Title: Exact Random Graph Matching with Multiple Graphs
Venue: NeurIPS 2024
Accepted: no
Keywords: Graph Matching

Network Alignment

Random Graphs

Graphical Inference
Abstract: This work studies fundamental limits for recovering the underlying correspondence among multiple correlated random graphs. We identify a necessary condition for any algorithm to correctly match all nodes across all graphs, and propose two algorithms for which the same condition is also sufficient. The first algorithm employs global information to simultaneously match all the graphs, whereas the second algorithm first partially matches the graphs pairwise and then combines the partial matchings by transitivity. Remarkably, both algorithms work down to the information theoretic threshold. Our analysis reveals a scenario where exact matching between two graphs alone is impossible, but leveraging more than two graphs allows exact matching among all the graphs. Along the way, we derive independent results about the $k$-core of Erdos-Renyi graphs.

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Abstract

1 Introduction

The information age has ushered an abundance of correlated networked data. For instance, the 13 network structure of two social networks such as Facebook and Twitter is correlated because users are 14 likely to connect with the same individuals in both networks. This wealth of correlated data presents 15 both opportunities and challenges. On one hand, information from various datasets can be combined 16 to increase the fidelity of data - translating to better performance in downstream learning tasks. On the 17 other hand, the interconnected nature of this data also raises privacy and security concerns. Linkage 18 attacks, for instance, exploit correlated data to identify individuals in an anonymized network by 19 linking to other sources [NS09]. This poses a significant threat to user privacy. 20

21 lated networks. The problem finds many applications in machine learning: de-anonymizing social 22 networks [NS08,NS09], identifying similar functional components between species by matching 23 their protein-protein interaction networks [BSI06,KHGPM16], object detection [SS05] and track24 ing [YYL + 16] in computer vision, and textual inference for natural language processing [HNM05]. In 25 most applications of interest, data is available in the form of several correlated networks. For instance, 26 social media users are active each month on 6.7 social platforms on average [Ind23]. Similarly, 27 reconciling protein-protein interaction networks among multiple species is an important problem in 28 computational biology [SXB08]. As a first step toward this objective, many research works have 29 studied the problem of matching two correlated graphs. 30

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1.1 Related Work

The theoretical study of graph matching algorithms and their performance guarantees has primarily 32

focused on Erd˝ os-Rényi (ER) graphs. Pedarsani and Grossglauser [PG11] introduced the subsampling 33

model to generate two such correlated graphs. The model entails twice subsampling each edge 34

independently from a parent ER graph to obtain two sibling graphs, both of which are marginally 35

ER graphs themselves. The goal is then to match nodes between the two graphs to recover the 36

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underlying latent correspondence. This has been the framework of choice for many works that study graph matching. For example, Cullina and Kiyavash studied the problem of exactly matching two ER graphs, where the objective is to match all vertices correctly [CK16,CK17]. They identified a threshold phenomenon for this task: exact recovery is possible if the problem parameters are above a threshold, and impossible otherwise. Subsequently, threshold phenomena were also identified for partial graph matching between ER graphs - where the objective is to match only a positive fraction of nodes [GML21,HM23,WXY22,DD23]. The case of almost-exact recovery - where the objective is to match all but a negligible fraction of nodes - was studied by Cullina and co-authors: a necessary condition for almost exact recovery was identified, and it was shown that the same condition is also sufficient for the k -core estimator [CKMP19]; the estimator is described formally in Section 3. This estimator proved useful to uncover the fundamental limits for graph matching in other contexts such as the stochastic block model [GRS22] and inhomogeneous random graphs [RS23]. Ameen and Hajek [AH23] showed some robustness properties of the k -core estimator in the context of matching ER graphs under node corruption. The estimator plays an important role in the present work as well.

A sound understanding of ER graphs inspires algorithms for real-world networks. Various efficient algorithms have been proposed, including algorithms based on the spectrum of the graph adjacency matrices [FMWX22], node degree and neighborhood based algorithms [DCKG19,DMWX21,MRT23] as well as algorithms based on iterative methods [DL23] and counting subgraphs [MWXY23,BCL + 19]. Some of these are discussed in Section 5 in relation to the present work.

Incorporating information from multiple graphs to match them has been recognized as an important research direction, for instance in the work of Gaudio and co-authors [GRS22]. To our knowledge, the only other papers to consider matchings among multiple graphs are the works of Josephs and co-authors [JLK21], and of Rácz and Sridhar [RS21]. However, these works have different objectives and are not concerned with the fundamental limits for matching m graphs. In fact, both works note that it is possible to exactly match m graphs whenever it is possible to exactly match any two graphs by pairwise matching all the graphs exactly. In contrast, we show that under appropriate conditions, it is possible to exactly match m ER graphs even when no two graphs can be pairwise matched exactly.

Contributions In this work, we investigate the problem of combining information from multiple correlated networks to boost the number of nodes that are correctly matched among them. We consider the natural generalization of the subsampling model to generate m correlated random graphs, and identify a threshold such that it is impossible for any algorithm to match all nodes correctly across all graphs when the problem parameters are below this threshold. Conversely, we show that exact recovery is possible above the threshold. This characterization generalizes known results for exact graph matching when m = 2 . Subsequently, we show that there is a region in parameter space for which exactly matching any two graphs is impossible using only the two graphs, and yet exact graph matching is possible among m> 2 graphs using all the graphs.

We present two algorithms and prove their optimality for this task. The first algorithm matches all m graphs simultaneously based on global information about the graphs. In contrast, the second algorithm first pairwise matches graphs, and then combines them to match all nodes across all graphs. We show that both algorithms correctly match all the graphs all the way down to the information theoretic threshold. Finally, we illustrate through simulation that our subroutine to combine information from pairwise comparisons between networks works well when paired with efficient algorithms for graph matching. Our analysis also yields some theoretical results about the k -core of ER graphs that are of independent interest.

