Recovering Communities in Structured Random Graphs

2026-01-23

Information Technology Convergence and Services

AI Summary

This work investigates whether the $d$ underlying coordinate sparse cuts of structured graphs—such as the $d$-dimensional hypercube—can be efficiently recovered when edges are randomly sampled with probability $p$. By integrating Fourier analysis of Boolean functions, particularly the Friedgut–Kalai–Naor theorem, with Karger’s cut-counting techniques, the authors establish strong sparsification bounds for cuts. Their main contribution shows that when the sampling rate is $p = C \log d / d$, which is asymptotically optimal, the sparsest balanced cut in the sampled graph approximates some coordinate cut within an error of $1/\mathrm{poly}(d)$ with high probability, thereby enabling simultaneous approximate recovery of all $d$ coordinate cuts. For hypercube-like graphs, exact recovery is even achievable.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

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.

Problem

• community recovery
• structured random graphs
• sparsest cuts
• hypercube graph
• overlapping communities

Innovation

• structured random graphs
• overlapping sparsest cuts
• hypercube graph
• community recovery
• cut sparsification