Algorithms for Computing Maximum Cliques in Hyperbolic Random Graphs

TL;DR

This paper introduces a new algorithm for finding maximum cliques in hyperbolic random graphs, providing theoretical analysis and empirical evaluation, and demonstrates its effectiveness on real-world networks.

Contribution

The paper presents a simple, efficient algorithm for maximum clique detection in hyperbolic random graphs with proven expected running time and improved practical performance over previous methods.

Findings

Algorithm achieves expected time complexity based on graph parameters.

Empirical results show superior performance over previous algorithms.

Effective in identifying large near-optimal cliques in real-world networks.

Abstract

In this paper, we study the maximum clique problem on hyperbolic random graphs. A hyperbolic random graph is a mathematical model for analyzing scale-free networks since it effectively explains the power-law degree distribution of scale-free networks. We propose a simple algorithm for finding a maximum clique in hyperbolic random graph. We first analyze the running time of our algorithm theoretically. We can compute a maximum clique on a hyperbolic random graph $G$ in $O(m + n^{4.5(1-\alpha)})$ expected time if a geometric representation is given or in $O(m + n^{6(1-\alpha)})$ expected time if a geometric representation is not given, where $n$ and $m$ denote the numbers of vertices and edges of $G$, respectively, and $\alpha$ denotes a parameter controlling the power-law exponent of the degree distribution of $G$. Also, we implemented and evaluated our algorithm empirically. Our algorithm outperforms the previous algorithm [BFK18] practically and theoretically. Beyond the hyperbolic random graphs, we have experiment on real-world networks. For most of instances, we get large cliques close to the optimum solutions efficiently.