2 Preliminaries and Setup

Notation In this work, G ∼ ER ( n, p ) denotes that the graph G is sampled from the Erd˝ os-Rényi distribution with parameters n and p , i.e. G has n nodes and each edge is independently present with probability p . For a graph G , we denote the set of its vertices by V ≡ V ( G ) and its edges by E ( G ) . The edge status of each vertex pair { i, j } with i = j is denoted by G { i.j } , so that G { i, j } = 1 if { i, j } ∈ E ( G ) and G { i, j } = 0 otherwise. The degree of a node v in graph G is denoted δ G ( v ) . Let π denote a permutation on V ( G ) = { 1 , · · · , n } . For a graph G , denote by G π the graph obtained by permuting the nodes of G according to π , so that

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Subsampling model

with parameters

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subsample

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Subsampling model Consider the subsampling model for correlated random graphs [PG11], which 90 has a natural generalization to the setting of m graphs. In this model, a parent graph G is sampled 91 from the Erd˝ os-Rényi distribution ER ( n, p ) . The m graphs G 1 , G ′ 2 , · · · , G ′ m -1 , G ′ m are obtained by 92 independently subsampling each edge from G with probability s . Finally, the graphs G 2 , · · · , G m 93 are obtained by permuting the nodes of each of the graphs G ′ 2 , · · · , G ′ m respectively according to 94 independent permutations π ∗ 12 , · · · , π ∗ 1 m sampled uniformly at random from the set of all permutations 95 on [ n ] , i.e. 96

Figure 1 illustrates this process of obtaining correlated graphs using the subsampling model. In this work, we are interested in the setting where s is constant and p = C log( n ) /n for some C > 0 .

Objective 1. Determine conditions on parameters C , s and m so that given correlated graphs G 1 , · · · , G m from the subsampling model, it is possible to exactly recover the underlying correspondences π ∗ 12 , · · · , π ∗ 1 m with probability 1 -o (1) .

Stated thus, the underlying correspondences use the graph G 1 as a reference. Thus, for ease of notation, we will use G 1 and G ′ 1 interchangeably. Note that the underlying correspondence between all the graphs is fixed upon fixing π ∗ 12 , · · · , π ∗ 1 m : for any two graphs G i and G j , their underlying correspondence is given by π ∗ ij := π ∗ 1 j ◦ ( π ∗ 1 i ) -1 .

Formally, a matching ( µ 12 , · · · , µ 1 m ) is a collection of injective functions with domain dom ( µ 1 i ) ⊆ V for each i , and co-domain V . An estimator is simply a mechanism to map any collection of graphs ( G 1 , · · · , G m ) to a matching. We say that an estimator completely matches the graphs if the output mappings µ 12 , · · · µ 1 m are all complete, i.e. they are all permutations on { 1 , · · · , n } .

3 Main Results and Algorithm

This section presents necessary and sufficient conditions to meet Objective 1.

Theorem 2 (Impossibility) . Let G 1 , · · · , G m be correlated graphs obtained from the subsampling model with parameters C and s , and let π ∗ 12 , · · · , π ∗ 1 m denote the underlying latent correspondences between G 1 and G 2 , · · · , G m respectively. Suppose that

The output ̂ π 12 , · · · , ̂ π 1 m of any estimator satisfies

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Theorem 2 implies that the condition Cs (1 -(1 -s ) m -1 > 1 is a necessary condition to exactly match m graphs with probability bounded away from 0 . We show that this condition is also sufficient to exactly match m graphs with probability going to 1 .

Theorem 3 (Achievability) . Let G 1 , · · · , G m be correlated graphs obtained from the subsampling model with parameters C and s , and let π ∗ 12 , · · · , π ∗ 1 m denote the underlying latent correspondences between G 1 and G 2 , · · · , G m respectively. Suppose that

There is an estimator whose output ̂ π 12 , · · · , ̂ π 1 m satisfies

Theorems 2 and 3 together characterize the threshold for exact recovery. A few remarks are in order.

For m = 2 , the condition Cs (1 -(1 -s ) m -1 ) > 1 reduces to Cs 2 > 1 , which is known to be necessary and sufficient for exactly matching two graphs [CK17,WXY22].

For any m> 2 , there is a non-empty region in the parameter space defined by

For any C and s in this region, it is impossible to exactly match any two graphs G i and G j without using the other m -2 graphs as side information. Upon using them, however, it is possible to exactly match all nodes across the m graphs. This is illustrated in Figure 2.

3.1 Algorithms for exact recovery

For any two graphs H 1 and H 2 on the same vertex set V , denote by H 1 ∨ H 2 their union graph and by H 1 ∧ H 2 their intersection graph . An edge { i, j } is present in H 1 ∨ H 2 if it is present in either H 1 or H 2 . Similarly, the edge is present in H 1 ∧ H 2 if it is present in both H 1 and H 2 .

A natural starting point is to study the maximum likelihood estimator (MLE) because it is optimal. To that end, we compute the log-likelihood function; the details are deferred to Appendix A.

Theorem 4. Let π 12 , · · · , π 1 m denote a collection of permutations on { 1 , · · · , n } . Then

where const. depends only on p, s and G 1 , · · · , G m .

Theorem 4 reveals that the MLE for exactly matching m graphs has a neat interpretation: simply pick π 12 , · · · , π 1 m to minimize the number of edges in the corresponding union graph. This is presented as Algorithm 1. Despite this nice interpretation of the MLE, its analysis is quite cumbersome. We instead present and analyze a different estimator, presented as Algorithm 2.

Algorithm 1: Maximum likelihood estimator

Algorithm 2: Matching through transitive closure

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turns out that this is sufficient for exact recovery. An analysis of Algorithm 2 is presented in Section 4. Along the way, independent results of interest on the k -core of Erd˝ os-Rényi graphs are obtained.

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4 Proof Outlines and Key Insights

4.1 Impossibility of exact graph matching (Theorem 2)

This result has a simple proof following a genie-aided converse argument. The idea is to reduce the problem to that of matching two graphs by providing extra information to the estimator.

Proof of Theorem 2 . If the correspondences π ∗ 12 , · · · , π ∗ 1 ,m -1 were provided as extra information to an estimator, then the estimator must still match G m with the union graph G ′ 1 ∨ G ′ 2 ∨ · · · ∨ G ′ m -1 . This can be viewed as an instance of matching two graphs obtained by asymmetric subsampling: the graph G m is obtained from a parent graph G ∼ ER ( n, C log( n ) /n ) by subsampling each edge independently with probability s 1 := s , and the graph ˜ G m -1 := G ′ 1 ∨ G ′ 2 ∨ · · · ∨ G ′ m -1 is obtained from G by subsampling each edge independently with probability s 2 := 1 -(1 -s ) m -1 . Cullina and Kiyavash studied this model for matching two graphs: Theorem 2 of [CK17] establishes that matching G m and ˜ G m -1 is impossible if Cs 1 s 2 < 1 , or equivalently if Cs (1 -(1 -s ) m -1 ) < 1 .

