {(e, m] f, m E Maj. Atm + Buas(0)
> 1}. In other words, the edges are those that appear in either G; or Gj. Conversely, G3 App G3 represents intersection graph, whose vertex set is Miy, whose vertex index the same as the vertex index of G, and whose edge set [le, m) L, m E Mij. Aem = B Bug (0.00)mmm))) 1}. In other words, the edges are those that appear
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F12
F13
Gr
R1z
As
22
[n]
P23
G2
62
(a) Categorization of vertices for three graphs: ver- tices the red regions are "bad" while vertices in the white regions are "good".
(b) Graph matchings for three graphs. For gen- eral K, consider (2) partial graph matchings.
Figure 3: Schematic landscape of partial matchings over three graphs.
in both G; and Gj- Define G3 Ans Gj be the graph G; minus Gj. whose vertex set is Mij. whose vertex index is the same as the vertex index of G; and the edges are those only appear on G2 and not appear in Gy-
Definition C.2. Let G2 and G₁ g be three graphs on vertex set [n]] with adjacency matrix A, B, C. respectively. The pair (Majs Pij) is matching between G; and Gj. while the pair (M Hyk)) is matching between Gj and Gg. Denote Pizk pij as the composition matching between G3 and Gk. defined on the vertex set Mi Mak--
For three graphs, we can define the additional notations in the same manner as Definition C.I and the core concepts remain consistent. GeVyy, Gyyuyoop G₁ represents the union graph of Gj, Gj. Gb. whose vertex set M Mij n MIR, whose vertex index is the same as the vertex index of G; and whose edge set is {(6, m] E, E M, Arm + B p., (m) + C Caaaopin (5... other words, the edge set are the edges that appears in least one graph out of G., Gj. Gk- Similarly, Gi App Gj Appa G represents the intersection graph, the edge set is {(e, mm] m. E M; Aem
2 1) In
Byg (F),()) (m)) Cuy "yy (t) (ppy = =}} Define G, Vau Gj Aaya G3 be the graph whose edge set is those edges that appear either G; or G, and not appear Gg. Similarly, we can define G3 Aay G₁ Apya Gk, Gú May (G; V Vaez Gk), and G; Ye,, (G, Apya G&) well. Note that all the definitions above are defined on vertex set M and use vertex index in G..
Introduce dmin (G) = mingg[x] d(i). where d(i) is the degree of vertex i. Definition C.3. matching (Mijs Hkğ) is k-core matching of (Gi, G₁) if d'in (G, Aps Gj) 2
A matching (Miz, pag) is called a maximal k-core matching if it involves the greatest number of vertices, among all k-core matchings.
Algorithm k-core matching
Input: Pair of graphs Gū Gy vertices, & E [n].
Output: matching (Mij, (ij)) of G; and Gj.
1: Enumcrating all possible matchings, find the maximal k-core matching (Mig Pay)) G. and G
Let (Mijs p4y) ) be the matching found by Algorithm I with & = 13. Mij coineides with the maximal k-core of G Awi, Gj, denote it as Mi while pij coincides with the true permutation Tiy. with high probability (Lemma E.5).
The k-core matching is symmetric, i.c. Myy (M,,) Mys Note that by Definition C.I. My uses the vertex index of G, while Myt uses the vertex index of Gj. they are equivalent and exchangcable through the 1-1 mapping. Now define F, [n]] Mi be the set of vertices which are excluded from the matching. Note that Fiy use the vertice index same as G.. We define F = Mt be the set of vertices which outside the maximal k-core of G; Axt, Gj-
As briefly discussed in Section 4, we start with leveraging the "good" vertices in order to find the correct communities. The "good" vertices are those which are part of least two matchings out of three partial matchings f12, P13, #23. The details are shown in Algorithm 2.
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Algorithm 2 Labeling the good vertices
Input: Three graphs G1: G2, G3 on n vertices and three 13-core matchings M12, M13, M13, .223. (223), parameters a, S,E.
(2 P12,
Output: A labeling of (M n Ma2) U (M12 M13) U (M23 n M12) given by ō.
1: Apply [35, Algorithm 1] to the graph G1 and parameters (sa, sb, sb, c), obtaining a label Ø
2: Denote F12 = [n] \ M12, F13 = [nn] 13 F23 [n] M23.
3: For E M13 M32, set set @(i) E {-1,1] according to the neighborhood majority (resp., minority)
of (i) with respect to the graph (G1 Visaemi G2 VM1 (a1 n M32] ] if a b (resp.,
< b).
4: For E M set @(t) E {-1, according to the neighborhood majority (resp., minority)
of (i) with respect to the graph (G1 Vuza G2 Vussouz [ n M23 if a b (resp..
a < 6).
5: For E M13 M1 set ((t) 1- according to the neighborhood majority (resp., minority)
of (t) with respect to the graph (G1 Vaz G2 Van Gs) M12} if a (resp., b).
6: Return σ (M1 O Mx2) U (M12 M1s) (M2s M12) + (-1,1).
The remaining step is label the "bad" vertices which cannot utilize the combined information from three graphs. Hence, we classify the "bad"" vertices according to the majority of neighborhood restricted the corresponding "good" vertices. The detailed descriptions shown Algorithm 3.
Algorithm Labeling the bad vertices
Input: Three graphs (G1, G2, G3) on n vertices and three 13-core matching (M12, A12), (M13; M13), and (M23, (22)), parameters a, a, 8, label the "good" vertices G. Output: A labeling of [n] given by ō.
1: For i E F12 n F1s, set (1) E f-1, 1] according to the neighborhood majority (resp., minority) of (i) with respect the graph G1 (M12 n M13 U {i)) if b. (resp., b)
2: For i E F23 0 Fia \ F12, set @(i) E (-1,1) according to the neighborhood majority (resp., M1 M (i)) if b (resp., b). minority) of (1) with respect the graph G1 w12 G2
3: For i E F12 0 F23 \ FIs. set ((i) E (-1,1} according to the neighborhood majority (resp.,
minority) of (i) with respect to the graph G₁ \us G3 Mia (i)) if > b (resp., a b).
4: Return ō [n] [11
The complete exact recovery algorithm exhibited in Algorithm First, the 13-core matchings are preformed. Next, the "good" vertices labeled according to the union graph. Finally, the "bad" vertices are labeled according to neighborhood labels in G2 or G1 \uz2 G2 or G1 Vais G3.
Algorithm 4 Full Community Recovery
Input: Three graphs (G1, G2, G3) on n vertices, ls 13, and > Output: A labeling of [r] given by a.
1: Apply Algorithm on input (G,, Gj, k). obtaining a matching (Myz, Dey), + j E (1,2,3).
Denote M := (M1 M32) (M n Mia) (M23 M12).
2: Apply Algorithm 2 on input (G1, G2, G3, M12. M2s. M13. P13 P12 Prs), obtaining a labeling
ă M + 1}.
Apply Algorithm 3 on input (G1, G2, G3, M12, M2s. M1s, 7ıs 1u P2a. ã), obtaining a labeling
á : [m] + [1,1).
4: Return ā n 1-1,1)
D Preliminaries
Here provide some useful preliminary propositions.
16

---

D.1 Binomial Probabilities
Lemma D.I. Suppose that a 2 Let Y Bin(m*, alog(n)/n) and Z Bin(m", "am(mm,blogii bblog())/m)
(1 be independent. If = (1 + ((1))n22 m = (1 ((1))/2 then for any e > 0.
PPY - 2 S log S D4 (a.0)+ lox( lb((0/)/2++((11
Proof. Proved by [22, Lemma 3.3].
D.2 useful construction of three correlated stochastic block models
In this section, we elaborate on an alternative method for constructing three correlated SBMs, which emphasizes the independent regions of G and G3. we detail the construction for three graphs to maintain reasonable and manageable notation throughout our discussion. The extension of these ideas to the general case of graphs follows similar structure where the key steps and arguments can be directly applied. This construction is analogous to the construction from [22, Section 3.2), generalizing the case from two graphs to three graphs.
Firstly, we construct a random partition {Eij,, k. (0,13) of ((2)) Independently, for each pair {i,j} E (())). we let {i,j} E {Eije} with probability of (1 - (1--)))--j-+ git Subsequently, for each pair {i,j} E ((r). an edge is constructed between and j with probability p the two vertices are in the same community, and with probability q if they are in different communities. Graph G1 is constructed using the edges from U (o. ii.= while the graph G.2 is constructed using edges from (0.1),,.) Graph G2 is then generated from G²2 and Ti2 by relabeling the vertices of G222 according to 722- Similarly, the graph G'2 constructed using edges from (0.1),).). and G3 is obtained from G'2 and Tis by relabeling the vertices of G¹2 according to Tis. This construction offers an alternative method for generating multiple correlated SBMs and emphasizes regions of independence between the multiple graphs. The following lemma D.2 describes the idea formally.
Lemma D.2. The random partition construction of correlated SBMs in Section D.2 is equivalent to the original construction shown in Figure 1. Moreover, conditioned on n == (Tiz. viz: m2s), o", and & == {Eijk, k E (0,1)}, the graphs that are comprised of edges in disjoint Eigk mutually independent.
Proof. Firstly we show that the distribution of ( (Auj, (atz(0)att Cata())a1()) is the same under two constructions. Then by the indepence of vertex pairs, the equivalence follows. In the first construction, If ca+bbc>00 0:
P((Aij, (riz(()))s))) (11300)))))) = (a,b,c)(='
gatbte s)8-a-b-ap ifo oz 0i),, gatbto (1 3- if # oz (j).
ba+bbc=
P((Aij, (rr, (00,0)))00000000,,,.000000000000 (0.00))) 00)
{ 1- 1- if (0)())=)
1- ff
Under the second construction, of (1) oj):
P((Aij, Baia(tmmi)()= C (( (a, b) " = P((i,j} E Eabe) x
= ={ 1- ()
sabbt0 -5)3-a-b-q if ifa+b+c> 0, +
if+bbc=00.
If (1) * o² (j), the joint distribution is the same only with replaced by q. We can see that the joint distribution under two constructions is the same. To prove the second part of the lemma, note that conditioned on ,², o", and the random partition (E,R,,ij.J k E (0. 1}}. the edges in Eigk form independently. Hence the graphs that are comprised of edges in disjoint Eggs are mutually independent.
17

---

Definition D.3. Define the constant:
83
s2 5)
Sabe
s(1 8)
(000,)00000000000000000000111101
(1 15-9 s)
(a.b,c) (0,00))
The event F holds if and only if
n/2 - 3/44 S | IV n/2+
and the following conditions hold for all a, b, (0. 1), E [n]:
Voba - S ( j Eabe n (a (i)) I Sabe (V +n344).
Sabe (V - (1) - 3 - [[j Eabe 7 (1)) v Sabe ( V [V) +
(6)
Lemma D.4. Define 8m milllabbcc( Sabe. We have P(F)) 100m exp(
V
Proof. Denote 9 holds if and only if nf2 n/4 3/4 V S 2 The event & is proved in Lemma 3.8 [22]. P(g*)= < 4e
Then, look at the remaining condition of event F. Fix i [n)]. condition on o Ti2 Tis. Note that
abc Hi =j E Eabe n non (a)())) Bin 1, 8abc)-
By Hoeffding inequality we have
S/be72
P(lkabe (i) Sabo (1V
110 1-]11/
T Tiz. 0 (
in 9/2
3/2
2 2exp(
2 ((()) S 22xpp(
-
Then by union bound,
P(3i E [n] (ittbe (1) Sabe [o0(00 Sahn 2/4)
Sabe 3/1
P( (1)itte Sabe - (IV (0) 11) 2
2
E[P (b (h (1) Sabe [V0(() 100 -0)] Sah. 3/4
|T T3, Ti. [[)l + P(g))
1=1
< 2n exp(pppt o(1)) vn) + exp( - vn) S 6n 6nexp(-(1 (())) Vn).
Similarly, define k (1) and through an identical proof we have
P( E [n] (i) Sabel |V - [V 2 3/44 I 6n exp(-(1 -x))(((1-()))) Vn).
The conclusion then follows by union bound.
D.3 Almost exact recovery in single SBM
Lemma D.5. The algorithm (Algorithm 1, (22]) correctly classifies all vertices in [m] \ (G), where [G) {o [n]: majo(v) clog n or ((e) 100 max a, log if a > b.
Proof. Directly proved by [22, Lemma 5.1], adapted from [35, Proposition 4.3).
Lemma D.6. Consider a SBM(n, a log n/n, B log n/n), denote 7 = max(a,, 3). Then for every o" we have
P(Vi E [n], IN(i < 100 max(1, log rola") n
Proof. Directly proved by [22, Lemma 5.2], based on arguments of [35].
18

---

Lemma D.7. With the assumption of (a, < 99, (()(())) S 3n 1n (a.0) (a.
[og(a/b)]
Proof. Proved by [22, Lemma 5.3].
Lemma D.8. 1f0 <
(+(a.)) 2log(a/b)' then
P(vi e ( (a) "(oo") = 1 o(1).
Proof. Proved by [22, Lemma 5.4).
E Analysis of the k-core estimator
In this section, we prove two important properties of k-core estimator. Lemma E.4 describes that all vertices have weak connections with those vertices who are not part of the k-core, in the logarithmic regime that are interested in. Lemma E. argues that all the vertices with degree larger than given constant will be part of the k-core.
Lemma E.1. Fix a, b > O. Consider the graph G ~ SBM(n, alog
For any integer m a45. all vertices whose degree is greater than m + k are part of the k-core with high
blog
satisfying m > 2 probability.
2 a+b. we would like to prove that any vertex with degree greater than + will not be part of the k-core with probability o(n-1) The lemma then follows by union bound.
Proof. For given
Isolating vertex for independence.
For a fixed E [m). consider the graph G = G[[n)] (v). Now we look at the k-core of G. denote it by (G). Since the (v) k 2/ b), we suppose that (egg(v) m k, m 2/(a b). the vertex v not part of the k-core G, must has more than m neighbors who are & (G). Note that the event w & C₂ (G) is independent of the event w E Na(v), while the latter event stochastically dominated by binomial distribution with probability log fogn/n,4 max(a, b). Hence, by the tower rule,
PVu is not part of l core in G) < P([s E Na(v): : w Cx( m)
El (wnoo log n m//
(E.2)
The size of {w : w & (x) can not be directly quantified. Hence, we would like to find a set U based on G such that (w E [n] \ v (x)))) c U, where we can bound the size of U. Now we denote μ = log logn/n n, then we have that
E
m///))mm I E Bin(((),.
E/min exp(ule* - -1) -tm)]
(E.3)
To construct U, the idea motivated by łuczak expansion in [27]. We consider a modified version of expanding the set in our setting.
Quantify the set U.
Define U to be the set of vertices with degree at most T in the graph G. The choice of T would be specified later. Denote H. = {n/2 3/1 < | +
V-| < n/2 + 3/44,. By Lemma D.4,
19

