Degree Heterogeneity in a Graph Facilitates Quicker Meeting of Random Walkers
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Multiple random walks is a model for movement of several independent random walkers on a graph, and it is applied to various graph algorithms. In order to design an efficient graph algorithm using multiple random walks, it is essential to study theoretical considerations for deeply understanding the characteristics of graph algorithms. The first meeting time is one of the important metrics for the multiple random walks. The first meeting time is defined as the time it takes for multiple random walkers to meet on a same node. The first meeting time is closely related to the rendezvous problem. In various works, the first meeting time of multiple random walk has been analyzed. However, many of these previous works focused on regular graphs. In this paper, we analyze first meeting time of multiple random walks in arbitrary graph, and clarify the effect of graph structures on its expected value. First, we derive the spectral formula for the expected first meeting time of two random walkers using the spectral graph theory. Then, the principal component of the expected first meeting time are examined using the derived spectral formula. The resulting principal component reveals that (a) the expected first meeting time is almost dominated by n/(1+d_ std^2/d_ avg^2) and (b)the expected first meeting time is independent of the beginning nodes of multiple random walks where n is the number of nodes. d_ avg and d_ std are the mean and standard deviation of the weighted degree of each node, respectively. n and d_ avg, and d_ std are related to the statistics of graph structures. According to the analysis result, the variance of coefficient for weighted degrees, d_ std/d_ avg (degree heterogeneity), facilitates quicker the meeting of random walkers.
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