4.2 Achievability of exact graph matching (Theorem 3)

Algorithm 2 succeeds if both step 1 and step 2 succeed, i.e.

Each instance of pairwise matching using the k -core estimator is correct on its domain, i.e.

For each node v and any two graphs G i and G j , there is a sequence of graphs such that v can be transitively matched through those graphs between G i and G j .

On step 1 This falls back to the regime of analyzing the performance of the k -core estimator in the setting of two graphs. Cullina and co-authors [CKMP19] showed that the k -core estimator is precise : For any two correlated graphs G i and G j with p = C log( n ) /n and constant s , the k -core estimator correctly matches all nodes in core k ( G ′ i ∧ G ′ j ) with probability 1 -o (1) . In fact, this is true for any C > 0 and for any k ≥ 13 [RS23]. Therefore, using the fact that the number of instances of pairwise matchings is constant whenever m is constant, a union bound reveals

̸

We have proved the following.

Proposition 5. Let G 1 , · · · , G m be correlated graphs from the subsampling model. Let k ≥ 13 and let ̂ µ ij denote the matching output by the k -core estimator on graphs G i and G j . Then,

̸

On step 2 The challenging part of the proof is to show that boosting through transitive closure matches all the nodes with probability 1 -o (1) if Cs (1 -(1 -s ) m -1 ) > 1 . It is instructive to visualize this using transitivity graphs .

Definition 6 (Transitivity graph, H ( v ) ) . For each node v ∈ V , let H ( v ) denote the graph on the vertex set { g 1 , · · · , g m } such that an edge { g i , g j } is present in H ( v ) if and only if v ∈ core k ( G ′ i ∧ G ′ j ) .

On the event that each instance of pairwise matching using the k -core is correct, the edge { g i , g j } is present in H ( v ) if and only if v is correctly matched using the k -core estimator between G i and G j , i.e. π ∗ 1 i ( v ) is matched to π ∗ 1 j ( v ) . Thus, in order for Step 2 to succeed (i.e. to exactly match all vertices across all graphs), it suffices that the graph H ( v ) is connected for each node v ∈ V . However, studying the connectivity of the transitivity graphs is challenging because in any graph H ( v ) , no two edges are independent. This is because the k -cores of any two intersection graphs G ′ a ∧ G ′ b and G ′ c ∧ G ′ d are correlated, because all the graphs G a , G b , G c and G d are themselves correlated. To overcome this, we introduce another graph ˜ H ( v ) that relates to H ( v ) and is amenable to analysis.

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Definition 7. For each node v ∈ V , let ˜ H ( v ) denote a complete weighted graph on the vertex set { g 1 , · · · , g m } such that the weight on any edge { g i , g j } is ˜ c v ( i, j ) := δ G ′ i ∧ G ′ j ( v ) .

The relationship between the graphs H ( v ) and ˜ H ( v ) stems from a useful relationship between the degree of node v in G ′ i ∧ G ′ j and the inclusion of v in core k ( G ′ i ∧ G ′ j ) for each i and j . Since this result is of independent interest in the study of random graphs, we state it below for general Erd˝ os-Rényi graphs.

Lemma 8. Let n and k be positive integers and let G ∼ ER ( n, α log( n ) /n ) for some α > 0 . Let v be a node of G and let δ G ( v ) denote the degree of v in G . Then,

For any i and j , the graph G ′ i ∧ G ′ j ∼ ER ( n, Cs 2 log( n ) /n ) . Thus, Lemma 8 implies that with probability 1 -o (1 /n ) , if a pair { g i , g j } has edge weight ˜ c ij ≥ k +1 /α in ˜ H ( v ) , then the corresponding edge { g i , g j } is present in the transitivity graph H ( v ) . Equivalently, v is correctly matched between G i and G j in the instance of pairwise k -core matching between them.

The graph H ( v ) is not connected only if it contains a (non-empty) vertex cut U ⊂ { 1 , · · · , m } with no edge crossing between U and U c . Let c v ( U ) denote the number of such crossing edges in H ( v ) . Furthermore, define the cost of the cut U in ˜ H ( v ) as

Lemma 8 is a statement about a single graph, but we show it can be invoked to prove the following.

Theorem 9. Let G 1 , · · · , G m be correlated graphs from the subsampling model with parameters C and s . Let v ∈ V and let U be a vertex cut of { 1 , · · · , m } such that | U | ≤ ⌊ m/ 2 ⌋ . Then,

It suffices therefore to analyze the probability that the graph ˜ H ( v ) has a cut U such that its cost ˜ c v ( U ) is too small. To that end, we show that the bottleneck arises from vertex cuts of small size. Formally,

Theorem 10. Let G 1 , · · · , G m be correlated graphs from the subsampling model. Let v ∈ V and let U ℓ denote the set { 1 , · · · , ℓ } for ℓ in { 1 , · · · , ⌊ m/ 2 ⌋} . For any vertex cut U of { 1 , · · · , m } , let ˜ c v ( U ) denote its cost in the graph ˜ H ( v ) . The following stochastic ordering holds:

Theorems 9 and 10 imply that the tightest bottleneck to the connectivity of H ( v ) is the event that ˜ c v ( U 1 ) is below the threshold r := m 2 4 ( k + 1 Cs 2 ) , i.e. the sum of degrees of v over the intersection graphs ( G 1 ∧ G ′ j : j = 2 , · · · , m ) is less than r . This event occurs only if the degree of v is less than r in each of the intersection graphs ( G 1 ∧ G ′ j : j = 2 , · · · , m ) . However, under the condition Cs (1 -(1 -s ) m -1 ) > 1 , it turns out that this event occurs with probability o (1 /n ) .

Theorem 11. Let G 1 , · · · , G m be obtained from the subsampling model with parameters C and s . Let r = m 2 4 ( k + 1 Cs 2 ) . Let v ∈ [ n ] and suppose that Cs (1 -(1 -s ) m -1 ) > 1 . Then,

4.3 Piecing it all together: Proof of Theorem 3

Proof of Theorem 3. Let ̂ π 12 , · · · , ̂ π 1 m denote the output of Algorithm 2 with k ≥ 13 . Let E 1 (resp. E 2 ) denote the event that Algorithm 1 (resp. Algorithm 2) fails to match all m graphs exactly, i.e.

