Social Networks: Enumerating Maximal Community Patterns in $c$-Closed Graphs

TL;DR

This paper extends the study of maximal community patterns in $c$-closed social network graphs, proving polynomial bounds on the number of such patterns for arbitrary fixed graphs and characterizing cases with induced blow-ups.

Contribution

It generalizes previous results by analyzing maximal blow-ups of arbitrary graphs in $c$-closed graphs and characterizes when induced blow-ups are polynomially bounded.

Findings

Number of maximal blow-ups of fixed graph $H$ is polynomially bounded in $n$.

Characterization of graphs $H$ with polynomially bounded induced blow-ups.

Analysis of infinite families of graphs for blow-up enumeration.

Abstract

Jacob Fox, C. Seshadhri, Tim Roughgarden, Fan Wei, and Nicole Wein introduced the model of $c$-closed graphs--a distribution-free model motivated by triadic closure, one of the most pervasive structural signatures of social networks. While enumerating maximal cliques in general graphs can take exponential time, it is known that in $c$-closed graphs, maximal cliques and maximal complete bipartite subgraphs can always be enumerated in polynomial time. These structures correspond to blow-ups of simple patterns: a single vertex or a single edge, with some vertices required to form cliques. In this work, we explore a natural extension: we study maximal blow-ups of arbitrary finite graphs $H$ in $c$-closed graphs. We prove that for any fixed graph $H$, the number of maximal blow-ups of $H$ in an $n$-vertex $c$-closed graph is always bounded by a polynomial in $n$. We further investigate the case of induced blow-ups and provide a precise characterization of the graphs $H$ for which the number of maximal induced blow-ups is also polynomially bounded in $n$. Finally, we study the analogue questions when $H$ ranges over an infinite family of graphs.