Maximal Cliques in Scale-Free Random Graphs

TL;DR

This work investigates the number of maximal cliques in three network models: Erdős–Rényi random graphs, inhomogeneous random graphs (IRGs), and geometric inhomogeneous random graphs (GIRGs), and gives super-polynomial lower bounds for these models.

Abstract

We investigate the number of maximal cliques, i.e., cliques that are not contained in any larger clique, in three network models: Erdős-Rényi random graphs, inhomogeneous random graphs (also called Chung-Lu graphs), and geometric inhomogeneous random graphs. For sparse and not-too-dense Erdős-Rényi graphs, we give linear and polynomial upper bounds on the number of maximal cliques. For the dense regime, we give super-polynomial and even exponential lower bounds. Although (geometric) inhomogeneous random graphs are sparse, we give super-polynomial lower bounds for these models. This comes from the fact that these graphs have a power-law degree distribution, which leads to a dense subgraph in which we find many maximal cliques. These lower bounds seem to contradict previous empirical evidence that (geometric) inhomogeneous random graphs have only few maximal cliques. We resolve this contradiction by providing experiments indicating that, even for large networks, the linear lower-order terms dominate, before the super-polynomial asymptotic behavior kicks in only for networks of extreme size.

Figures (6)

• Figure 1: Illustration of the gray shaded boxes $B_i$ on the 1 and 2-dimensional torus.
• Figure 2: Lower bound on the number of maximal cliques of Corollary \ref{['cor:maxcliquesgirg']} (with $b=2$, $\varepsilon\downarrow 0$), \ref{['thm:girg_non_torus']} (for $\varepsilon\downarrow 0$, $b=2$ and $C=1$), \ref{['thm:max_cliques_irg']} (for $\varepsilon\downarrow 0$ and $b=2$) against $n$ for different values of $\tau$. The black line is the line $n$.
• Figure 3: Clique minus a matching in the 2-dimensional GIRG.
• Figure 4: Scaling of the number of maximal $k$-cliques, and the total number of (not necessarily maximal) 3,4,5-cliques.
• Figure 5: The number of maximal cliques of the dense subgraph of GIRGs and IRGs. The considered subgraphs contain all vertices with weights in $[0.5 \sqrt{n}, \sqrt{n}]$ (left column) and $[0.5 \sqrt{n}, n]$ (right column). The top and bottom plots show the number of cliques with respect to the size of the full graph, and with respect to the size of the considered subgraph, respectively. All axes are logarithmic. Each point is the average of 10.0 sampled graphs.

Theorems & Definitions (34)

• Theorem 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Lemma 3.1
• proof