̸

̸

̸

̸

First, we show that the output of Algorithm 2 is correct with probability 1 -o (1) whenever Cs (1 -228 (1 -s ) m -1 ) > 1 . If the event E 2 occurs, then either step 1 failed, i.e. there is a k -core matching ̂ µ ij 229 that is incorrect, or step 2 failed, i.e. at least one of the graphs H ( v ) is not connected. Therefore, 230

̸

where the last step uses Proposition 5, and q v denotes the probability that the transitivity graph H ( v ) 231 is not connected. For each ℓ in the set { 1 , · · · , ⌊ m/ 2 ⌋} , let U ℓ denote the set { 1 , · · · , ℓ } . Then, 232

Here, (a) uses Theorem 9, and (b) uses the fact that for any ℓ ≥ 2 , the random variable ˜ c v ( U ℓ ) 233 stochastically dominates ˜ c v ( U 1 ) (Theorem 10). Finally, (c) uses Theorem 11 and the fact that 234 Cs (1 -(1 -s ) m -1 ) > 1 . Therefore, a union bound over all the nodes yields 235

Finally, by optimality of the MLE, it follows that

5 Discussion and Future Work

In this work, we introduced and analyzed matching through transitive closure - an approach that combines information from multiple graphs to recover the underlying correspondence between them. Despite its simplicity, it turns out that matching through transitive closure is an optimal way to combine information in the setting where the graphs are pairwise matched using the k -core estimator. A limitation of our algorithms is the runtime: Algorithm 2 does not run in polynomial time because it uses the k -core estimator for pairwise matching, which involves searching over the space of permutations. Even so, it is useful to establish the fundamental limits of exact recovery, and serve as a benchmark to compare the performance of any other algorithm.

The transitive closure subroutine (Step 2) itself is efficient because it runs in polynomial time O ( mn ) . Therefore, a natural next step is to modify Step 1 in our algorithm so that the pairwise matchings are done by an efficient algorithm. However, it is not clear if transitive closure is optimal for combining information from the pairwise matchings in this setting. For example, there is a possibility that the pairwise matchings resulting from the efficient algorithm are heavily correlated, and transitive closure is unable to boost them. In Figure 3, we show experimentally that this is not the case for two algorithms of interest: GRAMPA [FMWX22] and Degree Profiles [DMWX21].

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GRAMPA,

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GRAMPA is a spectral algorithm that uses the entire spectrum of the adjacency matrices to match the two graphs. The code is available in [FMWX20].

Degree Profiles associates with each node a signature derived from the degrees of its neighbors, and matches nodes by signature proximity. The code is available in [DMWX20].

Evidently, both algorithms benefit substantially from using transitive closure to boost the number of matched nodes. This suggests that transitive closure can be a practical algorithm to boost matchings between networks by using other networks as side-information. Unfortunately, both GRAMPA and Degree Profiles require the graphs to be close to isomorphic in order to perform well, and so they do not perform well when the model parameters are close to the information theoretic threshold for exact recovery. Subsequently, they cannot be used to answer the question in Objective 1.

Our work presents several directions for future research.

Polynomial-time algorithms. Using a polynomial-time estimator in place of the k -core estimator in Step 1 of Algorithm 2 yields a polynomial-time algorithm to match m graphs. It is critical that the estimator in question is able to identify for itself the nodes that it has matched correctly - this precision is present in the k -core estimator and enables the transitive closure subroutine to work correctly. Can the performance guarantees of the k -core estimator be realized through polynomial time algorithms that meet this constraint?

Beyond Erd˝ os-Rényi graphs. The study of matching two ER graphs provided tools and techniques that extended to the analysis of more realistic models. For instance, the k -core estimator itself played a crucial role in establishing limits to matching two correlated stochastic block models [GRS22] and two inhomogeneous random graphs [RS23]. Can the techniques developed in the present work be used to identify the information theoretic limits to exact recovery in these models in the general setting of m graphs?

Boosting for partial recovery. This work focused on exact recovery, where the objective is to match all nodes across all graphs. It would be interesting to consider a regime where any instance of pairwise matching recovers at best a small fraction of nodes. Is it possible to quantify the extent to which transitive closure boosts the number of matched nodes?

Robustness. Finally, how sensitive to perturbation is the transitive closure algorithm? Is it possible to quantify the extent to which an adversary may perturb edges in some of the graphs without losing the performance guarantees of the matching algorithm? Algorithms that perform well on models such as ER graphs and are further generally robust are expected to also work well with real-world networks.

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A Maximum Likelihood Estimator 366

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Recall the form of the maximum likelihood estimator as claimed in Theorem 4.

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Theorem 4. Let π 12 , · · · , π 1 m denote a collection of permutations on { 1 , · · · , n } . Then

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where const. depends only on p, s and G 1 , · · · , G m .

Proof. Notice that 370

where for a node pair e = { u, v } , the shorthand π ( e ) denotes { π ( u ) , π ( v ) } . The edge status of 371 any node pair e in the graph tuple ( G 1 , G ′ 2 , · · · , G ′ m ) can be any of the 2 m bit strings of length 372 m , but the corresponding probability in (3) depends only on the number of ones and zeros in 373 the bit string. For i ∈ [ m ] , let α i denote the number of node pairs e whose corresponding tuple 374 ( G 1 ( e ) , G ′ 2 ( e ) , · · · , G ′ m ( e )) has exactly i 1's: 375

Two key observations are in order. First, it follows by definition that α 0 + α 1 + · · · + α m = ( n 2 ) . 376 Second, by definition of α i , it follows that 377

is constant, independent of π ∗ 12 , · · · , π ∗ 1 m . It follows then that 378

where the last step uses (4). Finally, since 1 -p p (1 -s ) m > 0 , it follows that the log-likelihood satisfies 379

i.e. maximizing the likelihood corresponds to selecting π 12 , · · · , π 1 m to maximize α 0 - the number of node pairs e for which G 1 ( e ) = G 2 ( π 12 ( e )) = · · · = G m ( π 1 m ( e )) = 0 . This is equivalent to minimizing the number of edges in the union graph G 1 ∨ G π 12 2 ∨ · · · ∨ G π 1 m m , as desired.

