Finding Cliques in Social Networks: A New Distribution-Free Model

TL;DR

This paper introduces a new model for social networks based on triadic closure, proving that enumerating maximal cliques is fixed-parameter tractable in this model, with empirical evidence from real social networks.

Contribution

It defines the concept of $c$-closed and weakly $c$-closed graphs, and proves fixed-parameter tractability for maximal clique enumeration in these models.

Findings

Social networks are often weakly $c$-closed for small $c$.

Maximal clique enumeration is fixed-parameter tractable in $c$-closed graphs.

Empirical data supports the relevance of the model to real social networks.

Abstract

We propose a new distribution-free model of social networks. Our definitions are motivated by one of the most universal signatures of social networks, triadic closure---the property that pairs of vertices with common neighbors tend to be adjacent. Our most basic definition is that of a "$c$-closed" graph, where for every pair of vertices $u,v$ with at least $c$ common neighbors, $u$ and $v$ are adjacent. We study the classic problem of enumerating all maximal cliques, an important task in social network analysis. We prove that this problem is fixed-parameter tractable with respect to $c$ on $c$-closed graphs. Our results carry over to "weakly $c$-closed graphs", which only require a vertex deletion ordering that avoids pairs of non-adjacent vertices with $c$ common neighbors. Numerical experiments show that well-studied social networks tend to be weakly $c$-closed for modest values of $c$.