Approximating Sparse Graphs: The Random Overlapping Communities Model

TL;DR

This paper introduces the Random Overlapping Communities (ROC) model, a simple and effective way to approximate the spectral properties of sparse graphs, including real-world networks, by capturing their limiting spectra through a flexible random graph construction.

Contribution

The paper presents the ROC model, a novel random graph model that accurately approximates the spectra of sparse graphs and sequences, including hypercubes, using a simple sampling process.

Findings

ROC models converge to the spectra of many sparse graph sequences.

ROC can match specific triangle-to-edge and four-cycle ratios, unlike bounded stochastic block models.

ROC graphs show an inverse degree-clustering relationship characteristic of real networks.

Abstract

How can we approximate sparse graphs and sequences of sparse graphs (with unbounded average degree)? We consider convergence in the first $k$ moments of the graph spectrum (equivalent to the numbers of closed $k$-walks) appropriately normalized. We introduce a simple, easy to sample, random graph model that captures the limiting spectra of many sequences of interest, including the sequence of hypercube graphs. The Random Overlapping Communities (ROC) model is specified by a distribution on pairs $(s,q)$, $s \in \mathbb{Z}_+, q \in (0,1]$. A graph on $n$ vertices with average degree $d$ is generated by repeatedly picking pairs $(s,q)$ from the distribution, adding an Erd\H{o}s-R\'{e}nyi random graph of edge density $q$ on a subset of vertices chosen by including each vertex with probability $s/n$, and repeating this process so that the expected degree is $d$. Our proof of convergence to a ROC random graph is based on the Stieltjes moment condition. We also show that the model is an effective approximation for individual graphs. For almost all possible triangle-to-edge and four-cycle-to-edge ratios, there exists a pair $(s,q)$ such that the ROC model with this single community type produces graphs with both desired ratios, a property that cannot be achieved by stochastic block models of bounded description size. Moreover, ROC graphs exhibit an inverse relationship between degree and clustering coefficient, a characteristic of many real-world networks.