On the Number of Maximal Cliques in Two-Dimensional Random Geometric Graphs: Euclidean and Hyperbolic

TL;DR

This paper investigates the number of maximal cliques in Euclidean and hyperbolic random geometric graphs, providing probabilistic bounds that are significantly smaller than the worst-case scenario, thus explaining their practical tractability.

Contribution

It establishes probabilistic bounds on the number of maximal cliques in Euclidean and hyperbolic random geometric graphs, bridging the gap between worst-case and real-world network behaviors.

Findings

Number of maximal cliques in Euclidean graphs is bounded by exponential functions of |V|^{1/3}.

Hyperbolic graphs have bounds depending on the degree exponent gamma.

Bounds are tight with high probability, explaining practical enumeration feasibility.

Abstract

Maximal clique enumeration appears in various real-world networks, such as social networks and protein-protein interaction networks for different applications. For general graph inputs, the number of maximal cliques can be up to $3^{|V|/3}$. However, many previous works suggest that the number is much smaller than that on real-world networks, and polynomial-delay algorithms enable us to enumerate them in a realistic-time span. To bridge the gap between the worst case and practice, we consider the number of maximal cliques in two popular models of real-world networks: Euclidean random geometric graphs and hyperbolic random graphs. We show that the number of maximal cliques on Euclidean random geometric graphs is lower and upper bounded by $\exp(\Omega(|V|^{1/3}))$ and $\exp(O(|V|^{1/3+\epsilon}))$ with high probability for any $\epsilon > 0$. For a hyperbolic random graph, we give the bounds of $\exp(\Omega(|V|^{(3-\gamma)/6}))$ and $\exp(O(|V|^{(3-\gamma+\epsilon)/6)}))$ where $\gamma$ is the power-law degree exponent between 2 and 3.