First passage time of multiple Brownian particles on networks with applications

In this article, we study some theoretical and technological problems with relation to multiple Brownian particles on networks. We are especially interested in the behavior of the first arriving Brownian particle when all the Brownian particles start out from the source s simultaneously and head to the destination h randomly. We analyze the first passage time (FPT) Ysh(z) and the mean first passage time (MFPT) 〈Ysh(z)〉 of multiple Brownian particles on complex networks. Equations of Ysh(z) and 〈Ysh(z)〉 are obtained. On a variety of commonly encountered networks, we observe first passage properties of multiple Brownian particles from different aspects. We find that 〈Ysh(z)〉 drops substantially when particle number z increases at the first stage, and converges to dsh, the distance between the source and the destination when z→∞. The distribution of FPT Prob{Ysh(z)=t},t=0,1,2,… is also analyzed in these networks. The distribution curve peaks up towards t=dsh when z increases. Consequently, if particle number z is set appropriately large, the first arriving Brownian particle will go along the shortest or near shortest paths between the source and the destination with high probability. Simulations confirm our analysis. Based on theoretical studies, we also investigate some practical problems using multiple Brownian particles, such as communication on P2P networks, optimal routing in small world networks, phenomenon of asymmetry in scale-free networks, information spreading in social networks, pervasion of viruses on the Internet, and so on. Our analytic and experimental results on multiple Brownian particles provide useful evidence for further understanding and properly tackling these problems.