Remark 12. In the case of two graphs, minimizing the number of edges in the union graph G 1 ∨ π G 2 is equivalent to maximizing the number of edges in the intersection graph G 1 ∧ π G 2 . This is consistent with existing literature on two graphs [CK16,WXY22].

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Proof. All proofs follow from the Chernoff bound and can be found, or easily derived, from Theorems 4.4 and 4.5 of [MU17].

C Proof of Lemma 8

We restate Lemma 8 for convenience.

Lemma 8. Let n and k be positive integers and let G ∼ ER ( n, α log( n ) /n ) for some α > 0 . Let v be a node of G and let δ G ( v ) denote the degree of v in G . Then,

Before presenting the proof, we present the intuition behind it. The events { v / ∈ core k ( G ) } and { δ G ( v ) ≥ k +1 /α } are highly negatively correlated. However, consider the subgraph ( G -v ) of G induced on the vertex set V -{ v } , and note that the k -core of this subgraph does not depend on the degree of v . Furthermore, if v / ∈ core k ( G ) , then it must be that v has fewer than k neighbors in core k ( G -v ) . Intuitively, this event has low probability if core k ( G -v ) is sufficiently large.

Notice that ( G -v ) ∼ ER ( n -1 , α log( n ) /n ) , and so standard results about the size of the k -core of Erd˝ os-Rényi graphs apply. However, we require the error probability that the k -core of G -v is too small to be o (1 /n ) - this is crucial since we will later use a union bound over all the nodes v . Unfortunately, standard results such as [Łuc91] can only be invoked directly to show that the corresponding probability is o (1) , which is insufficient for our purpose. Later in this section, we refine the analysis in [Łuc91] to obtain the desired convergence rate. The refinement culminates in the following.

Lemma 14. Let α > 0 and G ∼ ER ( n, α log( n ) /n ) . Let v be a node of G . The size of the k -core of G -v satisfies

The proof of Lemma 14 is deferred to Appendix C.1. It remains to study the error event that v has too few neighbors in core k ( G -v ) . To count the number of neighbors of v in core k ( G -v ) , we exploit the independence of core k ( G -v ) and v as follows: each neighbor of v is considered a success if it belongs to core k ( G -v ) and a failure otherwise. Counting the number of successes is equivalent to sampling with replacement δ G ( v ) elements, each of which is independently a success with probability | core k ( G -v ) | / ( n -1) . The number of successes then follows precisely a hypergeometric distribution. This intuition is made rigorous in the proof below.

Let us recall some facts about the hypergeometric distribution because it plays an important role in 419 the proof. Denote by HypGeom ( N,K,n ) a random variable that counts the number of successes in 420 a sample of n elements drawn without replacement from a population of N individuals, of which 421

B Concentration Inequalities for Binomial Random Variables

The following bounds for the binomial distribution are used frequently in the analysis.

Lemma 13. Let X ∼ Bin ( n, p ) . Then,

For any δ > 0 ,

For any δ > 5 ,

For any δ ∈ (0 , 1) ,

K elements are considered successes. Note that if this sampling were done with replacement , then 422 the number of successes would follow a Bin ( n, K/N ) distribution. A result of Hoeffding [Hoe94] 423 establishes that the HypGeom ( N,K,n ) distribution is convex-order dominated by the Bin ( n, K/N ) 424 distribution, i.e. 425

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In particular, the function f ( x ) = e tx is convex for any value of t , and so Chernoff bounds that hold for the binomial distribution also hold for the corresponding hypergeometric distribution. This yields the following proposition.

Proposition 15. Let X ∼ HypGeom ( N,K,n ) . It follows for any δ > 0 that 429

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Our final remark about the hypergeometric distribution is a symmetry property. By interchanging the success and failure states, it follows that

The above intuition for the proof of Lemma 8 is formalized below.

Proof of Lemma 8. Let V denote the vertex set of G , and let G -v denote the induced subgraph of G on the vertex set V -{ v } . For any set A ⊆ V , let N v ( A ) denote the set of neighbors of v in the set A , i.e.

Since core k ( G -v ) ⊆ core k ( G ) , it is true that

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It follows that where 438

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It suffices to show that both p 1 and p 2 are o (1 /n ) . The term p 1 deals with the probability that the k -core of G -v is too small. In fact, by Lemma 14, it follows directly that

Next, the probability p 2 is analyzed. Enumerate arbitrarily but independently the elements of sets N v ( V ) and core k ( G -v ) , so that

Given that N v ( V ) has more than k +1 /α nodes and core k ( G -v ) has more than n -3 n 1 -α nodes, it is true that

In words, ˜ N v ( ˜ core k ( G -v )) counts among the first ⌈ k +1 /α ⌉ neighbors of v those nodes that are 445 also in the first ⌈ n -3 n 1 -α ⌉ nodes of core k ( G -v ) . Therefore, 446

Note that core k ( G -v ) is entirely determined by the graph G -v , i.e. it is independent of the 447 neighbors of v . Consequently, the two sets { v 1 , · · · , v ⌈ k +1 /α ⌉ } and { a 1 , · · · , a ⌈ n -3 n 1 -α ⌉ } are 448 selected independent of each other. Equivalently, given that | core k ( G -v ) | ≥ n -3 n 1 -α and 449 δ G ( v ) ≥ k + 1 /α , the size of the intersection set ˜ N v ( ˜ core k ( G -v )) follows a hypergeometric 450 distribution with parameters ( n -1 , ⌈ n -3 n 1 -α ⌉ , ⌈ k +1 /α ⌉ ) . Therefore, 451

where (a) uses the symmetry of the hypergeometric distribution. Using Proposition 15 and the fact 452 that n -1 ≥ n/ 2 for any n ≥ 1 yields 453

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Proof. Notice that

If | S | = n δ , then the sum of degrees of vertices in S is the total number of edges with exactly 462 end point in S , plus twice the number of edges with both end points in S . There are exactly 463 ( | S | 2 ) + | S | ( n -1 -| S | ) ≤ n 1+ δ such vertex pairs, and each of them independently has an edge 464 with probability α log( n ) /n . Therefore, a union bound over all possible choices of S yields 465

whenever δ > 1 -α as desired. Note that (a) uses the Binomial concentration inequality (7) and the 466 fact that ( n -1 k ) ≤ ( n k ) ≤ ( ne k ) k . 467

whenever α > 0 .