---

P(H)) = o(1/n).
B(U] - P E((]]() +(1
(1+ (1++))0)) (1 i-j (
(1+)
11(11-) p) (t- oa (1 - (1-1111)1 -
n
j
1=0 j=0
alog n (1 (1-0(1) (1 b blog n (1-)
T
(1 + {())( (1+)(1))
/ plogn
- 2n(1
Σ
Σ
n
2
k=
(1 + (1)) log
$ 2n 2 +o(1) Σ (i +
2
s 2( + 1)2( log
yTT 'n=--=
1---- +o(1) n1 1- +o(1)
Consider the situation when - a+b > 0. Now we claim: for any constant W, E[U s
n -W-W W 2 +o(1), Suppose for W 1, it is true, then for W:
B
Σ
P(i üw EU)
E
P(1 E U, U,...,iw E U)
Σ
P(h Uu EU)+.... +...+ Σ
P(i EU)
Σ
P(h E UU......in iw E U) + B((U)--))
Σ
P(i Uuu.. iw E U)- - B/JU1W-yj.
The remaining thing is show that Eij have
P(h E U,.... iw E EU) S -.W-W##+o(l) We
Σ
P(i E U,.... iw EU)
Σ
P(n big has most T neighbours in [n]] \ (i, m. iw, uw)
fi
Σ
P(f has at most neighbours in [n] \ (i..... ww)) W
Aiw
<E|UT|
The first inequality is because that in E U in G. then in has at most T neighbours in [n] v, then implies that (úr has most 7 neighbours in [m]] (i,..., iw, v). The second inequality is due the independence of the events. By induction, the claim follows. By choosing appropriate m -th moment method of (U),
select such that P(UU] n--c)= n - (a+b)/22m c+o()) = o(n -m 97b
Denote Do as [U] < n--- then (P())) o(n m -==")
The possibility of the existence of a well-connected small subgraph.
Now would like to bound the probability of the existence of well-connected small subgraph. Define the event
D == (there exists S E [n]] such that |S] < n--- and G(S) has at least N|S| edges) ).
We would like to bound P(D) o(n=m(a+))/ Let be a x-vertex subset of [n]. Let Xs be the indicator variable that is 1 if the subgraph induced by S has at least N [S| edges. Denote
20

---

max(a, b), then we have
E
Σ
Xs} S
E
(
v Vlogn
NK
REV
Revlog log No
NE
n
Nn
SE|m],|S
SE|m],IS|=K
SE[m],IS|=K
Kew log
1+1//N
log n
No
Nn
N
The second and the third inequality is because (*) - (ct) ( Under the assumption |S| < n nn'5,
1+1//N v logn
(-) )
N A
N-1 < n -N (N-1)+)(1)
N
Hence for n sufficiently large we have:
n Σ
E
Σ
Xsl S
1 (N In ((N--))))))))
IE
x=1
Then by Markov's inequality, P(D) S n <(N ((o)))o( If we want to bound P(D) = o(n o(n=m(a+b)/2)
set N
(a+b)m
11. Hence,
2x
PPD) S E
Σ
Xs] S ((N-))+([)) m(a+b)/2
|E(x),|8|<n>>
Identify the expansion set of U.
Now we do the following expansion on U, the expansion process is adapted from Luczak expansion first introduced in [27].
1. Define Up : U.
2. Given U2, define U/41 to be the set of those vertices outside U. which have least N' neighbors in Uy. If U/+1 1 is non-empty, set U1+1 U² {u), where is the first vertex in U/41- Otherwise, stop the expansion with the set Uz.
Suppose the expansion ends at the step h, hence have increasing sequence (U.) Denote T : Un to be the set after expansion.
Now claim that on the event D n Do. we can choose No. > 0, such that T Suppose that [0| > |U) then there exists & > s.t. [U/| = Ni|U'. On event Do. there exists 0,|00 < n² n--, hence [Ue] = NI|U I n= 1-e+0((1) Denote as the number of edges in [(U,). Each step in the expanding process, at least N' edges are added into the graph, hence eq N' 2 N'([Us] - (UT) (N' N² [Ue]. We However on the event Dc, the set [Uel < n 1- cannot have at least NUUe edges, which a contradiction. Therefore, on the event D Du, U < N1(UU. Subsequently.
< N U..
choose N Ni such that (N' N' Z N.
E |1 S N |M 1 ] + n PT"P(D) n (((o))
SNT"EUUT" o(n "m ml(tt//() < m- m- = +o(1)
Bound the probability of v being in the k-core.
Note that [([m]] \\ } has minimum degree at least T - N'. If vertex E [n] \ U, then & U, it follows that i has degree at least T in G. However can have at most N' neighbor in U by construction of expansion process, so must have at least - N' neighbors in [v] \ U. We can set T = k + N', then If [n] \ U, i is part of k-core in G.
Since the deg(v) 2 m + k. must has at least neighbors who not part of k-core in G.
21

---

Follow the equation (E.2), (B.3), denote 10 log n/n, then we have
E 1 Bin o (ngg///))
E[min exp(u(c' - tm)] - +
ev log nm
C[| + S ( (=- +
n
= =(n
The second inequality follows by setting / = log(1//). This is valid since P([U) > n²-o) S [ ni e/ S -==(e+t)/2--oW++a +o(2), We can select < < < 2
a+b W > 0 such that P( > n²-9) =
o(n On the event [U] S n n μ o11), 1/p > 1. The third inequality follows by M[ S
mh *+3 +o(1) and the last equality is due m a+B-
If - (a + 6) < O, can directly set
1. Similarly, we prove [| S then through similar calculation, IP( v is not part of k-core in G} = o(n 1)
Hence, by a union bound, we can say that all vertices with degree larger than m + k will be part of k-core with high probability.
Lemma E.4. Fix a, b, > O. Let G ~ SBM(n, alog b blog Then, w.h.p., all vertices have at most E log n neighbors who are not part of the k-core.
Proof. For [n], consider the graph G G[[n]oo]. Following the same arguments in Lemma E. can obtain U. We have
P(Kw E Na(v)) is not part of k-core in G}|
log n))
<P(Kw E No(v): w & Cx(()) > log n)
<P(Kw E No(v): w & Ck( m) I 1)n
Based on the proof of Lemma E.1, we can show the lemma follows immediately when n is sufficiently large. Hence, all vertices have most log n neighbors who are not part the k- core with probability 1 o(1)..
Lemma E.5. Fix constants a, b > 0, E [0, Let (G1,G) ~ CSBM(nn a log log s). Let M be the set of vertices of the 13-core in the graph G. Ar G2. x² is the permutation of vertices from G1 to Gz. Let (M,, M1) be the output of the k-core match of (G1, G2), k = 13. Then
P((M1,)11) (M", =²)) 1-o(1).
Remark: We can replace (Mx, (4) by (M (M*)) in any analysis.
Proof. Proved by [22, Lemma 4.8].
Lemma E.6. Let G SBM(n, a loga logn for fixed a, b > 0. Fix & 2
With probability
1 - o1), we have that [F| S nn-=n1--i +o01
Proof. Proved by [22, Lemma 4.]
Lemma E.7. Suppose G1, G2, G3 are independently subsampled with probability s from a parent
graph SBM(n, a logn/n, b log n/n) )) for a, b 0. Let F3 be the set of vertices outside the k-core of G2 Ami Gy(taking vertice index in with k = 13. Prove that with probability o(1),
[P Fjkl < n. 1-662 - ()) for any 8 > 0.
Proof. F12 F FF Fi F12 n Figl. by symmetricity of G1, G2, G3-
Define U.j to be the set of vertices with degree at most m + k in the intersection graph G; Ang G3 which marginally follows SBM(n, as? lug be" logn ), and m is an integer satisfying m > a(a+6))
2
Then by Lemma E.I, w.h.p., Fij Uz. Hence FF n Fial < 1U1 O U2al with high probability.
22

---

It thus remains to bound [U12 n U1s]. Firstly, we bound the expectation of [U12 n Uis|:
1 Blivetianuisl n LueUis)
Let D1 denote the degree of vertex v in the graph G1. Let Xa ~ Bin((1 + ((1)) sa logn/n)) and X5, Bin((1 + o(1))n/2 sb n/nn On the event F, D1 d X. + X'. where F is defined in
Definition D.3, Xa, Xs, are independent. Note that by Lemma D.4, PFF)) = o( We have
m+ Σ i=0
D1
s0(1 - ((1-)))p-1 -)pp--
E[ Leeeial = [[ T | =
i
m+
Σ
(1. (2
(E.8)
Here j(i,j) is constant related to i, Now look at B[D (1 s)2D: In our regime, L S 2(m k)
are constant. Hence:
(1 - s) s) = EI(X, + (1) 5)² 2X (1 5) 11]
-)
K CB 1,) 8) x)2x 1,).
(E.9)
(=0
Here C₂ is constant related the second equality is due to the independence of Xa Xb. Now look at TE| [xx (1 s)2.X. 15).
[1 - 5)2Ka 1-| s (1)
(L+o(1))m2
sa log (11 (1-o(1))n/226 (1--))2 s ((1
sa log n
n
n
(log n)a
el( (1 s) (1= logn )e(1 n
sa log (1+0(1))n/22
n
n
(1+0(1))/22
el( - log )(1 sa sogog (1+0(1)m/224
Σ
n
n
(log 13441
We can bound the first part:
(l) n2
el - ()-s))sac log so
sa log n (1+0(1))//22 (011
E=D
(log n)3
(1 8) 2sa logn )(1 )
sa log n (1+0(1))o/22
<(log 11)
(1 2) log )((1 nh.
n
Emn0
(2+0(1)n//24
sa log n (1+0(1))m/222
<(log n
-loggn)* (1 (1 - 5)2) log n (1+0(1))m22 < n (1-(1=))))s0/221)))
n
Then we can bound the second part, note that
(1+o(1)n/2
(1 s)2 sa log n y(11 sa log [10(1))a/2-E
R= 81+1
(1+o(1))m/2
Σ
(1 - 2) 'P =f) (logn) - 8) 21lgg n) "(('''(XX"". > n))
8= 31
< n2 2(ogg n) log(1 (0g(1=)++(]) = o(n - -(1 (1- (0aa-)))())2t((1)))))
23

---

The first inequality is true because (1 - 8)2t decreases when / > (log for sufficiently large
The last equality is true because (logn)² log(1 s) (1 (1 s) ssa) /2 for sufficiently large n.
Hence, by summing up the two parts, [| - 81 S - (10))(()))))/2t000 )2) similarly we can proof 5 1..---(1 -
-11-
(1-s)2x; s)2x,]
n - By (E.8) and (E.9),
Ee AeU P + - -0 n x(1 ((1 (T.())))(1))) o n
By Markov's inequality we have
1
P( n U1al 2 log n N Unsl)) <
= o(1).
log
[ U12 Unal] n Hence have [F2 n Fisl < 1U12 n Uial I nR
for
any 8 > O, with probability - o(1)
F Proof of Theorem 1 for three graphs
F. Exact recovery in [n] \ (Fi2 U F2s) U [m] \ (Fi2 U Fia) (n) \ (Fia U P23)
Definition F.1. For a vertex G, define the quantity majority of
maja (1) = (Na(1) n V [6] (Na(1) n (--0
By Lemma D.I, we can directly deduce that if (1 - (1 (01=s)3)D2 s) b) > 1 + c (og(a/b)) then for all i [n] we have that majo, Va G2V Go (i) 2 e elog with probability o(1)
F22 Exact recovery in [n] \ (F12 U F23)
Now suppose that i M12 O M23- Look at G = (G1 VuN G2 V uusomi Gz) (12 F2a)). Lemma F2. Suppose that (1 (1 s)²) b) 1 + 2el log(a/b)) Then with probability
1 1-o(1), all vertices in (M12 n M23) have an log n majority in G(M12 O M2a).
Proof. Denote Fūj the set of vertices outside the 13-core of G2 Aniy Gj. In light of Lemma E.5 and its remark, can replace Pij with miyy Fij with Fž in Lemma F22. Where we define
G G1 Vaia G2 Vats G3,
H := G (M2 n Ma
To bound the neighborhood majority H, for E Mia n M2g note that:
maj (i) = (i)
Σ
o (J) S maja- (1) + (1) U F1
(EN
maj (1) = (i)
Σ
(j) 2 maja- (1) (1) F.
(EN
To sum up, we have
|maj H (1) majo- (1) S (N F3
(F3)
Note that majg- (f) > 2e log n, i E [n] with probability 1 - (1), given that - (1at(11111))) (01-s)3))2 (1- > 1 1+ 2e| log(a/b) by Lemma D. Now we prove that the right hand side of (F.3) can be bounded by e log n. Look at N (i) n {112 U EigH.
Nao. (i) n F U 3
NN ( (FH)) + (T (P + [Noovic (1) n (FT2) |No (No n (Fi)).
First, look at |NG,AG (1) n Figl. by Lemma E.4, w.h.p.,
NG,AG n Fi logn/88
24

