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

TL;DR

This work extends the study of maximal pattern enumeration in social-graph models by examining maximal blow-ups of a fixed graph H in c-closed graphs. It proves a polynomial bound on the number of maximal non-induced H-blow-ups for any fixed H and provides a precise induced-blow-up dichotomy based on H's twin-structure, plus constructions showing exponential behavior for certain infinite pattern families. The results illuminate how triadic-closure constraints influence combinatorial patterns beyond cliques, and they establish a framework for analyzing infinite families of patterns with respect to polynomial vs exponential growth. Overall, the paper advances understanding of pattern enumeration in triad-closure networks and offers tools for predicting when such enumerations remain tractable.

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.

Figures (7)

• Figure 1: From Fox et al. paper, modeling triadic closure in Enron emails paper1
• Figure 2: Example of extending a blow-up to maximality when searching for a blow-up of $H$ in $G$: The blue circles represent the vertex groups in the blow-up that correspond to the vertices of $H$. The left diagram shows a proposed (non-maximal) blow-up, while the right diagram shows a maximal blow-up. Notably, reaching the maximal blow-up on the right requires regrouping the vertices from the smaller blow-up on the left; that is, the grouping in the maximal blow-up is not simply a superset of the original—it may involve reassigning vertices among groups.
• Figure 3: This example illustrates that for a general graph $H$, verifying maximal blow-ups of $H$ within a host graph $G$ can be challenging. The images below show two blow-ups: the left depicts a proposed (but not maximal) blow-up, while the right shows a maximal blow-up that includes all relevant vertices. Notably, a vertex set that is both contained in a blow-up and contains another blow-up is not necessarily itself a blow-up. For instance, $G[\{u_1, u_2, u_3, u_4, u_7\}]$ is not a blow-up of $H$, even though it is contained in the larger blow-up $G[\{u_1, \dots, u_7\}]$, and contains the smaller blow-up $G[\{u_1, \dots, u_4\}]$.
• Figure 4: Graph $H$. The vertices $v_1, \dots, v_b$ are bad and form an independent set. Each $N_1, \dots, N_b$ is a clique and is disjoint from $\{v_1, \dots, v_b\}$.
• Figure 5: Construction of a $k+1$-closed graph $G$ given $H$. For $i = 1,2$, the vertices $v_i$ is replaced by a clique $V_i$ of size $K$ if $i \in U_+$ or $i \notin U_-\cup U_+$, and respectively and the non-edge between $v_1, v_2$ in $H$ is replaced by a perfect matching $\{(a_1, b_1), \dots, (a_K, b_K)\}.$

Theorems & Definitions (74)

• Definition 1
• Theorem 1
• Remark 1
• Definition 2
• Definition 3
• Theorem 2
• Definition 4
• Theorem 3
• Definition 5
• Definition 6