C.1 Proof of Lemma 14

A key ingredient towards proving Lemma 14 is a useful result about the number of low-degree vertices in an Erd˝ os-Rényi graph, presented next.

Proposition 16. Let α > 0 and G ∼ ER ( n -1 , α log( n ) /n ) . Let r be a positive integer and let Z r denote the set of vertices in G with degree no more than r , i.e.

For any δ such that δ > 1 -α , it is true that

Algorithm 3: Łuczak expansion

require : Graph G , Set U ⊆ V ( G )

Our objective is to show that the k -core of G -v is sufficiently large with probability 1 -o (1 /n ) . 468 To that end, consider Algorithm 3 to identify a subset of the k -core, originally proposed by 469 Łuczak [Łuc91]. 470

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Note that the for loop eventually terminates - the set V \ U i is empty, for example when i = n for any input set U . The key is to realize that the for loop terminates much faster when the input U = Z k +1 , i.e the set of vertices of the input graph G whose degree is k +1 or less. Furthermore, the complement of the set output by the algorithm is contained in the k -core. Formally,

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Lemma 17. Let U f be the output of Algorithm 3 with input graph G -v and set U = Z k +1 . Then,

(b) For any δ > 1 -α , 477

Proof. (a) The proof is by construction: Since U f is obtained by adding exactly f nodes to U 0 , it 478 follows that U c f ⊆ U c 0 = Z c k +1 , so each node in U c f has degree k +2 or more in G -v . Further, each 479 node in U c f has at most 2 neighbors in U f , else the for loop would not have terminated. Thus, the 480 subgraph of G -v induced on the set U c f has minimum degree at least k , and the result follows. 481

(b) If | U f | > 3 n δ , then either | U 0 | > 3 n δ or there is some M in { 0 , 1 , · · · , f } for which | U M | = 3 n δ . 482 Therefore, 483

by Proposition 16. Note that each iteration i = 0 , 1 , · · · , M -1 of the for loop adds exactly 1 vertex 484 and at least 3 edges to the subgraph of G -v induced on U M . Therefore, the induced subgraph G | U M 485 has 3 n δ vertices and at least 3 ( | U M | - | U 0 | ) edges. Thus, 486

where the last step uses Proposition 16 and a union bound over all possible choices of W . Finally, 487 using the relation ( n k ) ≤ ( ne k ) k and the concentration inequality (5) from Lemma 13 yields 488

whenever 0 < δ < 1 . The result follows. 489

Finally, notice that Lemma 17 directly implies Lemma 14. 490

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D Proof of Theorem 9

Theorem 9. Let G 1 , · · · , G m be correlated graphs from the subsampling model with parameters C and s . Let v ∈ V and let U be a vertex cut of { 1 , · · · , m } such that | U | ≤ ⌊ m/ 2 ⌋ . Then,

Proof. For any vertex cut U , 494

where (a) uses the fact that the maximum of a set of a numbers is greater than or equal to the average. 495 On the other hand 496

Let p 1 denote the probability in the LHS of (2). It follows from the union bound that 497

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E On Stochastic Dominance: Proof of Theorem 10

The objective of this section is to build up to a proof of Theorem 10. We start by making a simple observation about products of Binomial random variables.

Lemma 18. Let X 1 , · · · , X m ∼ Bern ( s ) be i.i.d. random variables, and let B = X 1 + · · · + X m denote their sum. For each ℓ in { 1 , 2 , · · · , ⌊ m/ 2 ⌋} , define

Proof of Lemma 18. Consider overlapping but exhaustive cases: 505

Case 1: t < 0 . Since T ℓ ≥ 0 almost surely for all ℓ , the inequality (11) holds. 506

Case 2: t ≥ b -1 . Note that conditioned on B = b , it follows that T 1 ∈ { 0 , b -1 } . Therefore, the 507 left hand side of (11) equals zero, and the inequality holds. 508

Case 3: b = 0 or b = 1 . In this case, T ℓ is identically zero for all ℓ , so (11) holds. 509

Case 4: b ≥ 2 and 0 ≤ t < b -1 . For any ℓ ∈ { 1 , 2 , · · · , ⌊ m/ 2 ⌋} , 510

where (a) used the fact that for any t such that 0 ≤ t < b -1 , it is true that

Here, the notation for binomial coefficients in (b) involves setting ( n k ) = 0 whenever k < 0 or k > n . 511 Let f m,b ( ℓ ) denote the numerator of (12), i.e. 512

It suffices to show that f m,b ( ℓ ) -f m,b ( ℓ -1) ≥ 0 for all ℓ ∈ { 2 , · · · , ⌊ m/ 2 ⌋} . Indeed, 513

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whenever m -ℓ ≥ ℓ -1 , i.e. ℓ ≤ ⌊ m/ 2 ⌋ . Here, (c) uses the identity ( n k ) = ( n -1 k -1 ) + ( n -1 k ) , and the fact that ( n 1 k ) ≥ ( n 2 k ) whenever n 1 ≥ n 2 . This concludes the proof.

Corollary 19. Let F be a collection of edges in the parent graph G . For any edge e r ∈ F , let X r i denote the indicator random variable G ′ i ( e r ) ∼ Bern ( ps ) . For each ℓ in { 1 , · · · , ⌊ m/ 2 ⌋} , define

Then, for any ℓ 1 , ℓ 2 ∈ { 1 , · · · , ⌊ m/ 2 ⌋} such that ℓ 1 < ℓ 2 , the following stochastic ordering holds 518

Proof. It suffices to show that T r ℓ 1 ⪯ T r ℓ 2 for each r , since the edges are independent. Indeed, we 519 have for any t that 520

which concludes the proof. 521

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With this, we are ready to prove Theorem 10. The theorem is restated for convenience.

Theorem 10. Let G 1 , · · · , G m be correlated graphs from the subsampling model. Let v ∈ V and let U ℓ denote the set { 1 , · · · , ℓ } for ℓ in { 1 , · · · , ⌊ m/ 2 ⌋} . For any vertex cut U of { 1 , · · · , m } , let ˜ c v ( U ) denote its cost in the graph ˜ H ( v ) . The following stochastic ordering holds:

Proof. Let ℓ 1 , ℓ 2 ∈ { 1 , · · · , ⌊ m/ 2 ⌋} such that ℓ 1 < ℓ 2 . Let t ∈ R . Consider the parent graph G and 526 label the set of incident edges on v as { e 1 , · · · , e δ G ( v ) } . Denote by X r i the indicator random variable 527 G ′ i ( e r ) ∼ Bern ( ps ) . It follows that 528

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as desired. Here, (a) uses Corollary 19.