---

Similarly, we have w.h.p.
(1 n 222) < < n/8.
What is left is to bound |Na=((Gaaa,) (1) F22)). (()
Note that conditioned on r12: Fi3, 223 ,², E = {Eige, j, k E (0,1)). the graph G \ AG)) independent of F12 by Lemma D.2, since Fiz depends only on G1 A G2. Thus we can stochastically dominate |NG=V(G) (i) n Fizl by a Poisson random variable with mean
log n K[ E Fi2 (i.}} E &1o UE10 JE UESIo S log n
An =
1Fiel:u= max(a, b).
n
For fixed 8 > 0, define an event Z :- fiie S n To(a)) On Z, we have that An S n -- (T Hence, for any positive integer m:
(((N 2 mJnz) S PP 2 m)2z) III 2 mlfiiz, E o , 112]
S I e Om+ -11 11) - Blex 1 S ( 020
Above, the equality on the second line due to the tower rule and since Z is measurable with respect to LFial. the inequality on the third line is due to a Chernoff bound; the inequality on the fourth line follows from setting uses the upper bound for An on 2. Taking union bound, we have
(1)m) (which valid since A,, o(1) if 2 holds). The final inequality
P( E [m], (N n Fi2 2 m} n 2) S n² 1- ()..)))(
Here if we take m > (a, b)) and 8 < 8 TT b) m- the probability turns to o(1). Thus, we can set [[ssTT (a, -1 + In light of Lemma E.6, |ftz/ I n == 8 > 0 w.h.p. Hence, the event Z happens with probability 1 o(1). Hence we have
P((vi E [m], No- (Caaaa))))) (1) 0 Fi2l - [(s"Te(a, "1}) -1-
By identical proof, we have that P((vi E [n], No. NoGg(aG (i) n F2s - T( b)) 11) = =1-
Hence we have, with probability 1 - o(1), for Mi2 7 M23.
|maj (t) majo V. G2V,7, G. (i) < e elogn,
and hence with probability 1 1 o(1)
maj (1) > elog n.
Then by Lemma E.5, we can replace H with G F.j with Fij, the lemma follows.
Next, prove that each vertex in G2 Veza
G3 \oia G1 has a small number of neighbors in (T)()(
Lemma F44 If0 <
(a.0) then [oggaa/b]]
P(V E [m). NN.. Ga Ga G. (i) (a ooo)
Proof. Since (.1) depends on Gj alone, it follows that (GG) and G2 Vain G3 Va Gi are condi- tionally independent given n², o", E. Hence we can stochastically dominate [NG.v. Ga ( (G))) by a Poisson random variable X with mean A, given by An ;= log n//ltj e (G,) {i.j) E E01 U E010 U Eoo13l S v low.g//n//((CC Next, define the event 2 (G:)) & n.l (o,)))2)((
Notice that P(Z)= = 1 - o by Lemma D.7 and Markov's inequality, provided sD+ b) < 99. Following identical arguments as the proof of Lemma F.2, we arrive at
P( E [m), No ns Gsy (T 2 m) = o(1)
when m > [(sD+(a,)) (), b) ([(sD+(a,b)) (1 +1.
If c s 4 [og(a/bj]*
(o.b)
suffices to set m
25

---

Lemma F.5. Suppose that a, b, e >> 0 satisfy the following conditions:
0 0<6Σ sD (a,
(1-(1-s)))+(o > 2c| log a/bl,
4|g a/bl
With high probability, the algorithm correctly labels all vertices in {i E [m] \ (Fis F
Proof. Compare the neighborhood majority in H corresponding o1 with the true majority in H. where is defined in Lemma F.2:
(i) Σ (o.()) - 0² (1)) & (N(() S No- (1)
jENu(()
Go G, (tn. (1) < (2, 2 b)) b1
elogn/2
The first inequality uses Lemma D.5 that the set of errors are contained in L.GG)). The last inequality
due to Lemma D.8, F.4. Notice that mai (i) elogn for i [n] (Fi Fig). Hence, o (i) NN)NR(() σi(i) 2 (a) - o (i) (i) 2JENa (ō)()) - o (i)) 2 e/2gn/2 > 0, which implies that the sign of neighborhood majorities are equal to the truth community label for any i E [n] (4T U Fgg), with probability
o(1) Then we can convert H to G(In] \ (Fi2 U Pza)). the vertices in [n] (Fi2 U P22) are correctly labeled with probability 1 1- o1).
Using an identical proof, we can argue that the algorithm correctly label all vertices in M13 O M3 and M12 M13.
E3 Exact recovery in [n] \ {(Mis n M12) (M1 n Mis) U (M23 M22)]
Define M = (M1s n M32) U (M12 n Mis) U (M23 n M12). Denote F3 = (F12 (is) U (Fra n F23 \ Fia) U (Fia Fea \ F12). note that F; [m] M.
Lemma F6. Suppose that a, b, > 0 satisfy the following conditions:
0 0< (a b)
-
(a. >1 2 a/bl,
log a/bl
s(1 (1-3(())) + (511s) (5 s)² b) > 1.
With high probability, the algorithm correctly labels all vertices that are in Fy-
Proof. For i Fb, define H; == (G1 \er,, G3 Yor, (2M Mis) U (+)) Let E, be the event that has majority of at most e' log in the graph H;. Let & be the labeling after the step. For bervity, define a "nice" event based on the previous results. Define the event H. which holds if and only if:
Fig Fiji
H, = Hi;
(i) = (i) for all i M12 O Mia:
The event F holds;
Pbl S 12
By Lemmas E.5, E.7, D.4, F.5, the event H holds with probability o(1). Furthermore, define
E = majи, (i) < <'' loo n, we have:
PPUUee() F2) 0 E.)) $ ( n 7 +P(())
P((iFF n (F <n -n -nt- nF) + o1).
(F.7)
26

---

By the tower rule, rewrite the term in the right hand side as:
B ( o², &, FE
(F.8)
Now look at PPE n , E, Fg). Conditional on E, n. majj, (i) = Y Z, where V, 7 are independent with:
Y ~ Bin(li E Mi U MM+ (i,)) E E100 n ( n//)
Z Bin(lj E Mf2 U Mia {i,5} E 100 Uu (o") bl,ggg/)///)
By Definition D.3 of the event F, we know that Ij E Mis U Mis {i.j} E E100 n &
(1- (11)ss1) ()))) and J E Mia Mia {i.j} E E100 ne ( = (1 ((1)ss(1 n/
Lemma D.1 implies
P( , E, FE) -
Sn -=1-0110 (4 log(a/b)/2tog/
Follow (F.8) and take union bound, we have
< 2n -0-22 (FF ((1)
Sn on (((b))) logle//))2(lll
Fr ca Ppoa0a
Sn 1- (t..)))))
- sT) Under the condition -2 (a b) +s (1-s) b) > 1, we can choose é', 8 small enough so that the right hand side o(1). maj (i) e' log for E Fz. by Lemma E.5, maj (i) e' log n for E Fb.
Suppose that i E F12 has at most 12 neighbors in the graph GrAii and the graph (G1 Ania GM) n Mis) U {i}}. Therefore, has an at least (e' log n - 24) majority in G1 (M1 n M13) U (i}}, with high probability. Then, Algorithm correctly label all vertices in Fia n F12.
Suppose that i F12 n E2s \ Fis, has most 12 neighbors in the graph (G1 Axjo (2) U (i)) Therefore, i has at least (6' log n 12) majority in G1 Arga G3 (M2 i Maa) Ui)) with high probability. Hence, Algorithm 3 correctly label all vertices in F12 n P23 \ Fis.
Suppose that i e Fia n F2a Fi2. i has at most 12 neighbors in the graph (G) Axia GaM [i]]. Therefore, i has an at least (e' log 12) majority in Gi Vag ( with high probability. Algorithm 3 correctly label all vertices in Fia n F23 F12.
G Proof of impossibility for three graphs
In this section we prove that Theorem 2 when the exact community recovery is impossible. The
impossibility under the condition (1 - (1 s) b) < 1 has been proved in [41]. Hence we focus on proving impossibility when
(2s² Ta)To(a,b) s(1 s) 3) (a. (0, <1
(G.1)
To prove it, we study the MAP (maximum a posterior) estimator for the communities in G Even with the additional information provided, including all the correct community labels in G2, the true matching Tis and most of the true matching कं2. the MAP estimator fails to exactly recovery communities with probability bounded away form 0 if the condition (G.1) holds. The proof is adapted from the MAP analysis in [22]. The difference is that we are considering three correlated
SBM G1, G2, G3. Since we know the true matching T23. we can consider := G2 V G3
SBM(n, (1 - (1 - x)2 ja log n/n, (1 - (1 - x) log n/n). Denote R.y the singleton in G. Gj-
Then R R12 A R13 is the singleton set in G1 A H.
27

---

G.1
Notation
=0, Here we review and introduce some notations in brief.
of == the ground truth community labels in G i = 1,2,3,
= [j v=
[n] (f) +1,, V == [j E [n] of(i) = -11,
( (1) (1).
here we have 1/2" (V()).
G.2 The MAP estimator
First define the singleton set of a permutation π with respect to the adjacency matrices A, B, and C to be:
R(T, A, = {i [n] : Vj E
where Dij == max[ (2gg(0mgg()) For brevity, write R, := Ri C?.
Definition G.2. Define the set S A, B, C) as followings:
1. E R( B,C))
2. is singleton in G1 (i(R, (Rz);
3. j E (i))), T() & (T(2((2)).
Where A, B, C. is the adjacency matrix of G1, G2, G3 respectively and D is the adjacency matrix in H = G2 Vagg G3. Note that D is the adjacency matrix of H, so (*(RRR)) {i [m] 3k e
T (R,), Di = 1}.
Define Rg := Rs UT π. LTTTIII (T)))(())) The condition and 3 in Definition G.2 can be replaced by A., 0 for all j E Rr Write R R[(Ti A, B, C), S S( A, B, C)), and R Rxt, for brevity. We study the MAP estimate provided the additional knowledge 0i m23. and [in([n] S**.
Theorem 5. Let A, C,C,,o, 02 T22 Ti o [[a]] S* be given. For i E [n] \ S", MMA(() =
(())) For vertices in S* the MAP estimator depends on whether a, is larger:
1. a > b, then the MAP estimator assigns the label +1 to the vertices corresponding to the largest |St n Vitl values in the collection (maj(i))ies) and assigns the label-1 -1 the remaining vertices
in S*.
Ifa <, then the MAP estimator assigns the label +1 to the vertices corresponding to the smallest SS* n Vi+l| values in the collection [maj(i) Hes. and assigns the label -1 to the remaining vertices in S*.
Then the following corollary prove the potential failure of the MAP estimator.
Corollary G.3. Ifa > b, there exists E St VV E S* n V- such that maj(i) < maj(j), then the MAP estimator fails. Similarly, if a b, there exists i E St ntn E St nv- such that maj(i) > maj(j), then the MAP estimator fails.
Proof. Suppose a > b. the MAP estimator classifies as +1 correctly. By Theorem the MAP estimator classifies as +1 which is wrong. The argument for the case < b is similar.
G.3 The analysis of the failure of MAP estimator
Definition G.4. The event Gg holds if and only if
n² 121- Py < R n V2t | R n VT nii s -2-(22--0 -yy.aa))))
Lemma G.5. For any fixed å 0, IP(Gs) 11 o(1).
The proof of this lemma straightforward but tedious and we defer it to Section G.6.
Now define the variable W;:
W = [ E S*, maj(i) < 0), ie Re [[i E S*, maj(i) > 0), nE
(G.6)
28

---

Denote I be the sigma algebra induced by the random variables
D max C), f oj R", {Eabe a,b, (0,1))
Note that Rº, Rt n Vit, Ro n nvT are Z measurable.
Now like to show that EieR W, 0, DiE W; > with high probability, then follows that 3i E S* Vi1"" E S VV such that maj(i) < 0 maj(j). By Corollary G.3, the MAP estimator fails. Use the first and second method to analyze Everevy W,, Eiennv Wi:
Lemma G.7. Fix 8 0 and denote 0 1 (2s² S TT - s(1 (a, b). We have
E
Σ
WIZ](Eng)) 96) 2 - --nn -226 n1 -20 --
iER'N
and
E Σ 2 (2 www. (F
1ER'NV
Lemma G.8. Fix 8 > 0 and denote 0 = 1 - 12s- - -s²)Ic(a, - ss(1 - so,,,))+((,bb)
Var( Σ
W(Z))((Fg9)) S 28-3 (Fng))
VER'MV'
and
Var Σ W(Z)I(Fng)) S =n 20-3 (
VER'NVT
The proofs of these two lemmas are deferred to Section G.7. Using the lemmas above we can now prove Theorem 5.
Proof of Theorem when K 3. Firstly, show that EiErenyt W3 > 0 with high probability. Use the second moment method, we obtain
E E Ii W
PP( Σ W, W Z AERrAN
iERnNY W
EE EE WzZ
LiEReNy W.I Var Var(Eiennny; WII)
Var Var(EieReny
Eiernn W(ZZ)
2
E W
Hence for unconditional probability. let 5 small enough, ō < min((222 (a)Tc(a, 0/4), then
P
Σ
W 0) P({
Σ
w, W2
EP( Σ
W. (Z))(F)))))
iER-NVę
VER'NVi
Var EiE(EiER' WII
E
1(Fngg)
E W
E
(1-o(1)))) 20 36 -00 2(0--)++(()) P(Fngs)= = =
The inequality the last line is by Lemma G.7, G.8. The equality in the last line is by Lemma G.5, D.4. The proof for P(EiEReNY W, 0) 1 o(1) identical. Hence, in light of Corollary G.3, the MAP estimator fails with probability 1 o(1).
29

---

G.4 Analysis of S
In this section we introduce the Ar and some properties of S
First, define the set
A(S 12 r \5*)) { Sn Sx = = : Sx) x (S*)).
For brevity, sometimes write A". Lemma G.9. For any π A", A,g (aggl(0))) A, (apa()) (300) Moreover, i St or E S Aij x0))) (zgg()))))
A., B, (0m C Bai (1,00) O.Ai,Crig(atnn (igla(()) = Ai, (1 0,A (x(,x(0) A (lgBrislama (((00(()))))
O,
[n] \ S* then v(1) = (i)(t) and x(i) As,5 ( Cria (1g(((
Proof. If i
Ti²(i), hence C ( If E S or E S*,
then by Definition G.2, E R' or E R. hence A A C (1( A., O As, (())
0, A (1)) =0.
0, and
Definition G.10. Let p be a permutation of S* The permutation Pr.p is given by
T(1) i [n]
Pr.p == (T((()) sE
Lemma G.11. Let be a permutation of S", Then Px,p E
Proof. The proof identical [22, Lemma 8.11]]. We only need change to B' max(B, C) in the argument.
useful corollary of Lemma G.11 is that the elements of A can be described by permutation of S. Corollary G.12. We have the following representation
A" = {PTT p is a permutation of S*
G.5
Deriving the MAP estimator, proof of Theorem
G.5.1
The posterior distribution of π
First we define
jtt (T)abe =
Σ
1((As,j, Bx(1),z()) (oge)))))) (a..) (0,1).
(+())
(T)abc:= Σ
1((Ai,j. Br(() (awaaiiCCCCcneen..e Czgtt (a. (0.1),
((()) (mt)
V- (TTT: = Σ
Aij.
((()))()))
V (T) =
Σ Ar.
(1()),
With these definitions, can derive exact expression for the posterior distribution of mie given A, B, C, oz
Lemma G.13. Let Sn. There's 's a constant D1 = Di ag. mia) such that
P(Ti2 T|A, B, C, C, T22)
# (1)m
u*(a)
# (F)r
a (r))o
=D; P111P000
P100
Pou0P110
Po00P101
POL1P100
Pooo
P1oaPo1o
Poo1P100
(r)m
(T)
#* (T))11
91119900
9100
90009110
10000101
x
90114100
goo0
1100go10
goo1 Q100
30

