Graph Degree Heterogeneity Facilitates Random Walker Meetings

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