Bimodal distribution of path multiplicity in random networks

Abstract

Erdös–Rényi (ER) random networks have long been central to the study of complex networks, providing foundational insights into network structure and behavior. Despite extensive research on their structural properties, the exploration of path multiplicity in ER random networks — quantifying the number of shortest paths between a random node pair — remains limited. In this paper, we systematically investigate the path multiplicity in ER random networks, including exploring its distribution, average, variance and coefficient of variation through both simulation and analytical approaches. We first observe a bimodal distribution of shortest path amounts between node pairs in ER random networks. As the connection probability $p$ increases, the left part steepens and the right part forms a bell-shaped distribution, gradually separating from the left. The mean and variance of path multiplicity reach their maximum values at approximately $p=2/3$ and $p=5/6$, respectively, while the coefficient of variation peaks at low $p$ values and then increases monotonically before $p=1$. These statistical properties highlight significant variations in path multiplicity under different connection probabilities. Furthermore, we examine the behavior of other network metrics in ER random networks, including resistance distance, efficiency, and natural connectivity, and identify distinct differences compared to path multiplicity. These results shed new light on the intricate structural patterns that emerge in ER random networks and provide a deeper quantitative understanding of the factors that govern shortest path multiplicity, contributing to the broader study of random network theory.