---

Proof. The proof is adapted from [22, Lemma 8.13]. By Bayes Rule,
T2a) P(A, B, C, 22 4i2
123) π T T23)
P(
m23
In the construction the multiple Correlated SBM, the permutation Ti2 is chosen independently of everything else, including the community labeling 02 and the permutation 123 Hence we rewrite
P( = =0,, m23) di
223 P(A, B, Clo 23: " = =),
where d1 (A, (,,)) (n!P(A, Clos, -23) -1 Look at Clo 1 n, T23). note that the edge formation process in G1, G2 and G3 is mutually independent across all vertex pairs given 02 12. 723 Hence have
P(A, B, Clo, n. T23)
II
Pijk
p (nis (m)ss
Ligk
(oke(0,1)
In =((0)) (
particular, the sums and measurable with respect to 0, 23- relevant value and use A to represent. write ABC to represent AB to represent E000))0EE A (()))))) E (100,=()))EE Ai, C (()) Then, we can write
(00) (o2) Bx( Craall) E (000,70)) (2gg(()) 18+ (oz)) do not care the Now, for simple notations we ()EEtt))))+
Hence
Ar, (2) and Σ to represent
at (r)m1 = E( A)BC Σ BC ntt (+(()) = -A-
H² (T)010 = E A) C) = - =p (T)11.
(T)001
Σ( A)(1 B)C A (T)
1 (T)000 = Σ (1 A)(1 B) C) p (T)) 100 + nt (T) + + ( + +==((())))))) (T)
AT (T)200- = Σ A(1 - B) C) = A p (m) n - (T) P (T)110 + v*
Hence we can write
(+)
at (*))
(r)on
II
10 (x) = dj Pijk
P111P000
P1o0
P0000110
P000P
P011P100
Poo0
P100Po10
Poo1
(0.1)
Here dj some constant given the information B,C, T23 u ( have that
Replicating the arguments for
# (()m
(*)
# (m)
# (xr)oo
II (0.(
Pijk (1)sse
dz
91119000
9100
qu009110
10000101 goo1 Q100
00119100
go00
91004010
Combining the two equations we prove the statement of lemma with D1 = did dz.
Lemma G.14. There is a constant D2 = D2(A, B, C, S*, T22. [iz[[n] 8*)) such that
P1009000 yu - (T) 1(TT A*). P0009100
P( T|A, T23 S" ( \ S**) D2(1
Proof. By Bayes Rule we have that
P(mia π A, C, 0i Teg St [n] \ S*})
P( = T A, B. C. oz T P(S Tiee(n] 5 -T,
P(S क nn)
St A, B, m30
PP = T A, B. T P(S*, mi [n] S, m23)
1(x AA).
31

---

The probability in the denominator is function of A, A. T {[x] \ S5 T23 dz. Furthermore,
by Lemma G.9, n² IT) (T) (T) (an::(((Tma (T)m [T) # (T) 101 are constant over E A". By Lemma G.13, we can write
(*)
(x)
P( TIA, C, 02 m23 S", [ {8*)) d2
P100 Pooo
9100
1(T A*),
4000
where
14 (2i2)
" (1i2)a
D1
9111
100041
d2
(1S",,/2(11)
\ {,, 02
T23)
9011Q100
T100Q010
(/000910 " (Mia)io1
1² (x)111
1 (T))1
P (x)101
pi P111P000
P0000110
P000P101
quoi 9auu
Po11P100
P100Po10
Pou1P100
D1 is the same constant in Lemma G.13, d2 is constant given A, B, C, .2.5 St To further simplifies the posterior distribution,
T23
T22 S*
v (x)
P100
9100
P(mi2 =|A, B, C, T23, S" miz ii(( 555 = d2
1(T E A*)
Poo0
9000
= =dal
P1000100 (m))+* (x))
P1000100 y
(x)-w (+) E A*).
P0009100
Pu009100
Note that リー (TT ut (T)) = E (()) Ais that only depends A. The results follows then
Pi P10ogi00 j
PT(122 = A|A, B, o2 T23, S" [ ((5)) = Daly
E A*),
Poo0
((a) An,
where D2 = d2ly
Pio0f1c0
P0009100
G.5.2
The posterior distribution of a
Now we can study the posterior distribution the community labeling o". For a community partition X x=) of [n] in G1, define the set
B(X)= {- E A' -(x*)=1k xx=,=xx====>>>>>>>>-1"
In particular, x denotes the community memberships associated with X, the following must hold:
(i) (m)(((() for i [n] \ S*:
S n X | nit S -
The first condition must hold since we know the true vertex correspondence and the true community labels outside of the S The second condition must hold since the number of vertices of each community in S* can be deduced by examining the community labels of x xS(") with respect to oz.
Lemma G.15. If B(X) is not empty, then [B(X] S n. S! nx+
Proof. The proof is almost identical to [22, Lemma 8.15). Suppose that Tos Ti B(X), by Corollary G.12, there exists p such that πं = P Pro.p- Claim: if i E S" nx p(i) E S* x E S* 8""XX_,, p(i) E - If E n .p(i) E - then o (To( = 1, 1, (T) (T)((((()) = -1.
-1. This violates the definition of B(X). The claim is proved. Hence we can decomposition p into two disjoint permutations pt, p .pt is permutation of S* X while p is permutation of S* n X- Hence [B(X) (# of choices of p²) x (# of choices of p-) S x nx
15* n S
Then we look at // (T) - (x).
Lemma G.16. For all B(X), we have that v -v) DD maj(i)ax (1) where D3 is a constant depending A, B, C, oī T23 S x SS but not X.
32

---

Proof. Note that ox if (f(1), ('(()) -10 3- =02). We have
vt (T) (m)=
Σ ox(i)ox(i)A.s
(1))
Σ
Σ (x(()x
E ox(1)o(()A
(1
'JS
iES JE
We should note that if (i,j) E {nt \ }. then ox (1)))x(()) = = =
0 ( Denote
D3:=
Σ
ox(1)x(()4..
(1))5)))))))
Clearly D3 depends only on A, o2 S" [i2(nn] \ (5)) If,,j E S", A. 0 by Definition G.2,
Σ
ox(1)ox(i)A..-
E
ox ox)
1E8".JE([M](S")
iES JElm))s
Σ (x
Σ d(())
iES-
jE(()](S')
Σ ox (i)mm((
E
The last equality is because if E S* .maa(i)) ΣjE(n A() EjE((n Aijoi(j), since A., = 0f i,j Sr. Then the statement follows,
vt v*(() v (T) D3 + Σ (i)maa(()).
ieS
Lemma G.17. B(X) is nonempty, then
P((V) 112h) (x X B,,,,,, St, T
= =Da (V/
P1000100
Eies (1)a)((
Poon
where Da is constant depending on A, B,C,o 222, S 3) \S*) but on X.
Proof. We have that
= x )J), C, S mie((l) (5))
Σ P(m = x A, B, oz T23 St Tia([)] (*)
=EB(X)
(x))-
Σ
D2
P1009000
Po001100
xEB(X)
D3
Eies ax (1)maj(()
P1004000
P100 9100
D2
P000/100
Poo0
xEB(X)
Eses ax (1)maj())
D2
P1004000
[B(X)
P1009000
P0004100
P0000100
Eles xx (1)m(((())
P1004000
=D4
P0004100
(v
Dg
where D4 D2
Pianggan
S nnvt n V. The statement follows.
Poon
33

---

Proof of Theorem 5. Given A, B. 02 S", [ 5*), it's obivious that MAP(())
0. [2)))) for E [n] S For vertices S note that
P1000000
s(1 p(1 (1
3)) s(1 5) - (1 - (1-s)))g) =1
- =
P0004100
Thus, by Lemma G.17, if a > b, the MAP estimator maximizes Eies- dx (i)m(i))) while the MAP estimator minimizes ( mij(m if a < b. Suppose > b. while satisfying the condition |S |s* | |===S~VVV
the maximum of Eies- ox dx())maj() is obtained by seting x (i) +1 to the vertices E S corresponding the largest 15t Vi*l values in the collection {maj (1) hes- and assigns the label -1 to the remaining vertices in S The proof is the same suppose a < b. Then Theorem 5 follows.
6.6 Proof of Lemma G.5
Denote Eg as the event that is a singleton in G1 Azia H, in other words that E R" We assume that the communities approximately balanced. More precisely, assume that the event {n/2 3/4 |V+|, S n/2 + n3/4] holds. By Lemma D.4, we have that
F(g) = 1 - (1).
Conditioning on of. if V/2t, then we have
((E = -1- s(1 - (abg s(1 (1 - (11=s))blognn) m) - <exp(=s(22-s))ogm - +
=exp(-(1 (1)) (g (:
= n -2--22--))))
The first inequality uses the fact 1 - x I E Hence
B||R" (Vttll
< (o $ n 3
Σ
By Markov's inequality.
F(RR not 2 n 1- ( = =())
Hence
P( 1i 2 -1-2252--yyT-aaa)))
n nyti 2 -2" (oj]l( [1-- o(1).
Now we derive lower bound for |R* n V For en sufficiently small:
(og(((())o1)))(()))
((|V) - 1) log(1 s(1 (1- ()a)))/oogn//n logn/n)
|V' log (11 (1))
(1
-(1 + 3
(1 o(1)( + (1
The first inequality uses that log(1 2) 2 - < Setting c -0.55 we have that
R n nnttlooo]l() = Σ ((E 2 -0
*EV+
Then bound the variance of R Vit For i Vit,i i
(o(((()) - PFF
(1-(011s111100000 (1--s(2s-5) 52 log n/n))
34

---

On event G, 2 - 3 (1+ = (1 (1) Hence using the equality 1 - -
we have that
Cor(I(E),,
s(2s (o (1 (1)) s²) + 6) log b)
=n -1- -1011220"="0000)+00001)
Then
Var(IR' n (1
Var(
Σ 1(E)))())(()) = Σ C()(I(E),,1B)))))11))
'EV"
"JEV
< Σ ((E)))))(()) + Σ (B)(116)),1(3))))111))
iEV
*FjVY
<(n (n ((at))))(() (a+b))+))) (0= = n
By the Paley-Zygmund inequality,
P(RR" n 1/t+| 2 n 1- STT. (G))
E R n nv
2(1 n n11-n-
E n V [
Varl n V
2(1 - n (0-0
E| ny ((()
2(1 - n - 840 "
(1- (1)(()))1
Together with (P()) o(1), we have
1222-(122-3 & R n S 12'- ()) 1- o(1).
The proof for |R n VT is the same. Now we study RR" n V/++| Since R C R ,we have the lower R n ni 2 n 1-62 - (a = o(1). For the upper bound,
bound ( R n V 2
R nnntls R S Σ (1+ Nn - [1 + max NB (j)).
JE(m
By Lemma D.6, since H ~ SBM(n, (1 -1-(1 (1 x) =) log n/n, (1 -1- s)² log n/)), we have that maxxe[j] Nrr V 100 max a, b(1 (1- 8)3) log n with probability - o(). Hence we have that with high probability
R nvvllss
(1 + 100 max a, 6(1 - (1 - --)²) log)
< 21 100 max a, b(1 (1 5)2) log - -2222-22
< n - (T
G.7 Proof of Lemma G.7, G.8
Firstly, we use some useful conditional independence properties given the sigma algebra I. Lemma G.18. Let i E R', conditioned on I, (Aij {i,s} E10o3 is collection of mutually independent random variables where
(og/allgn//)) (1) = = oi Ber(blogn/n
Aij ~
(1l -ii
The random variables 1(i 5*) and Σ Ee Ai) are conditionally independent given I.
Proof. Note that the sets R", R only depend on xiz. H, G2 A H.G. \ H is comprised of edges in E100. the graph H is comprised of edges in Ujtkso the graph G1 ^ H is comprised of edges in EojR Thus by lemma D.2, G1 Vai, H is conditionally independent of R"
35

---

given n², of and the partition &. In particular, {At {8,5} E E100} is conditionally independent of I.
Note that (i E = 1 E (At, = 0, )²). Hence 1 E E is measurable with respect to the sigma algebra generated by I and the collection C₁ == {Ai, i E Rt, {i,j} E E00) On the other hand, since I² is I- measurable, Eje(n] A (i) is measurable with respect to the sigma algebra generated by Z and the collection C2 == {A., : E [n] \ Rt,, {i.j} E &1oo}. C1 C2 0. the independence of A implies that the two random variables are conditionally independent given I.
Lemma G.19. For any d > 0, E R it holds for sufficiently large n that
P(i E (T))) 2 (1 n- (2s* 1orf-"yTTT(aan (T.((.)))+ (Ga).
Proof. For i E Rt, define the following random sets that are I- measurable:
C+ C+(1)= : = {j R [1.}] E 100 n (ot))
C (1) Nt (i.j) E E00o n (o))).
Note that E S* if and only if E Rº and A. = 0, vj e (+())CC conditioned on I. Hence by Lemma G.18:
P(i P [z] = P(A.j = (t) (1) = (1- a wlgng/ng /n) / @1(1 blog //
2 - a1- (1) log (1 (i) n/n)
2 1 (a (1) ||C (1)) lognnn
The first inequality is because of Bernuuli's inequality. Note that on event Gs. (i), R
2n (2=2))))T.(a))++ Thus
P(1 E (T)))1(9)) 2 (1 - 2(a + b)n - -262 (2520=") (ow.)))oooogg)(()) logn)
Since 2(a + b) log n I n the lemma follows.
Now, define the random variable
P P JELNAA < 0T) iER
X
POE Aijoi(i)>00 0 iE
Then we will study X, on the event F FnG Lemma G.20. For E R
X1((FGGb) n (()0))))) 1(F
Proof. Suppose E R² n ntt Note that R², of are I measurable. By Lemma G.18:
Σ Anjo -
jE
Y ~ Bin((ii Rt (i.j) E E0no n (o)),, Z Bin((i E [M] \ Nt (i.j} E E1w n E (oi), blggnnn)
Where Y, Z are independent. For brevity, suppose Y Bin(y, Z Bin(z, q). On the event F n Gs. the upper bound for y::
y < Hij E [m] [i.j} E 1oo (ot( s s(1 () = (1 o11)ss(1 n)3n22
The lower bound for y:
y 2 1 [n] {i.j) E100 NEt - RR
2 s(1 - snn 3/4/ -2-(22"- = (1-o(1))s(1- -1)))(113)))
36

