Mean First Hitting Time of Searching for Path Through Random Walks on Complex Networks

TL;DR

This paper derives an exact formula for the average time to find a fixed path via random walks on complex networks, introducing a new measure called RWPM that optimizes routing by balancing network load.

Contribution

It provides an analytical expression for mean first hitting time, decomposes it into path-dependent and network-dependent parts, and proposes RWPM for improved routing strategies.

Findings

Mean first hitting time is divided into path and network components.

The proposed RWPM effectively balances network load.

RW optimal routing outperforms shortest path routing.

Abstract

We study the problem of searching for a fixed path $\epsilon_0\epsilon_1\cdots\epsilon_l$ on a network through random walks. We analyze the first hitting time of tracking the path, and obtain exact expression of mean first hitting time $\langle T \rangle$. Surprisingly we find that $\langle T \rangle$ is divided into two distinct parts: $T_1$ and $T_2$. The first part $T_1 =2m\prod_{i=1}^{l-1}d(\epsilon_i)$, is related with the path itself and is proportional to the degree product. The second part $T_2$ is related with the network structure. Based on the analytic results, we propose a natural measure for each path, i.e. $\varphi=\prod_{i=1}^{l-1}d(\epsilon_i)$, and call it random walk path measure(RWPM). $\varphi$ essentially determines a path's performance in searching and transporting processes. By minimizing $\varphi$, we also find RW optimal routing which is a combination of random walk and shortest path routing. RW optimal routing can effectively balance traffic load on nodes and edges across the whole network, and is superior to shortest path routing on any type of complex networks. Numerical simulations confirm our analysis.