Title: Correlated Stochastic Block Models: Exact Graph Matching with Applications to Recovering Communities
Authors: Miklos Z. Racz
Anirudh Sridhar
Venue: NeurIPS 2021
Keywords: Stochastic block model
community recovery
graph matching
correlated random graphs
information-theoretic limits

Abstract: We consider the task of learning latent community structure from multiple correlated networks. First, we study the problem of learning the latent vertex correspondence between two edge-correlated stochastic block models, focusing on the regime where the average degree is logarithmic in the number of vertices. We derive the precise information-theoretic threshold for exact recovery: above the threshold there exists an estimator that outputs the true correspondence with probability close to 1, while below it no estimator can recover the true correspondence with probability bounded away from 0. As an application of our results, we show how one can exactly recover the latent communities using \emph{multiple} correlated graphs in parameter regimes where it is information-theoretically impossible to do so using just a single graph.

TL;DR: By determining the information-theoretic limits for exact graph matching in edge-correlated stochastic block models, we show how to combine information from multiple networks to improve community recovery algorithms.

Review discussion:
This paper is clearly written. It provides a nice descriptions of the problem and the results are easy to follow.
Overall, I enjoyed reading it.
The key part of the proof requires a careful analysis of the conditions for finding the graph matching.
While it seems the main approach builds upon previous work, there are a number of technical details which are well explained in the supplementary material. I think this analysis should also be viewed as a useful contribution of the paper.
The significance and novelty are on par with related work in the literature.
Also, similar to other work in the literature, the model is not very realistic so these results are more an exercise in mathematical analysis rather than shedding new insight into community detection problems in practice. Along these lines, one may question whether exact community recovery is a realistic objective, as opposed to approximate recovery. The authors do mention approximate recovery in the future work section.
No
Thank you for your review! The question of approximate community recovery is an interesting one. A related question of possible interest to you is whether approximate recovery of the vertex correspondence is possible. Our results imply that in the logarithmic degree regime we consider, it is always possible to correctly match a (1 - o(1)) fraction of vertices - see Lemma C.9 in the Supplementary Material. (Since our focus is on exact graph matching and community recovery, we did not highlight this as a main result, however.)
Coming back to the approximate community recovery problem, we conjecture that one can use an approximate matching (one that correctly matches a (1-o(1)) fraction of vertices) instead of an exact matching to recovery communities either approximately or exactly -- see Conjecture 3.5 and the discussion following it for details. We remark that it is unclear how one can leverage an approximate matching to obtain exact or approximate community recovery -- we shall investigate this in future work.
The main contribution of the this paper is the analysis on phase transaction on the possibility on exact graph matching in correlated stochastic block model. Existing literature only solves the problem with correct community labels. The result is interesting and substantial in network analysis.
The paper is well organized and reader-friendly.
No
Thank you for your review!
While your assessment of the limitations of our work is correct, we would like to add a few related remarks. For certain parameter regimes, our converse and achievability results actually yield matching bounds -- see for instance Figure 2a in the Supplementary Material.
You are also correct in stating "given the true matching function of the nodes, community recovery can be obtained by existing algorithms". In Conjecture 3.5 on page 6, we conjecture that exact community recovery is possible even when exact graph matching is impossible. To handle this case, we may need to design new algorithms for community recovery using a partial (rather than exact) matching between vertices of G1 and G2. For more details, see the discussion after Conjecture 3.5 and in lines 323-329, page 9. We leave this as an objective for future work, however.
The paper is very clear, well written and easy to read. I think that this line of work is a very natural extension for the SBM and will lead to subsequent works. As the authors claim, their work is a first step and I suspect that it will trigger interest in the crowd working on the theory of ML.
No
Thank you for your review!
This is a well written paper that not only is a precursor for using multiple correlated graphs to learn underlying latent communities, it demonstrates the advantages of using multiple graphs instead of a single graph (Theorem 3.6). The authors have shown robust and novel understanding of the subject as well as described conjectures on where future research work in this area could potentially head. I find that this adds to the significance of their results. Clear accept.
No
Thank you for your review!

Meta-review:
role: PC
data: comment: This paper presents a fine theoretical contribution showing how an initial step of graph matching of correlated graphs yields improved results to subsequently perform community detection, assuming graphs sampled from a correlated stochastic block model. The reviews are quite positive. Indeed joint consideration of multiple graphs to perform tasks such as community detection is a very natural strategy, and does not seem to have received much attention in the past. This makes the paper's contribution all the more valuable, hence the recommendation to accept. 

Decision: Accept (Spotlight)