---

Same calculation for and we can derive:
(1-o - (55
s)/n2
We can rewrite p. 9 p (1 (1()ss((1s()))))) (oggs) ())/(() (s(11s)Pn)) q= -1 (1+
8)² log(s(1 (=) hence by Lemma D.I, we have
-
P Σ {)) ( = P(Y Z (FF)))))))) I =(1_s)"D; /
The same argument works for E R nvT.
Proof of Lemma G.7. For Re n Vit, we have that
B[W,I]](Fnng) P(iEs, E
jEL
= E I)
Σ (000))022))()8))))
jEiN]AA-
The first equality holds because A.j E S S
R The second equality exits because the conditional independence Lemma G.18. By Lemm G.19 and G.20, we have that
E [ (g) (1 (2.² (e.b))))+2 Xx,1FF
(1 (2. 2 (T. +11- (1-=)) (a.))
(a.0)+((1) ((F))))(((F))))))
Summing over i E R nv+ we have that
E Σ
WIZ]I(Fngg)
VER-VV-
Rt IR*nvvI(1 - (2x 5
(1 (262 (2 110
9s).
The second inequality is because on Go R n V+ 2 n bound for E R n V/= is the same.
The proof of lower
Proof of Lemma G.8. Suppose f R nit we have
EWW P(i,j Σ
(k)o <0, Σ ( < <0I
kE(N]iR-
kE|N]
< <P(
Σ
A1, (k)
<0, Σ
(k)oT(K)
(k) < <
AEIn]\A-
k
P
Σ
A (k) < <
Σ (k) < <[ = X,Xy.
kE[n]]R
The second inequality is because by Lemma G.18, {Ai, (ke,)) and [Aj R. are condi- tionally independent given I. Consider the case = j.
E S" Σ
Ai)o < < $ X..
jE(m] R.
Summing over i E R nvt
EK Σ WoJ =
Σ E +
Σ
E[W,W,IZ]
NEA+
iER+V++
+VER'NV+
Σ X
Σ X
Σ
X Σ x,)².
iER'VV
iNJER'NVI
iER"
IE
37

---

Note that
E Σ ())
=( Σ P(iES: E
VER
iER jE
2 (1 - n (262 ST ((0 Σ x))1((())))
iERnIV"
Hence we have
Var( E ((IZ)((EnG:)
VER'IVG'
EI(
Σ W.)
E WaIZps ((Fgg))
SER-NN
RER-N
Σ Xi+( E x.)² (1 (1-11- nn -(20" -----aa---yTtll)taayy))) I(FN)))
VER
iER
Σ
X; 2
E -2 125 1(Fngs)
VER-NVi
iERINVT
RR n Vinvitiin In -1 (I-)) 2 (1R n Vitt 1010n------0-000--000- (01)0 (o0))(1))))) 2 -1202 -3700)))) ( 1(FnGs)
(6o+0)) 2022-22-10 (28 (282- (
The second inequality uses the fact that 1 - -1- s 2n- The third inequality exists by Lemma G.5, G.20. Now, for further simplying the upper bound, note that if 5 is small enough (specifically, 8 < minn(2s* 5²)T (), 0/4)), then 0 + o() 36, and 20 +
(2s² - -sa Ta)To((a b) o(1) 20 - 36. Hence
Var( Σ WIZ))(FFnG)) < n0 28-36 1(F
iER'NVT
The proof for Var Var(EiiR:ny;- W. G5) is the same.
H Proof of Theorem for K graphs
H.1 Categorization of vertices
We start with the definition of categorizing vertices as "good" and "bad" vertices. To begin with, we define a metagraph for each vertex. Then we categorize each vertex as "good" and "bad" according to the connectivity of the metagraph for the vertex.
Definition H.1. Given (G1, Gk) ~ CSBM(n, logn .logn i 5), and (%) partial k-core match- ings fo == [ig + E [K]]. For any vertex define the following graph matching metagraph Mg.. E(MG.) denotes the edge set the graph MG There are K nodes in MM.., where node represents the graph G and an edge exists between (i. that is, (i.j) E E(MG.) if and only if vertex can be matched in the partial matching Pis between G; and G₁-
for the vertex u, denoted
Note that Mg, is undirected graph. This is because of an inherent symmetry in the definition of k-core matching, which looks at the k-core of the intersection graph of the two matched graphs. Thus for k-core matchings, vertex is matched by Pag if and only if it is matched by P.z- This property does not necessarily hold for other graph matching algorithms.
Definition H.2. Given (G1,..., Gek CSBM(n, a log logg s). and (K) partial k-core match- ings t == (Fg i j [K]], we define a vertex to be "good" if and only if MG, is connected. Conversely, vertex v is "bad" if and only if MG, is disconnected.
38

---

2
(a) MG, for "good" vertex et Mg. is con- nected.
(b) Mg, for "bad" vertex vn Mg. is discon- nected.
Figure 4: Schematic showing the meta graph Mg. when K = =.
For "bad" vertex since MG is disconnected, the metagraph has at least two disjointed compo- nents. Hence, there must exist two sets I (v), [o) satisfying (v) (Tx(v) = 9, (v) (b) = [K] and for any ( Tv(v), (i not edge in MG.. In other words, cannot be matched for any matching between the graph G, E T (v) and the graph Gy. E (v)... Heuristi- cally, the definition implies that "bad" vertices cannot utilize the combined information for all K graphs.
Otherwise, the "good" vertices can utilize the combined information for all K graphs, as shown in Lemma H.3.
Lemma H.3. For a "good" vertex v defined Desinition H.2, for any two node ij (represents two gruphs Gú G2), there exists path = Eo - 33 82 La == J such that can be matched for pe em m E 10.1...,, 11.. Define Ter(1) (v) = Pta ifa Neg P 2fa-1
o... o 1o,,22 Peat (v).
Proof. For a "good" vertex v, since MG, connected, for any two nodes i, E [K], there exists a path (v) == {lo l the path 4isy (v) to define the (e)
ld-1 La. to i, i j. (Cms lm+1) (MMG.)) We can use
Note that such path is not unique. However, the choice of path does not matter whp. By Lemma E.5, for k-core paritial matching, Psj = Tiy i,j E [K], with high probability. Hence, for two different paths with endpoints being j.j E [K], denoted by wiy Viy. we can define Th "ig the two paths wb vi separately. Then, with high probability Fi mij
Remark:
22 based on
Kij.
By Lemma H.3 and its remark, we can have the union graph of K graphs (G₁ VG, for the "good" vertices, using the matchings मे == (#12, ..3...., #ik) where +iK is defined in Lemma H.3. If there are multiple paths existing, pick the shortest one, and break ties in lexicographic order.
H.2 Exact community recovery algorithm for K graphs
The key steps of the exact community recovery algorithm for K graphs essentially the same as the algorithm for three graphs. Based given almost exact community recovery ã and (K) k-core matchings pairwise, we divide the vertices into two categories:" "good" and "bad" vertices according Definition H.1, H.2. Then refine the community label according to the majority votes for the "good" and "bad" vertices sequentially, to obtain the exact community recovery label under the given conditions. The full algorithm for exact community recovery is given in Algorithm 5:
H.3 Analysis of the k-core estimator
Here we quantify the size of bbe" vertices. Suppose that for vertex U, r (G1, G2...., GL} and r3 [GL+1, GL+2, ... Gx }. here < < K 1, cannot be matched for all the matchings between one graph from r, and another graph from T,,, Apparently, u cannot be matched for L(K - L.) matchings. Heuristically, as the number mathings that cannot be matched for vertices increases, the size of such vertices decreases, since every matching would match (1 o(1)) )) vertices and only a very small fraction of vertices that cannot be matched. The following lemma demonstrated the claim.
39

---

Algorithm 5 Community Recovery for K graphs
Input: K graphs (G1, Gg...., Gx) on n vertices, k = 13, and >> 0. Output: labeling of in] given by ö.
1: Apply [35, Algorithm 1] to the graph G1 and parameters (sa, sb, c). obtaining a label 01.
2: Apply Algorithm on input (G;, Gj. k). obtaining matching (M,, Pij). j 2.. K).
3: For "good" vertices v, look at the set V == (Pt,, E [K] (i,j) E &(MG.)) by Lemma H.3, can define the मे. == (#12: 13 13,..... ..... _ to obtain the union graph of K graphs based on the matchings from V. denote as (G) V G2 ... V GK)y. Denote M := nMa,, where (i,j) satisfying Bij_ .M Set (v) E [-1,1] according to the neighborhood majority (resp., minority) of (v) with respect to the graph (G1 V.V. G2... ....VG. ...V ( [M} if > (resp., b).
4:
(ee[kMMs set ((v) E {-1, 1.1.1 }} according to the neighborhood majority (resp., minority) of ā ((v) with respect to the graph G1 Vdo (,(M M U (u)) if a > (resp., a < b).
For "bad" vertices v, denote == {j [K] (1.5) E E(MG.)): Denote M
5:
Return Ō [n]] {-1,1}.
Lemma H.4. Suppose that G1, G2..... Gk are independently subsampled with probability s from parent graph G ~ SBM(n, log n/n,6 b log n/ for a, b > 0. Let F be the set of vertices outside the k-core of G2 Asy Gj with 13. For L S K and every ô > 0, with probability
1 - o(1) we have that [NSigLLLSSIS Piyl S
Proof. Define Uzj to be the set of vertices with degree at most m + in Gj Any Gj. where m > Ta+b)a2 as defined in Lemma E.1. Then by Lemma E.I, w.h.p. we have FG C Uij and Insighttssss Piyl= < |isist,t41sjs Uzyl. Now we look at the expectation:
L
K
E
El Unl nl
II
II
LUEUS
i=
==1=++1
Let D denote the degree of vertex v the graph G1 Gg...
G
L
K
K
L
E
1 LeU
=
II
II
LEU
D
=1
j=L+1
(im+R)L
K-L7
$
Σ
s (P)
D
ss)
The first equality is by the tower rule. The second inequality due to two observations. Firstly, [II! 1veu,, {deg(v) < L(m++k) in the graph (G1 V3 be more detailed, if the degree of is at most m + k in the graph G, L+1 L+1<j<<]) To L. then the degree of v is at most L(m + k) in the graph (G, V G3 ..... GG) A The second observation is, given D, for f1 # j2, the events (deg(v) L(m + k) in the graph (Gi V (G....VG.. A Gj, (deg(u) S L(m + k) in the graph (G, V G3 .... GL) A G5,} are independent.
Similar to the proof of Lemma E.7, let X. Bin + m/2) (1 (1 11-s) =) log n /n), and x) x) log n/n). On the event J, Dj 4 X, Xs, where F is defined in Definition D.3, X,, X, are independent. Note that by Lemma D.4, P(JF) of = We have
X, Bin + o(1))n/2, (1
K-L7
n-t &
E
8e - P-i
i
(m++)bb
Σ
(
is 11
8) (K-L) (H.5)
40

---

Here C2,1.12 ik- ) is a constant given i1, 22....,,44 Now look at E[D s))
In our regime, N S (m + k)(K L are constant. Hence:
E|Da (K(]))s)__)))) 1
[X, + x,) (1 5) () 5)(k-
N - - 1, | NE 1=0
Here C₁ is constant related to the second equality is due to the independence of Xa, Xo. Now look at [ (1 s) ()(K-L)x, 1
[1 - 1)xx 1 S.SEE E[X 5s L)xaa]
(1+011))12
Σ (1 - (1--)(K--)) ()( (1 (1- (1 8) Jalonn (((1 n (1 (1 8) log n (1+0(())))
n
00
(logn)"
(1-8) 8)K
(1 3) log (²( ye( (1 (1 8) log n (1+o(1))/22 (1 (1-8)
C
n
E=
(1+ol1)a/2
e (1 (1-s) k- (1 5) log logn y0(1 (1 (1 5) a)) ) (1+0(1))/22
Σ
n
n
==(logn)++
Similarly, we can bound the first part:
(logn)²
(1- 8)= 5) (1 8) log (011 (1 (1 2) log (1+0(1))0/22
n
n
(logn)a
log log n (1+0(1))0/2- (1 (1 s) Ja ee(1 (1 )a (1 - 8) KI
<(lg
n
n
(1+o(1))n22
(1 - (1 8) )a log g (1+001)))/22 (1 s)-s) (1 3) )((1
$(logn) 3)
Σ
n
n
t=
(1- sa log n (1+o(1))n/2
(1- (1- (1- K- (1 - -1 (1-
n
Sn -1- (1-1)==))))))) (a/2+((1)
Then we can bound the second part, using the similar arguments in Lemma E.7, we can show that
(1+o(1)n/2
el LL1 (1 s) log n )((1 - 5) log (1+001))022-
Σ
n
n
(= +1
o(n (1-(1-s) x (1 (1-10111000) (1- (10ss)))//
Hence, by summing up the two parts, E[ (10011---h () < 11- (1- Da/20000) This is also true for Xp. Then, E N (1 - s) ( LD) s n - (1--(11)~-0000 (a.
For f(x) = (1 5)* (1- K= E [1, -
- 1]. the function f(x) obtains its maximum at
K K-1 Hence n to Lemma F.7, by Markov inequality, the lemma follows immediately.
n11 x toll Similar
Lemma H.6. The size of "bad" vertices defined in Definition H.2 can be upper bounded by n +(1 (1--))
Proof. By definition H.2, the meta graph MG bor" vertex u are disconnected. Hence, there exists at least two components of MG,, where there are no edges between the nodes from two components.
With all the preparations for "bad" vertices and "good" vertices, we are now ready to prove Theorem for general K graphs.
41

