Graph Degree Heterogeneity Facilitates Random Walker Meetings

Abstract: Various graph algorithms have been developed with multiple random walks, the movement of several independent random walkers on a graph. Designing an efficient graph algorithm based on multiple random walks requires investigating multiple random walks theoretically to attain a deep understanding of their characteristics. The first meeting time is one of the important metrics for multiple random walks. The first meeting time on a graph is defined by the time it takes for multiple random walkers to meet at the same node in a graph. This time is closely related to the rendezvous problem, a fundamental problem in computer science. The first meeting time of multiple random walks has been analyzed previously, but many of these analyses have focused on regular graphs. In this paper, we analyze the first meeting time of multiple random walks in arbitrary graphs and clarify the effects of graph structures on expected values. First, we derive the spectral formula of the expected first meeting time on the basis of spectral graph theory. Then, we examine the principal component of the expected first meeting time using the derived spectral formula. The clarified principal component reveals that (a)the expected first meeting time is almost dominated by $n/(1+d_{\rm std}^2/d_{\rm avg}^2)$ and (b)the expected first meeting time is independent of the starting nodes of random walkers, where $n$ is the number of nodes of the graph. $d_{\rm avg}$ and $d_{\rm std}$ are the average and the standard deviation of weighted node degrees, respectively. The characteristic(a) is useful for understanding the effect of the graph structure on the first meeting time. According to the revealed effect of graph structures, the variance of the coefficient $d_{\rm std}/d_{\rm avg}$(degree heterogeneity) for weighted degrees facilitates the meeting of random walkers.