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Recovering Communities in Structured Random Graphs

Michael Kapralov, Luca Trevisan, Weronika Wrzos-Kaminska

TL;DR

This work addresses recovering multiple overlapping communities in structured random graphs by analyzing subsampled hypercubes and their generalizations. It develops a provable pipeline that combines Fourier-analytic structure (FKN-type results) with robust cut-counting arguments (Karger-style bounds) to show that the sparsest balanced cuts in a sub-sampled graph remain close to coordinate cuts, enabling simultaneous recovery of all such cuts. The main results prove that for a hypercube Q_d and its k-distance generalization Q_{d,k}, one can exactly recover d coordinate cuts with high probability under suitable sampling, and approximately recover them under sparser conditions; exact recovery is attainable for hypercube-like graphs with near-logarithmic degree, and exact-threshold results extend to the connected components in even k. The methods highlight a bridge between geometric/overlapping community detection and classical graph sparsification, with potential implications for efficiently uncovering overlapping structures in high-dimensional discrete spaces. The work further identifies open questions about polynomial-time recovery and suggests SOS-based approaches as a promising direction for future work.

Abstract

The problem of recovering planted community structure in random graphs has received a lot of attention in the literature on the stochastic block model, where the input is a random graph in which edges crossing between different communities appear with smaller probability than edges induced by communities. The communities themselves form a collection of vertex-disjoint sparse cuts in the expected graph, and can be recovered, often exactly, from a sample as long as a separation condition on the intra- and inter-community edge probabilities is satisfied. In this paper, we ask whether the presence of a large number of overlapping sparsest cuts in the expected graph still allows recovery. For example, the $d$-dimensional hypercube graph admits $d$ distinct (balanced) sparsest cuts, one for every coordinate. Can these cuts be identified given a random sample of the edges of the hypercube where each edge is present independently with some probability $p\in (0, 1)$? We show that this is the case, in a very strong sense: the sparsest balanced cut in a sample of the hypercube at rate $p=C\log d/d$ for a sufficiently large constant $C$ is $1/\text{poly}(d)$-close to a coordinate cut with high probability. This is asymptotically optimal and allows approximate recovery of all $d$ cuts simultaneously. Furthermore, for an appropriate sample of hypercube-like graphs recovery can be made exact. The proof is essentially a strong hypercube cut sparsification bound that combines a theorem of Friedgut, Kalai and Naor on boolean functions whose Fourier transform concentrates on the first level of the Fourier spectrum with Karger's cut counting argument.

Recovering Communities in Structured Random Graphs

TL;DR

This work addresses recovering multiple overlapping communities in structured random graphs by analyzing subsampled hypercubes and their generalizations. It develops a provable pipeline that combines Fourier-analytic structure (FKN-type results) with robust cut-counting arguments (Karger-style bounds) to show that the sparsest balanced cuts in a sub-sampled graph remain close to coordinate cuts, enabling simultaneous recovery of all such cuts. The main results prove that for a hypercube Q_d and its k-distance generalization Q_{d,k}, one can exactly recover d coordinate cuts with high probability under suitable sampling, and approximately recover them under sparser conditions; exact recovery is attainable for hypercube-like graphs with near-logarithmic degree, and exact-threshold results extend to the connected components in even k. The methods highlight a bridge between geometric/overlapping community detection and classical graph sparsification, with potential implications for efficiently uncovering overlapping structures in high-dimensional discrete spaces. The work further identifies open questions about polynomial-time recovery and suggests SOS-based approaches as a promising direction for future work.

Abstract

The problem of recovering planted community structure in random graphs has received a lot of attention in the literature on the stochastic block model, where the input is a random graph in which edges crossing between different communities appear with smaller probability than edges induced by communities. The communities themselves form a collection of vertex-disjoint sparse cuts in the expected graph, and can be recovered, often exactly, from a sample as long as a separation condition on the intra- and inter-community edge probabilities is satisfied. In this paper, we ask whether the presence of a large number of overlapping sparsest cuts in the expected graph still allows recovery. For example, the -dimensional hypercube graph admits distinct (balanced) sparsest cuts, one for every coordinate. Can these cuts be identified given a random sample of the edges of the hypercube where each edge is present independently with some probability ? We show that this is the case, in a very strong sense: the sparsest balanced cut in a sample of the hypercube at rate for a sufficiently large constant is -close to a coordinate cut with high probability. This is asymptotically optimal and allows approximate recovery of all cuts simultaneously. Furthermore, for an appropriate sample of hypercube-like graphs recovery can be made exact. The proof is essentially a strong hypercube cut sparsification bound that combines a theorem of Friedgut, Kalai and Naor on boolean functions whose Fourier transform concentrates on the first level of the Fourier spectrum with Karger's cut counting argument.
Paper Structure (24 sections, 32 theorems, 149 equations, 1 figure)

This paper contains 24 sections, 32 theorems, 149 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Exact recovery.
  3. Related work on community detection on graphs with overlapping communities.
  4. Related work on community detection in geometric random graphs.
  5. Open problem.
  6. Proof of Theorem \ref{['thm:cube_main']}
  7. The algorithm.
  8. Preliminaries.
  9. Every sparse cut is close to a coordinate cut.
  10. Cut counting on the difference from a coordinate cut.
  11. Running time.
  12. Correctness.
  13. Proof of Theorem \ref{['thm:general_main']}
  14. Case 1: $2 \leq s \leq 2^k$.
  15. Case 2: $2^k \leq s \leq d/2$.
  16. ...and 9 more sections

Key Result

Theorem 1

Let $Q_d$ be the $d$-dimensional hypercube, and let $Q'_d$ be obtained by including each edge with probability $p \geq C \cdot \log d / d$, where $C$ is a sufficiently large constant. There exists an algorithm with running time $2^{O(n \log n)}$ that, given $Q'_d$, recovers $d$ orthogonal balanced c

Figures (1)

  • Figure 1: Illustration of the sets $A$ (red), $S$, $A^+$ and $A^-$, and the edges incident on $A^+$ and $A^-$ (blue).

Theorems & Definitions (71)

  • Theorem 1
  • Definition 2: Hypercube
  • Definition 3: $k$-distance hypercube
  • Theorem 4
  • Lemma 5: Folklore, see e.g., Example 4.1.3 in west2001introduction
  • Lemma 6
  • proof
  • Theorem 7: FKN theorem, Theorem 1.1 in fkn
  • Lemma 8: Sparse cuts are close to coordinate cuts, Corollary 1.2 in fkn
  • proof
  • ...and 61 more