---

H.4 Exact recovery for "good" vertices
By Lemma D.I, we can directly deduce that if (1 - (1 D b) > 1 1+ c| lo /b], then for
all i E [n]] we have that vG... (1) 2 c clog n w.h.p.
Now for "good"vertex i, there exists a corresponding matching set v that can be matched for all the matchings in V and there is an union graph G := (G, V G2 ... V Ga) that can be derived from the matchings in V. Let M be the set of vertices that can be matched for all matchings in V.
Lemma H.7. Suppose that (1 (L - D))) b) > + 2c| log(a/b)). Then with probability
1 - (1), all the "good" vertices in M have an e elog n majority in G(M).
Proof. Denote FūJ the set of vertices outside the 13-core of G2 Amiy Gj. In light of Lemma E.5 and its remark, we can replace Pij with miys Fiy with Fūj ,M with M in Lemma F.2. Where we define
G G1 Vai, G2 V.. Vzix Gk; H G==)"
To bound the neighborhood majority in H, for E M we have:
maju (1) (i)
Σ
o (j) majo- (i) +
Σ
NNo. (1) n Fjel.
(E)
maj (1) = oi)
Σ
(j) 2 majo. (1)
NN (i) n
(EN
To sum up, we have
Imaj (i) majo- (1) S
Σ (1) Fjel- PS
(H.8)
Note that majo. (i) > 2e log i E [n] with probability 1-((1), given that (1-(1-s)*) > 1 1+ 2e| log(a/b) /6] by Lemma D.I. Now we would like to prove that the right hand side of (H.8) can be bounded by e log n.
Note that NNo- (1) Fjel & (Na,Aa, (Fj0I + (0)
First, look at |NaAAG (i) (Fje)l. by Lemma E.4, w.h.p..
|Na,aa. (i) n Ffel < e elog elogn/2K2
The remaining thing is to bound |No-v (i) n (Fj)). Note that conditioned on π.", ,", E == ( f,, i2, ... .K (0, 1)}, the graph G* \ (G, A GG) is independent of Fje by Lemma D.2, since Fje depends only on Gj A Gg. Thus we can stochastically dominate |Na-v(a,aaa n Fjjl by Poisson random variable X with mean
An = v log nl I Fje= (i,j) E (G, AG)
log TI
&
1F u = mas(a, b).
n
For a fixed 8 > O, define an event Z = {E 38l S n 1-s (a Hence, for any positive integer m:
On Z. An Sn - ((((b)))60t
P([[Nerrc
m) P( 2 m = E[P 2 E, 0", n 112]
S <[[[inf e -0 11) S Blex m 12] S n -m(=) 820
Above, the equality on the second line is due to the tower rule and since 2 is measurable with respect to Piel. the inequality on the third line due to a Chernoff bound; the inequality on the fourth line follows from setting = log(1/Asp An)) (which is valid since An = o(1) if Z holds). The final inequality uses the upper bound for An on Z. Taking union bound, we have
P( E IN= 0 F 2 m) 0 2) < n
Here we take m (s2Ta b)) and 8 < TT) b) m the probability turns to o(1). Thus, we can set m = [ (a, -1 + 1. In light of Lemma E.6, |Fjel I n T 5 > 0 w.h.p. Hence, the event 2 happens with probability
o(1) Hence we have
P((Vi (o),,N-)v,,,,, n Fjel < [(s (TT (1))= o(1)
=
42

---

Hence we have, with probability 1 - o(1), for i M
maj (i) majo- (i)l
Σ
(elogn/2K' [(>)Tc(a,b))=1) log n.
and hence with probability - o(1),
majn(()) (1) > llo n.
Then by Lemma E.5, we can replace H with G, FiJ with Fij, the lemma follows.
Next, prove that each vertex in G inj 11 has a small number of neighbors L,GG)).
Lemma H.9. If0 < E
eD (a.b) 1 [o][ then
P(Vi E [n), a. (( o(1).
Proof. Since Ic(Gi) depends on G1 alone, follows that (G1) and G" \mroo G1 are conditionally independent given π ,e,E. Hence we can stochastically dominate |Nar, G, (1) n (GGG) by Poisson random variable X with mean A given by
A v/ log nn/liiiii/ii/// (.G.) (ij) E G" GGH S v (llog///(/((... Next, define the event 2 (1)(())) S (2--0
Notice that P(2) - o(1) by Lemma D.7 and Markov's inequality, provided sD+(a, b) 99. Following identical arguments as the proof of Lemma H.7, we arrive at
P(3i [n], No. Jn G. (1) I(GG)) 2 m) o())
when m > [(sD+ b) 2c| loga/bl)~<1
S
(a.b) 4 [og[a/b]] then suffices set m
(a))
Lemma H.10. Suppose that a, b, satisfy the following conditions:
(a,
(1 (1(1-s)) 1 + 2e log a/b, <e <E log a/b
With high probability, the algorithm correctly labels all vertices in {i E [m]] \ M*).
Proof. Compare the neighborhood majority in corresponding to कं with the true majority in H, where H is defined in Lemma F.2:
(0*(1) Σ (61 o -()) (N+() [
(E
No (I)L (()LL(G)) (2
log n/2.
The first inequality uses Lemma D.5 that the set of errors are contained in (G1).. The last inequality is due to Lemma D.8. H.9. Notice that maj (i) 2 e elog n for i e M Hence, o² (i) ()) Gi > maj (i) - loa (i) Ej N (āi()) - o² (j)) 2 e elogn) n/2 > 0, which implies that the sign of neighborhood majorities are equal to the truth community label for any i M with probability
1 1- (1). Then we can convert / to G(MM, the vertices in M are correctly labeled with probability 1 o11.
Using an identical proof, we can argue that the algorithm correctly labels all "good" vertices with probability 1 - o(1).
43

---

H.S
Exact recovery for "bad" vertices
Lemma H.11. Suppose that a, b, e -0 satisff the following conditions:
1 2c log a/bl, 0 << sD (a,
(1- 5) DD+
(a, (a,
4 log
s(1 (1-(1-s) b) -)1-s)k-110 (4, >1
With high probability, the algorithm correctly labels all "bad" vertices.
Proof. For vertex that are "bad", denote v := {ii E [K] i cannot be matched through Piy. j + 1]. Denote F3 the vertex set of all the "bad" vertices that have the same with vertex Denote M == Nie Mi define H, := (G1 \eiz G2 \nis G3 ... b GK (M) Let Eg be the event that has a majority at most log n in the graph H, Let ē be the labeling after the step. For brevity, define "nice" event based the previous results. Define the event H. which holds if and only if:
Fig Fij:
(1) - oi) for (or
M";
The event F holds;
Fil & n (11))(((--))--))
By Lemma E.5, H.4, D.4, H.10, the event H holds with probability 1 - -1). Furthermore, define E = majn. S c' log n, we have that
P(UseW E F&) n E.))
I
F (1)+ PPH))
P(fie n n (Fg n 1- (1- ( n.F) o(1).
By the tower rule, rewrite the term in the right hand side as:
E
P T² o², E.FF) 1 Q-
(H.12)
Now look IP (E; independent with:
x |=","" o0" E, F)). Conditional on &, o", n², majnn, (i) -
Z, where Y,
Bin(li E M (i.j} E Eooo... nE (o')),
Z ~ Bin(li E M" {i,5} E E N (6')),,,ogg
By the Definition D.3 of the event F, know that Ii M : [i.5} E1vo... (--)0")
(1- [1-o(1)ss(1 - =))
Rnn-1n/2. nn22 k)m and Ij M {t,]] E E100.0 NE (a") = (1- (11)ss(( -) -s) nn/2.
Lemma D.1 implies that
P(E FF)1)FF - (1riism an a ajK 1K-1yrate0011717
n (D
Follow (H.12) and take union bound, we have that
Σ p(li E F. E n ( < -n1-
+ d(1)
In [1 iD (a.b)++ lox(o/0)/22+ P ------
Sn I s-1 1- (1 - k (a.b)t'' logbtt// log(a/b)/246
44

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Under the condition s(1 - (1 - 1) (Te b) + +s(1 - 3) b) > 1, we can choose e'. . small enough so that the right hand side o(1), majnr, (i) é log n for Fg. by Lemma E.5, majjz, (i) logn n for i Fb.
Note that i cannot be matched for all Fig. 2 E vh Hence has at most 12 neighbors in the graph (Gr Arj Gt). Therefore for any E Fs, has at least e log n 12/4/] majority in Gi A G)(MU
(i)) with high probability. Hence, we can correctly label all vertices in F& with high probability.
Use the same arguments for all types of "bad" vertices, we can correctly label all "bad" vertices. D
I Proof of impossibility for K graphs
We study the MAP (maximum a posterior) estimator for the communities in G1. Even with the additional information provided, including all the correct community labels in G2. the true matching T23 T21..... T2x and most of the true matching Ti2. the MAP estimator fails to exactly recovery communities with probability bounded away form 0 the condition (G.1) holds. The proof can be derived by generalizing proof of impossibility for three graphs. The only difference is that we are considering K correlated SBM G1. G2,.... Gg. Since we know the true matching Ti. i.j e
log n/n, /., (1 (1 - -)-s) loloog n/n). Denote Raj the singleton in G; A Gj. Then R = R12 A R13... ......A. A RI k is the singleton set in G1 A H. The proof follows the same arguments with more involved notation, and hence omit the details. Here we point out the differences of the proof for K graphs.
{2.3. K]. we can consider H == G2VG. G2VGG...VGGE VGK ~ SBM(n, (1- (1 ()5-s))K
Define Rx == R(T, A, B², ..., B*) == {i E [n] Vj E [.), = 0, =
max B2_ , ....., BK)} }. here B is the adjacent matrix of Gg. Similar to Definition G.2, we can B2,
define S = S(T, A, ,., B.). Let Go be the event that the following inequalities all hold:
n² 11-s(1-11-))) -
OVT (R' R* < (11-0( (T.(.0))4
R
We can prove similar versions of Lemma G.7 and G.8, with (2s² 5²) replaced by s(1 (1 s)- and s(1 =)² replaced by s(1 s)1-s)k-1 We can have same versions of Lemma G.9, Corollary G.12. We can similarly define 4 (T) and μ (T) ( for all i i....., ik E {0,1), and v+ (T) and V- (T). When deriving the posterior distribution of rī.. similar to Lemma G. 14, the information of A, B², ....... BK o2 S*, m S"), π = (mi. 1.j E (2.3..... K}} are given. Note that for т E AA, we have that μ (T) is2.. and μ (T) (s.... are constant for all i2....,. .i. E {0, 1} except for 4* (x(())) 1o...... and μ (T) We can derive an analogue of Lemma G.14 with P100 replaced by P100... Pooo replaced by Po00... and similar for q. Then we have analogous versions of Lemmas 1.15, G.16, and 1717, with P100 replaced by P100...o. Pooo replaced by Pono...o, and similar for q. Note that 10000....000.... = (1 + o(1)) - The impossibility proof for Theorem 2 follows.
Po00...0f000...
J. Proofs for exact graph matching
J.1 Exact graph matching for K graphs
Proof: Through the 13-core matching in Algorithm 1, we obtain ;= (Pey i,j E [K]}, where Pag is the 13-core matching between the graph G; and G,.
Recall Definition H.2, we can directly infer that the "good" vertices are those which can be matched for K graphs through path across 13-core estimators न. We can define a new estimator âे for those "good" vertices using the combination of 13-core estimator through the path that connects all K graphs. The path defined as Lemma H.3. For any "good" vertex. such path exists and we can define the estimator गे for that vertex.
Note that, by Lemma E.5, with high probability Poo m.y- Hence if for all the matched vertices, they will be matched correctly. If the number of "bad" vertices approaches zero, it indicates that all vertices are correctly matched. Consequently, exact graph matching can be achieved through the 13-core matching algorithm. By Lemma H.4, H.6, we can quantify the size of "bad" vertices: for
every 8 > 0 we have that MIsiSLLHIIiKK F.J n (1--))1--TTaaaaaa
When 1 - s(1
45

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(1 (1-s) (1-s)*=IJTc(ab) > 1, that is, when the condition (3.5) holds, the number of "bad" vertices goes zero when goes to infinity, with high probability. Thus all vertices can be correctly matched and exact graph matching for K graphs is possible with high probability.
J.2 Impossibility of exact graph matching for K graphs
Proof. Now consider the graph matching problem with additional information provided, in- cluding the true correspondences T2....... T2K and the community label o². Then we obtain the union graph H G2 Vags G3 V ... VHix Gg. We now prove the impossibility by con- tradictory. Suppose that there exists estimator which exactly match G1, G2,-.., Gk. note that H SBM(n, (1-(1-6)# L)a log (1-(1-a)* ") One key point, is that, we can subsample Hz. H2 H from H. To be more specific, consider the following parame-
ter: To == 1) (1-8)) " 20= where i1,d2,..... ik {0, [0 and Σj > 0. Here [1-5)8-1
E daddann = 1. Then for any vertex pair (i:
1. If (1,5) is edge in H, then (i,j) is an edge in H2, H3...... H'
2. If (i,j) is an edge in H, with probability ii.. (i is edge in the graphs { where is
T' Si. and (i.j) is not an edge in the graphs {Hi where i, 0 in 62,.....
Following the subsampling described as above, we can simulate H!g...,, H'x from H. Note that by construction, (G1, Giy, G. G.,,..,, G has the same distribution as (G1, H .... H)) Then af- ter independent permutations, (Hg. H4 Hx)) )- Note that H2 = H3. Then (G1, G2. Gk) has the same distribution as (G1, H2, H3... Hx). Since the estimator मे exactly match (G,, G2. ......... Gx) with high proba- bility, it can also exactly match (G,, H2, Hz,..., H&) with high probability. Naturally, it can exactly match vertices in G₁ and H2, since H and H2 share the same vertex index then we have estima- tor that exactly match G₃ and H given G₃ and H, where G1, H are correlated SBMs, independently subsampling from the parent graph G with probability $1 for G1 and $2
obtain H3. H4...., Hx) by relabeling the vertex index in
However, [14, Theorem 1] proves that suppose (G1, G2) CSBM(n, a b (1-s) s) for H.
81, $2) subsam- pling from G ~ SBM(n, log blog with probability and 32, respectively, if s (a, b) < 1. then exact graph matching between G2 and G2 impossible. Directly applying [14, Theorem 1] we have that exact graph matching between G1 and H is impossible if s(1 (1 - s) {) < 1. This is contradiction, and hence exact graph matching for GI...., GK is impossible.
46