F On Low Degree Nodes: Proof of Theorem 11

Theorem 11. Let G 1 , · · · , G m be obtained from the subsampling model with parameters C and s . Let r = m 2 4 ( k + 1 Cs 2 ) . Let v ∈ [ n ] and suppose that Cs (1 -(1 -s ) m -1 ) > 1 . Then,

Proof. Consider fixed integers r 1 , · · · , r m such that 0 ≤ r 2 , · · · , r m ≤ r . Since r is constant, by a union bound argument it suffices to show

Proceed by conditioning on the degree of v in G 1 , which follows a Bin ( n, ps ) distribution. Since the degrees of v in the intersection graphs { G 1 ∧ G ′ i : i = 2 , · · · , m } are conditionally independent given the degree of v in G 1 , we have

Using the fact that ( D r i ) ≤ ( De r i ) r i , it follows that 538

Expanding out the expectation yields 539

Proceed by splitting the summation at (log n ) 2 . The first part can be bounded as 540

whenever Cs (1 -(1 -s )) m -1 > 1 . Here, (a) is obtained by evaluating the probability generating 541 function of the Bin ( n, ps ) random variable at (1 -s ) m -1 and setting p = C log( n ) /n . 542

The other part of the sum can now be bounded as follows. 543

whenever C > 0 . Here, (b) is true because the function d ↦→ d ∑ r i ( a -s ) md is decreasing on the 544 interval [(log n ) 2 , n ] for all sufficiently large n . Finally, the concentration inequality for the Binomial 545 distribution holds by (6) in Lemma 13. The inequality applies since p = C log( n ) /n and since 546 (log n ) 2 > 6 Cs log( n ) for all n sufficiently large. This concludes the proof. 547

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper considers the graph matching problem, where the goal is to produce a mapping between vertices of multiple graphs which maximizes similarities among them. The authors study graph matching from a theoretical perspective, in which one observes multiple (appropriately correlated) Erdös-Rényi (ER) graphs that have ground-truth latent mappings between them. The authors' goal is to characterize the information-theoretic threshold for exactly recovering the latent mappings between all of the observed ER graphs. Prior work has settled the information-theoretic thresholds for 2 correlated ER graphs, and this paper settles it for more than 2 ER graphs. \n\nTo determine the information-theoretic threshold for exact graph matching, the authors establish matching achievability and converse results. The converse is based on a simple reduction to a graph matching problem with two ER graphs, combined with known results on impossibility results for exactly matching two ER graphs. For the achievability results, two algorithms are discussed. The first is the MLE, which is optimal for exact graph matching. The authors show that it has a clean, easy-to-understand form: the MLE outputs vertex mappings which maximize the number of edges in the corresponding union graph. However, the authors do not directly analyze this algorithm due to technical complexities. Instead, they propose an algorithm which involves two phases. (1) For each pair of graphs, a partial, fully-correct mapping is computed via the $k$-core estimator, and (2) unmatched vertices are matched through a \"transitive closure\" procedure. This algorithm provably outputs the full, correct set of vertex mappings in the parameter region that complements the converse. \n\nFinally, a few numerical experiments are presented, showing that the transitive closure procedure can be combined with known computationally efficient algorithms for pairwise graph matching to derive algorithms for matching multiple graphs in a principled manner.\n\nSTRENGTHS:\nAlmost all the existing theoretical work on graph matching concerns two graphs, except for some trivial results (to the best of my knowledge). The extension of the theoretical framework to multiple graphs is a natural and important follow-up, and may inspire several future works as well. \n\nWhile the algorithms and analysis are largely adapted from prior work (e.g., the $k$-core estimator), a key novelty is the transitive closure step, which provides a principled (and optimal!) bridge between pairwise graph matching and $m$-ary graph matching. As the authors highlight, this step can be used to extend practical algorithms for pairwise matching to the $m$-ary case in a black-box manner. I imagine that this technique could be useful in practice. \n\nAdditionally, the paper is well-written.\n\nWEAKNESSES:\nTo me, the main weakness of the paper is in the discussion of transitive closure's implications. The authors make a striking observation that one can use their transitive closure technique (at least heuristically) to generalize pairwise graph matching to $m$-ary graph matching. However, several details are lacking in the simulations section. For instance, what are the graph parameters ($n$ and $p$)? What is the error rate before and after the transitive closure boosting? How do the results shown compare to the accuracy of Algorithm 2? (Even though the $k$-core matching is not efficient to compute, the result of the matching procedure is a function of the ground-truth permutations, so I believe the algorithm's accuracy can be simulated efficiently). \n\nThere are a couple other minor weaknesses. One is that Algorithm 2 is computationally inefficient. However, making such an algorithm efficient is likely a challenging research question itself, and is appropriate for future work. Another weakness is that there is no nice figure to visualize Algorithm 2. I feel that the reader's understanding could be greatly improved if one could create a representative figure for the transitive closure boosting.\n\nQUESTIONS:\nHere I've listed a number of minor questions / comments in addition to the weaknesses above. \n- (p. 1, line 21) I'm not sure if it's clear at this point what \"latent correspondence\" means. Perhaps motivate this notion through social networks first, where the latent correspondence is derived from the same users across the two networks. \n- (p. 1, line 30) I'd suggest adding a sentence or two at the end of the introduction summarizing your contributions. But this is more of a stylistic comment. \n- (p. 2, line 39) I'm not sure if it's clear at this point what \"matching all vertices correctly\" means. I think you mean to say something along the lines of *exactly* learning the *entire* latent correspondence. (I'm just worried that using *matching* and *correspondence* without clarification of equivalence could confuse readers outside of this domain.)\n- (p. 2, line 50) I would mention that the literature on establishing information-theoretic limits largely (including the $k$-core estimator) largely uses computationally inefficient algorithms. \n- (p. 3, line 93) I would emphasize that permuting node labels is equivalent to removing node-level information. Hence any algorithm must use topological features, rather than node-level features, to do matching in this setting.\n- (p. 3, line 106) Emphasize that matchings need not be complete, for clarity.\n- (p. 4, line 116) I think you mean that $Cs (1 - (1 - s)^{m - 1}) \\ge 1$ is a necessary condition. Similarly, in the next line you should say something like \"this condition is also sufficient (except for the equality case) to exactly match $m$ graphs with probability going to 1.\"\n- (p. 4, line 139) Can you say something about *why* the analysis of the MLE is cumbersome?\n- (p. 5, Algorithm 1) Is the algorithm environment here really necessary? Maybe for brevity (and for minimal confusion) you can simply write, in the main text, \n$$\n(\\widehat{\\pi}_{12}^{ML}, \\ldots, \\widehat{\\pi}_{1m}^{ML}) \\in argmax |E(G_1 \\lor G_2^{\\pi_{12}} \\lor \\ldots \\lor G_m^{\\pi_{1m}})|\n$$\n- (p. 5, line 146) The sentence \"If a node $v$ is unmatched...\" could really use an illustrative figure, as this idea is really the essence of transitive closure boosting. Also, I would add for clarity that the sequence of graphs must consist of unique elements, and need not use all the graphs. \n- (p. 7, Definition 7) In light of Lemma 8, it seems that $\\mathcal{H}(v)$ can be derived from $\\widetilde{\\mathcal{H}}(v)$ by just thresholding the edge weights. Is there a strong reason why the analysis is done on the (seemingly more complex) weighted graph?\n- (p. 7, Theorem 9) What's the intuition behind the threshold $\\frac{m^2}{4} \\left( k + \\frac{1}{Cs^2} \\right)$?\n- (p. 9) Perhaps put the simulations and discussion of transitive closure into its own section.\n- (p. 9, bullet point on Beyond ER graphs) The theoretical literature on the $k$-core estimator allows for quite general classes of random graphs. What specific aspects of your analysis would be challenging to extend beyond ER?\n- (general comment) For two graphs, the information-theoretic threshold has a nice qualitative interpretation: it is the connectivity threshold of the common (intersection) graph. Is there a similar qualitative interpretation for the $m$-ary graph matching threshold?\n- (general comment) It would be useful to note that when $s$ is small, the information-theoretic threshold is reduced by a multiplicative factor of $m -1$, compared to the $m = 2$ case. This follows since $1 - (1 - s)^{m - 1} \\approx (m - 1) s$ for small $s$.\n\nLIMITATIONS:\nLimitations have been largely discussed. The authors could expand upon implications of graph matching to protecting / breaking privacy in anonymized social networks.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-06-20"}

