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

2025-06-13

AI Summary

This paper studies the enumeration of maximum blow-ups (i.e., maximal homogeneous extensions) of an arbitrary fixed graph $H$ in $c$-closed graphs. We generalize classical maximal clique and biclique enumeration to arbitrary patterns $H$, establishing a unified framework. Our method integrates structural analysis of $c$-closed graphs, extremal graph theory, combinatorial counting, and computational complexity theory. We prove that, for any fixed $H$, the number of maximal blow-ups of $H$ in an $n$-vertex $c$-closed graph is $O(n^{f(c,H)})$, i.e., polynomially bounded. We fully characterize the necessary and sufficient graph-theoretic conditions on $H$ for which induced blow-ups admit polynomial-time enumeration. Furthermore, we extend our results to infinite families of graphs, revealing a complexity phase transition: polynomial-time enumeration remains feasible under a precise density constraint on the family, but becomes superpolynomial otherwise. These results provide theoretical foundations and algorithmic guarantees for generalized community detection.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Fox, Seshadhri, Roughgarden, Wei, and Wein (SICOMP 2020) 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.

Problem

• Enumerate maximal blow-ups of arbitrary graphs in c-closed graphs
• Characterize graphs with polynomially bounded induced blow-ups
• Study blow-up enumeration for infinite graph families

Innovation

• Enumerates maximal blow-ups in c-closed graphs
• Polynomial bounds for fixed graph patterns
• Characterizes induced blow-ups for specific graphs