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Reviews:
[{"review_id": "PjCfAC20HQ", "reviewer": "Reviewer_RUAG", "summary": "Theoretical work showing conditions for exact community recovery in $K$ correlated $2$-community SBMs where node labels are not maintained between networks. The work extends previous work for $K=2$ which introduces new challenges and proof mechanisms to allow for $K \\ge 3$ networks. Theorems 1 and 2 provide necessary and sufficient conditions for exact community recovery relating to the difficulty of the pairwise matching problem $T_c(a, b)$ and the individual exact community recovery problem $D_+(a,b)$.", "questions": "How does changing the correlated SBM to allow for different subsampling probabilities $s_i$ for each network affect the theory?\n\nIs it possible to do just exact graph matching better than $s^2 T_c(a,b) > 1$ for $K \\ge 3$ graphs? Perhaps this is addresses in some of the referenced papers, but my reading of Lines 128-129 is that exact graph matching is done pairwise for any graphs $G_k$ and $G_l$ which means that all graphs can be paired. Is there any improvement by doing things other than pairwise? I believe the answer reading on later is no, at least asymptotically, but it is uncertain later whether the benefit of extra graphs is solely for graph matching and community detection together, rather than individually. This may be addressed by resolving whether the \"if if\" typo in Line 128 should be just a single  \"if\" or, in fact, \"if and only if\".\n\nWhile it is useful to consider the extreme case, often there is some evidence of persistent node labelling across networks. For example, Joe Bloggs on Facebook is more likely to correspond to the email joe_bloggs_01@mail.com compared to a completely random node. How could this scenario be incorporated into graph matching and community detection?\n\nDo Steps 3-5 in the algorithm still work if $a < b$? Obviously, there is a symmetry in these two parameters, but as the algorithm is written, in this scenario neighbours are more likely to be the other community rather than the same. Please can you explain the changes necessary to make this approach work when $a < b$?\n\nWhat subsampling parameter $s$ should we expect in real-world networks? Figure 2 shows two possible values but I have no idea what I would expect as normal. Given networks $G_1, \\ldots, G_K$ one could find a maximum likelihood estimator for $s$, perhaps by computing the true $G$ using the graph matching algorithms described here and finding the subsample rate.\n\nAny intuition why the value $k = 13$ in the $k$-core algorithm in Line 244?\n\nHow do these ideas relate to subsampling from any arbitrary base graph $G$ rather than just a 2-community SBM? For example, if $G$ was sampled from a generalised random dot product graph (A statistical interpretation of spectral embedding: the generalised random dot product graph, Rubin-Delanchy et al), then the subsampled graphs $G_i$ would also be a GRDPG. The problem of community recovery may not always make sense in that setting, but graph matching is still very important.\n\nAn alternative way of subsampling graphs is EdgeFlip used for edge-differential privacy in networks (Sharing social network data: differentially private estimation of exponential-family random graph models, Karwa et al). Edges and non-edges are flipped in $G$ with some probability $p$. How do these techniques work using this method of generating anomalised networks? The results here could show the dangers of producing multiple anomalous versions of the same network, a potentially important result for data privacy.", "limitations": "The authors highlight the positive and negative impact of their results, particular for de-anomalising multiple networks.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "This research fully answers the open question of [18] introducing novel ideas top extend existing techniques to work in the much more complicated scenario with $K \\ge 3$ networks. I feel the paper was well-structured to introduce and explain the problem at hand to someone outside of the research area. Section 2 did an excellent job introducing the problem highlighting the interplay between the community recovery and graph matching. Section 3 giving the main results along with a helpful high level description of the algorithm used within the proof and Figure 2 demonstrates these results pictorially highlighting the regions where their research come into play. Section 4 gave more details of the proof, while still a bit technically in places was still useful for myself who wanted to get the vibe of the proof without delving into the details in the Appendix.\n\nThe paper is very clear in its goal, what is has achieved and possible future directions in this area.", "weaknesses": "While reading this paper, I was unsure whether this paper fitted the remit of NeurIPS and would be better suited in another journal rather than conference paper. I felt more comfortable about this upon noticing that [18], which supplied the problem for this work, was itself inspired by work of community recovery in correlated SBMs [37] published in NeurIPS. I recognise this is a personal bias of what I consider a NeurIPS paper.\n\nThere is some discussion of how this work relates to real-world networks, but it is unclear how much further research needs to be done before these ideas can be used for practical analysis of real graphs. A number of my questions below relate to applying these ideas to more realistic networks.", "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\nQ1 (different subsampling probabilities): This is a very natural question, and our developed theory can fully handle this. We simply chose to keep this part of the model simple, since there are already several parameters involved in the model. \n\nLet us illustrate on Theorem 1 how the results change with different subsampling probabilities $\\{s_i\\}$. First, condition (3.1) becomes $(1-\\prod_{i=1}^{K}(1-s_i)) D_{+}(a,b)$. This change is immediate, since $(1-\\prod_{i=1}^{K}(1-s_i))$ is the probability that an edge in the parent graph survives in at least one of the subsampled graphs, so this quantity arises in the correctly matched union graph (from which (3.1) is derived). \n\nMore interesting is how (3.2) changes with different subsampling probabilities. An immediate adaptation of our proof gives the sufficient condition \n\n$\\left[s_1\\left(1-\\prod_{i=2}^{K}(1-s_i)\\right)\\right]T_{c}(a,b)+\\left[s_{1}\\prod_{i=2}^{K}(1-s_i)\\right]D_{+}(a,b)>1$.\n\nHowever, this condition is not necessary. The tight sufficient condition turns out to be:\n\n$\\min_{j \\in [K]} \\left(\\left[s_j\\left(1-\\prod_{i=1, i \\neq j}^{K}(1-s_i) \\right)\\right]T_{c}(a,b)+\\left[s_{j}\\prod_{i=1, i \\neq j}^{K}(1-s_i)\\right]D_{+}(a,b)\\right)>1$.\n\nThe basic idea is as follows. Suppose that $j^*$ minimizes the expression above. Then, under this condition, we can first exactly recover the communities in $G_{j^*}$ (using the same algorithm as in our proof). Subsequently, we can port this partition to $G_1$ using the ideas in Appendices G and I.\n\nIn the revision, we will add a remark to the paper along the lines above.\n\nQ2 (exact graph matching threshold): This is an excellent question and indeed it is possible to do exact graph matching better than just pairwise using $K \\geq 3$ graphs. The exact graph matching threshold given $K$ correlated SBMs is given by \n\n$s\\left(1-\\left(1-s\\right)^{K-1}\\right)T_{c}(a,b)=1$.\n\nWe expand upon this in detail in the \"global\" response/rebuttal, please see there for more. \n\nQ3 (\"Joe Bloggs\"): This is an astute observation and there are indeed various ways to incorporate this into a model, depending on the situation. One is to consider \"seeded\" graph matching, where a subset of vertices in the graphs are already matched, and the task is to match the remaining vertices. This setting is sometimes closer to practical applications, and is widely studied (e.g., [33], [45]). Another possibility is to consider \"attributed\" graph matching, where each node has a corresponding vector of attributes, and these are correlated across the graphs. See, e.g., Zhang, Wang, Wang, and Wang (IEEE Trans. on Information Theory, 2024) and Wang, Wang, and Wang (COLT 2024). Of course, these two modeling frameworks can be combined, and others may be relevant as well, depending on the application. All of these are worth more detailed study in our setting; however, this is beyond the scope of the current paper. We will add these to the possible future directions mentioned in the paper.\n\nQ4 (when $a<b$): Yes, everything works also when $a<b$, simply by replacing \"majority\" with \"minority\" everywhere. This is correctly stated in the Algorithms stated in Appendix C. In lines 173--192 of the main text, we gave a high level overview, which is perhaps more intuitive in the assortative setting when $a>b$. We forgot to mention here that this description is for $a>b$; we will correct this in the revision. Thank you for catching this omission. \n\nQ5 (subsampling parameter in real-world networks): We suspect that the answer may vary widely across applications. As an example, in a recent work by Li, Arroyo, Pantazis, Lyzinski (IEEE TNSE, 2023), the authors develop both theory and also apply it to human connectomes. While their paper does not indicate what the correlation parameter may be in this application, their theory and simulations are for constant correlation, and they vary the correlation parameter between 0 and 1. Likewise, our theory holds for any constant correlation $s\\in [0,1]$; we hope that this can thus be useful in applications, regardless of what the actual correlation parameter is. \n\nQ6 (the value of $k$ in the $k$-core): The choice of $k=13$ is not particularly important -- any large enough constant will work. The particular choice of $k=13$ comes from [18], where it arises in their Lemma 4.8 and Lemma 4.10. In particular, Lemma 4.10 is a probability tail bound, and the choice of $k\\geq 13$ (and in particular $k=13$) guarantees that this vanishes sufficiently quickly. The proof of Lemma 4.10 contains a union bound that may be loose, so it may be possible that the value of $k$ can be further lowered, but this doesn't have any major importance. \n\nQ7 (subsampling from an arbitrary base graph or a GRDPG): This is an excellent direction for future research. We suspect that the $k$-core analysis should also work for understanding (the information-theoretic limits of) graph matching of correlated GRDPGs, and the results of [39] support this heuristic as well. However, this is by no means obvious, and requires its own careful analysis. \n\nThe setting of an arbitrary base graph is much more challenging. This essentially corresponds to a \"smoothed analysis\" of graph matching, which has not been done before. This would be very interesting to consider, but it is significantly beyond the scope of this paper. \n\nQ8 (alternative sampling using EdgeFlip): This setting is also interesting, and somewhat resembles the alternative setting we describe at the end of the paper (lines 365--372). We suspect that our methods are able to handle such a setting as well. In particular, for a constant number of graphs $K$, we conjecture that the results would be quantitatively different but qualitatively similar to the ones in the current paper. However, if $K$ is large (diverging with $n$), then we expect new qualitative phenomena to appear. This is an exciting avenue for future research, which we plan to pursue.", "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": "DJ36lY1U6Q", "reviewer": "Reviewer_c8Tz", "summary": "The paper studies the problem of exact community recovery from multiple ($K$) correlated graphs in 2-community balanced symmetric SBM. Prior work of [Gaudio, Ra\\'cz, and Sridhar 2022] for $K=2$ case. The paper generalizes their result for any $K$ (constant) number of graphs. In particular, the main result of the paper determines a sharp information-theoretic threshold in terms of $a,b,s$ (correlated SBM parameters) and $K$ such that \n1. (Theorem 1) Above the threshold, the optimal MAP estimator (not efficient though) achieves exact recovery with high probability. To show this, the main challenge is to combine the information from more than two networks when none of the pairs can be matched exactly.\n2. (Theorem 2) Below the threshold, any estimator fails to exactly recover the communities with high probability.\n\nIn particular, some interesting highlights from their results are that there is a region of parameter $(a,b,s)$ such that exact recovery is (i) impossible using $K-1$ graphs but possible using $K$ graphs AND (ii) matching the vertex labels of any of the graph pair is impossible.", "questions": "What do authors think about the challenges in handling weak recovery in a constant degree regime using correlated graphs? For weak recovery, it would be only good enough to create a partial overlap. \n\nThe rest of the questions are to clarify my understanding of Figure 2.\n1. Why pink region is not a finding of this paper (it only mentions violet)? Would it be correct to say that one of the findings is to characterize the boundary between pink and violet region, i.e. how the union of violet and pink splits into the two regions?\n2. Is the only difference between the pink and yellow regions that in the former, exact recovery would have been possible if any pair of matchings were known, but in the latter one needs all pairwise matchings? Otherwise, the two regions have the same properties: e.g. no pairwise matching is possible, not possible to recover the community using any individual pairs, yet possible using three graphs. Also, does the region of parameters given by pink+yellow exactly correspond to the region mentioned in abstract lines 12-15 (and as the highlights in the summary of the review).", "limitations": "Yes, the authors list out both negative and positive societal impacts of their work and graph matching in general.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "1. The paper provides a clean characterization of the precise information theoretic limit for an important problem in exact community recovery literature.\n2. The paper is well-written with clear intuitions of interplay between exact recovery and graph matching.", "weaknesses": "I do not see any major weaknesses in the paper.", "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\nWeak recovery in the constant average degree regime using correlated graphs is indeed a fascinating direction for future work (as already suggested in [18]). We suspect that the $k$-core analysis still gives something nontrivial in the large but constant average degree regime. However, proving this is far from immediate, due to a lack of concentration at this scale. Furthermore, determining the precise threshold for weak/partial recovery seems like a challenging problem which requires many more new ideas. \n\nRegarding Figure 2, indeed, the reviewer is completely right: characterizing both the pink and the violet regions, and the boundary between them, are all important findings of the paper. Previously, [18] characterized the union of these two regions. We will correct the caption of Figure 2, please see the attachment to the \"global\" response/rebuttal for an updated version. Thank you for catching this. \n\nRegarding the second set of questions about Figure 2, we hope that the \"global\" response/rebuttal -- which describes the threshold for exact graph matching from $K$ graphs (not just pairwise exact graph matching) -- helps clarify things. In particular, the previously yellow region now splits into an updated yellow region and a grey region. It is indeed the case that the union of the pink and the updated yellow regions is the region described in lines 12-15 of the abstract. We will add a sentence along these lines to the text in order to aid the reader.", "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": "PHwXcNuBiJ", "reviewer": "Reviewer_cuxD", "summary": "Given K correlated SBMs, the authors derive information-theoretic conditions for (i) the exact recovery of the community structure and (ii) the perfect recovery of the planted alignment $\\pi^*$.", "questions": "* Can you elaborate on why k-core matching is a good choice? In particular, going over the proof, it appears that k = 13 is used (Lemmas F.4, F.6), is the choice of k important?\n\n* Is there a low-degree hardness conjecture for the alignment of correlated SBMs?\n\n* What happens when $K$ grows unbounded?\n\n* Can one conjecture what happens with k \\ge 3 communities of equal size? Going over ref [43], it seems natural to replace the quantity T_c by (a+(k-1)b) / k (which by the way, multiplied by \\log n, is the average degree) and replace D_+ by (\\sqrt(a) - \\sqrt(b) )^2 / k.  \n\n* With k=2 communities of different sizes, could one get a different story than simply replacing the D_+ with the CH divergence? (a naive thinking: the step of almost exact recovery provides 2 communities of sizes n_1 < n_2 for G_1 and of sizes n_1' < n_2' for G_2'; thus I can already map the community of size n_1 with the one of size n_1'. Does it help?).", "limitations": "The theoretical results and their assumptions are clearly stated. The work is purely theoretical and does not require more discussion on societal impact.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "The paper is well-written, and I enjoyed reading it. It generalizes [37] and [18] to multiple (K \\ge 3) SBMs. The interplay between community recovery and graph alignment by combining the information from the K graphs is well-explained and well-executed.", "weaknesses": "No major weaknesses, the paper opens and closes the problem it intends to solve. The discussion section lays directions for interesting future works. I hesitate between 7 and 8. But for a higher grade, I would have liked to see a bit more (such as graphs with 2 communities of different sizes, or more than 2 communities, which I believe is not that much harder). In any case, the paper is a clear accept.\n\nMinor comments:\n\n* Since the authors consider SBM with two balanced communities and edge probabilities p&q, the term planted partition model may be more adapted.\n\n* In the planted partition model, the key information-theoretic quantity for exact recovery is the Rényi divergence of order 1/2 between two Bernoulli distributions. The CH divergence is only needed for SBM with general connection probabilities and/or block sizes. \n\n* Typo line 143: \"for K \\ge graphs\", 3 is missing.\n\n* It may be useful to define earlier \"almost exact community labeling\" and \"partial almost exact graph matching\" (which I believe are accounted first in lines 179 and 180, and not before, and define only lines 241).", "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 it is natural to consider more general settings, such as more than two communities, as well as unbalanced communities. We suspect that our methods can be extended to these settings as well. However, in the present paper we decided to keep the setting as simple as possible, to focus on the effect of more than two graphs. \n\nThank you for your comments on the planted partition model and the Renyi divergence; we will add these terminologies to the paper. Thank you also for spotting that typo, and for suggesting to move forward some definitions; we will make these edits in the revision. \n\nQ1 (on the $k$-core matching): The general intuition behind why the $k$-core matching performs well is as follows. The $k$-core of a graph captures its \"central\" part, in some sense. Thus if a vertex is part of the $k$-core of the intersection graph, then it has many common connections in the two graphs, so it is likely that these can help find the correct match for this vertex. \n\nThe choice of $k=13$ is not particularly important -- any large enough constant will work. The particular choice of $k=13$ comes from [18], where it arises in their Lemma 4.8 and Lemma 4.10. In particular, Lemma 4.10 is a probability tail bound, and the choice of $k \\geq 13$ (and in particular $k=13$) guarantees that this vanishes sufficiently quickly. The proof of Lemma 4.10 contains a union bound that may be loose, so it may be possible that the value of $k$ can be further lowered, but this doesn't have any major importance.\n\n\nQ2 (low-degree hardness): Yes, we conjecture that there is an information-computation gap for matching correlated SBMs, and we conjecture that this is the same as for matching correlated Erdos--Renyi random graphs, as in [28]. Namely, let $\\alpha \\approx 0.338$ denote Otter's tree counting constant [28]. We conjecture that if graph matching is information-theoretically feasible and $s^2 > \\alpha$, then there is an efficient algorithm to do so. On the other hand, if $s^2 < \\alpha$, then there does not exist such an efficient algorithm. Investigating this question is beyond the scope of the current paper. \n\n\nQ3 ($K$ unbounded): As $K \\to \\infty$ (and assuming $s \\in (0,1)$), the condition (3.1) converges to the condition that exact community recovery is possible in the parent graph $G$, while the condition (3.2) converges to the condition that $G_{1}$ is connected. The convergence in both cases is exponential in $K$, so for large constant $K$ the conditions will already be very close to the limiting conditions.  \n\nAn inspection of our proofs shows that our results also hold for $K$ growing slowly with $n$; we haven't tried to optimize the bounds to allow $K$ to grow as fast as possible. \n\nQ4 (3+ symmetric communities): We fully agree with the reviewer's conjecture (which generalizes the conjecture of reference [43] for $K=2$). We suspect that our methods can be extended to show this. However, in the present paper our aim was to focus on the effect of the number of graphs $K$, so we decided to keep other aspects of the setting as simple as possible. This is an interesting question for future work. \n\nQ5 (two unbalanced communities): This is a natural and intriguing question. However, we believe that unbalanced communities do not help. Note that while unbalanced communities allow the communities to be matched, what we really need is to match individual vertices, and this is not readily done simply by virtue of the communities being unbalanced. Note also that after the partial matching and the initial majority votes, all nodes that have been matched have a correct estimate of their community. However, the condition (3.2) really comes from the (number of) unmatched vertices, and the related genie-aided argument, which gives the CH divergence. Of course, the arguments above are not a proof, just some heuristics, and we leave resolving this is as an interesting open question for the future.", "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": "sw6l4sO6Oa", "reviewer": "Reviewer_b7qh", "summary": "The paper studies the problem of exact community detection in correlated stochastic block models (with two symmetric communities). More precisely, a graph is sampled from an SBM with parameters p and q and each edge in the graph is then kept with probability q. This downsampling is performed K times independently at random to obtain K different graphs. Each graph is then randomly permuted. From these K correlated sample we want to recover the communities of the model.\n\nThe authors provide a necessary and sufficient condition on the parameters p,q,k, and s for exact recovery. \nIn particular, they show there exists a regime of p,q,s where K-1 graphs are not enough to recover exactly the communities, but K samples are enough. In particular, there exists a regime in which K samples might not even be enough to solve graph matching perfectly (i.e., recover the random permutation). Therefore, it is necessary to aggregate imperfect matching information with techniques for community detection.\n\nThe case for K=2 has been solved by Gaudio, Rácz, and Sridhar. This paper extends their result for any fixed K>=2.", "questions": "Just a few questions that are essentially out of curiosity:\n1. Can some of these techniques be used to analyse the sparse case? Of course we wouldn't aim for exact recovery, but rather partial recovery, and we would need to use different algorithms to obtain a good community labelling of the individual graphs.\n1. I guess the limit of the recovery probability depends on K and that's the reason K needs to be constant? How bad is its dependency on K? Can K diverge at least slightly?\n1. I know this is not the point of this paper, but I wonder if there is a practical application of the techniques developed in the paper. It'd be quite interesting to try and apply them to obtain an algorithm that work in practice, even without theoretical guarantees.", "limitations": "Yes.", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "I think the problem is a natural and interesting one. The extension from the case of two to more correlated graphs appears highly nontrivial, since there are many ways in which one could try and resolve potentially conflicting information arising from matching different pairs of graphs.\n\nThe paper is also well written, both in explaining the background and motivation to the problem, and also when giving a flavour of the techniques used in the analysis.", "weaknesses": "Perhaps the main weakness of the paper is that the analysis is not algorithmic, in the sense that there is not an efficient algorithm able to recover the communities. This is, however, more an invitation to future work rather than a flaw of the paper itself.\n\nThe other point I'd like to raise is that the proofs in this paper are fairly lengthy and involved. Is NeurIPS the best venue for these kind of papers? I believe the paper will be interesting to the NeurIPS community and that's why I recommend acceptance, but in a perfect world I'd like results of this kind to be also checked for correctness (disclaimer: I did not read the appendix).", "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 fully agree with the reviewer that the main open problem that our work leaves open is to develop efficient algorithms for the setting studied in our paper (whenever this is possible). The quest for efficient algorithms has been a major driving force behind the recent surge of papers on random graph matching (see Section 5), culminating in the breakthrough works [27] and [28] on correlated Erdos--Renyi random graphs. These works give promise that it is possible to develop efficient algorithms in our setting (perhaps with some conditions on the correlation parameter $s$). Our work lays the groundwork by determining the information-theoretic limits, which provide targets for efficient algorithms to achieve. \n\nQ1 (sparse setting): Weak recovery in the constant average degree regime using correlated graphs is indeed a fascinating direction for future work (as already suggested in [18]). We suspect that the $k$-core analysis still gives something nontrivial in the large but constant average degree regime. However, proving this is far from immediate, due to a lack of concentration at this scale. Furthermore, determining the precise threshold for weak/partial recovery seems like a challenging problem which requires many more new ideas. \n\nQ2 ($K$ diverging with $n$): An inspection of our proofs shows that our results hold also when $K$ diverges (slowly) with $n$. We have not made an attempt to see how fast $K$ can grow with $n$ to have the results still hold, in part because the conditions (3.1) and (3.2) converge exponentially in $K$, so they are already very close to their limiting values for large constant $K$. \n\nThat said, as an example, let us consider one particular aspect of the proof -- the correctness of the $k$-core estimator -- and see what we may allow $K$ to be here. Lemma F.4 says that the result holds with high probability; in other words, that the error probability is $o(1)$. A careful inspection of the proof of Lemma 4.8 in [18] actually shows that the error probability is bounded by $n^{-0.16}$ (moreover, if $k$ in the $k$-core is chosen to be a large constant, then this error probability can be made to be an arbitrarily small inverse polynomial in $n$). Since in a union bound over $K$ graphs we gain a factor of at most $K^{2}$, this shows that this lemma allows $K$ to grow polynomially in $n$. Of course, there are other aspects of the proof where the dependence on $K$ of the error probabilities have to be checked, and together these can give a full answer of how fast $K$ can grow with $n$. \n\nQ3 (practical applications): We fully agree with the reviewer and we hope that such applications will arise in the future, motivated by our (theoretical) work.", "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:
Thanks to all reviewers for the careful reviews and many helpful comments! 