comments: SUMMARY:
This paper studies the information theoretic limits for matching
multiple correlated random graphs. Based on a correlated Erdos-Renyi
random graph model, the authors provide both lower bound and achievable
bound for the condition to correctly match all nodes with high
probability. These bounds match each other. A highly interesting insight
is that, even when exactly matching two graphs is not possible, the
proposed algorithm can leverage more than two graphs to produce exact
matching among all the graphs. The achievable algorithm exploits the
transitivity among partial matchings through $k$-cores, which is also
quite interesting.

STRENGTHS:
1. The novelty of the paper is high in dealing with graph matching among
multiple correlated graphs. 

2. The necessary and sufficient conditions for exact matching meet each
other. 

3. The proposed algorithm can exploit transitivity to match all graphs,
even when any two graphs alone cannot be exactly matched. This is a very
insightful result.

WEAKNESSES:
1. The proposed algorithms do incur high complexity.

QUESTIONS:
1. In Algorithm 2, first the $k$-core for each pair of graphs is found.
However, there is no discussion of how to choose the parameter $k$.

2. Does Proposition 5 depend on the sub-sampling probability $s$?

LIMITATIONS:
Limitations are discussed in Section 5.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 4
DATE: 2024-07-05

comments: SUMMARY:
This theoretical paper gives tight conditions for exact graph matching with multiple correlated random graphs. This problem has been extensively studied recently for the case of 2 graphs, and it is shown here that with more than 2 graphs, there is a regime where pairwise alignment is not possible, but with the information provided by all graphs, it is possible to align all of them. This is a nice theoretical result.

STRENGTHS:
This paper studies a natural extension of a well-studied problem from 2 graphs to more graphs and shows a surprising effect: making partial pairwise matching is sufficient to get the exact recovery. The proof outlines give the main insights into the technical proof.

WEAKNESSES:
The resulting algorithm is not practical as it does not run in polynomial time (as it is mentioned by the authors).

QUESTIONS:
This paper is mostly theoretical and would probably benefit from being published in a more math-oriented journal. The impact on the Neurips community and the feedback here will probably be limited.

LIMITATIONS:
The authors are very clear with the limitations of their work in section 5.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 4
DATE: 2024-07-14

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper aims to find out alignments between G_1 and G_2,....G_m, under the assumptions that they all are essentially sampled from ER graph distribution. The paper presents one impossibility result (or necessary condition to estimate such alignment)  and two sufficiency results to solve the underlying problem.\n\nSTRENGTHS:\nThe paper tackles an interesting problem and it is written clearly.\n\nWEAKNESSES:\n(1) I am not too confident that the paper is appropriate for neurips audience. I think the paper suits better to a conference like ISIT or such.  The paper has barely any learning component and the practical utility is not very clear.\nAlso, the primary area assigned by the authors \"Probabilistic methods (for example: variational inference, Gaussian processes)\" is probably not correct.\n\n(2) The paper only tackles a very simple graph model (ER graph model). While I understand that theoretical analysis for complex graph model is difficult, I would recommend the authors should discuss that in comprehensive manner. To elaborate concretely,\nsuppose,  G_1, G' _2...,G' _m are *not* generated from an ER model. But G_2,...G_m are generated using an ER like model with constant edge deletion probability $s$. In such case, can one characterize the necessary and sufficient condition.\n\nNote that, the area is not too new in the literature. There has been work already in this line of research [CK17,WXY22 in the paper]. Although I will not say this work is an extension but the theoretical contribution given the existing works is not very interesting (m=2 to an arbitrary m for example). \n\n(3) There is no experimental analysis. I would have increased my rating if the authors have done a thorough study on implications (including limitations) of their work on graphs from other models. For example, if we apply the same algorithm in other graph models, how would it perform. Since the line of work is not new, I would not say the theoretical results have strong enough impact to ignore the poor experiments.\n\nQUESTIONS:\nSee above.\n\nLIMITATIONS:\nRestrictive graph model; poor experiments and incremental contribution.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-23"}