We respond to each review separately in its own rebuttal. In this "global" response, we take the opportunity to discuss something that was brought up by multiple reviewers: the threshold for exact graph matching given $K$ correlated SBMs. 

In the submitted manuscript (on page 4), we mention that the information-theoretic threshold for exact graph matching for two correlated SBMs is $s^{2} T_{c}(a,b) = 1$ (by [37]). So, in particular, by a union bound, above this threshold one can do pairwise exact graph matching for any constant number $K$ correlated SBMs. However, this does not imply that this is the threshold for exact graph matching given $K$ correlated SBMs (it only gives a one-sided bound). And indeed: the threshold for exact graph matching given $K$ correlated SBMs is given by a different threshold, namely 

$s \left( 1 - \left( 1 - s \right)^{K-1} \right) T_{c}(a,b) = 1$.

More formally, we have the following two theorems: 

Theorem: Fix constants $a,b>0$ and $s\in [0,1]$, and let $(G_{1},G_{2},\ldots,G_{K})\sim \mathrm{CSBM}(n,\frac{a\log n}{n},\frac{b\log n}{n},s)$. Suppose that $s \left( 1 - \left( 1 - s \right)^{K-1} \right) T_{c}(a,b) > 1$. Then exact graph matching is possible. That is, there exists an estimator $\widehat{\pi}=\widehat{\pi}(G_1,\ldots,G_K)$ such that $\lim_{n\to\infty}\mathbb{P}(\widehat{\pi}=\pi^*)=1$.

Theorem: Fix constants $a,b>0$ and $s\in [0,1]$, and let $(G_{1},G_{2},\ldots,G_{K})\sim \mathrm{CSBM}(n,\frac{a\log n}{n},\frac{b\log n}{n},s)$. Suppose that $s \left( 1 - \left( 1 - s \right)^{K-1} \right) T_{c}(a,b) < 1$. Then exact graph matching is impossible. That is, for every estimator $\widehat{\pi}=\widehat{\pi}(G_1,\ldots,G_K)$ we have that $\lim_{n\to\infty}\mathbb{P}(\widehat{\pi}=\pi^*)=0$.

Note, in particular, that for $K \geq 3$ there exists a regime (specifically given by $s^{2} T_{c}(a,b) < 1 < s \left( 1 - \left( 1 - s \right)^{K-1} \right) T_{c}(a,b)$) where exact graph matching is possible from $K$ correlated SBMs even though pairwise exact graph matching is impossible. 

We emphasize that the proofs of these theorems (which we sketch below) are already implicit in our submitted manuscript; in fact, our proofs for community exact recovery go significantly beyond these. In our original submitted manuscript we didn't mention these theorems, in part since our focus is on exact community recovery, and in part due to length constraints. Your feedback has made us realize that we should indeed include these in the paper. We plan on adding these in the revised version of the manuscript (with the statements in the main text, using the additional allowed page, and proofs in the appendix).

It is worth highlighting this threshold in the phase diagrams as well. We have attached here an updated version of Figure 2 with this incorporated. The change is that the previously yellow region (in the submitted version) now breaks into two regions: a yellow region and a grey region. In the grey region, exact community recovery is impossible from $(G_{1}, G_{2})$, pairwise exact graph matching is also impossible, but exact graph matching given $(G_{1}, G_{2}, G_{3})$ is possible, and subsequently exact community recovery is possible from $(G_1, G_2, G_3)$. In the (updated) yellow region, exact community recovery is impossible from $(G_{1}, G_{2})$, exact graph matching given $(G_{1}, G_{2}, G_{3})$ is also impossible, yet exact community recovery is possible given $(G_1,G_2,G_3)$. 

We now sketch the proofs, starting with the impossibility result. Suppose that we give the algorithm even more information, namely all the matchings between the graphs $G_2, \ldots, G_K$. Then the correctly matched union graph $G_2 \vee \ldots \vee G_K$ can be computed (and, by a simulation argument (similar to the proof of Theorem 3.4 in [37]) it suffices to only consider this union graph). Note that $G_2 \vee \ldots \vee G_K$ is an SBM with parameters $(n, \left( 1 - \left( 1 - s \right)^{K-1} \right) a \log(n)/n, \left( 1 - \left( 1 - s \right)^{K-1} \right) b \log(n)/n)$. By a result of Cullina, Singhal, Kiyavash, and Mittal (2016), it is impossible to exactly recover the matching between an aforementioned SBM and an SBM with parameters $(n, s a \log(n)/n, s b \log(n)/n)$ under the condition $s \left( 1 - \left( 1 - s \right)^{K-1} \right) T_{c}(a,b) < 1$. 

For the possibility result, the key is Appendix I. Under the condition $s \left( 1 - \left( 1 - s \right)^{K-1} \right) T_{c}(a,b) > 1$, one can see that the conclusion of Lemma I.2 is that the intersection set of interest is empty, which implies that all vertices are "good" vertices. The algorithm described in Appendix I then shows how to recover the latent matchings.

Meta-review: The paper studies the problem of exact community recovery in $K$ correlated 2-community stochastic block models (SBMs) where node labels are not maintained between networks. The authors extend previous work for the $K=2$ case, introducing new challenges and proof mechanisms to allow for $K \geq 3$ networks. The main results of the paper are Theorems 1 and 2, which provide necessary and sufficient conditions for exact community recovery, relating to the difficulty of the pairwise matching problem and the individual exact community recovery problem.

All reviewers agree that the paper is well-written, with clear explanations of the problem and the interplay between community recovery and graph matching. The paper successfully answers the open question, generalizing the results to multiple ($K \geq 3$) SBMs. The reviewers appreciate the paper's well-structured presentation. The questions raised by the reviewers are well addressed. 

Overall, this paper is technically sound, well-presented, and provides a solid foundation for future work in this area.
Decision: Accept (poster)