Comparison of Multiple Random Walks Strategies for Searching Networks

Zhongtuan Zheng,Guoqiang Wang,Shengguo Gao,Hanxing Wang

doi:10.1155/2013/734630

Abstract

We investigate diverse random-walk strategies for searching networks, especially multiple random walks (MRW). We use random walks on weighted networks to establish various models of single random walks and take the order statistics approach to study corresponding MRW, which can be a general framework for understanding random walks on networks. Multiple preferential random walks (MPRW) and multiple simple random walks (MSRW) are two special types of MRW. As search strategies, MPRW prefers high-degree nodes while MSRW searches for low-degree nodes more efficiently. We analyze the first passage time (FPT) of wandering walkers of MRW and give the corresponding formulas of probability distributions and moments, and the mean first passage time (MFPT) is included. We show the convergence of the MFPT of the first arriving walker and find the MFPT of the last arriving walker closely related with the mean cover time. Simulations confirm analytical predictions and deepen discussions. We use a small random network to test the FPT properties from different aspects. We also explore some practical search-related issues by MRW, such as detecting unknown shortest paths and avoiding poor routings on networks. Our results are of practical significance for realizing optimal routing and performing efficient search on complex networks.

Highlights

A random walk on a network [1] is precisely what its name says: a walk X0X1 ⋅ ⋅ ⋅ operated in a certain random fashion
We focus on the first passage time (FPT) and the mean first passage time (MFPT) of multiple random walks (MRW), which are closely related to the distance between a selected pair of nodes and the cover time (CT) of corresponding single random walks
1(b) and 2(b), with the same walker number z, we find that the distribution of Ts(hz) of multiple simple random walks (MSRW) is more decentralized than Multiple preferential random walks (MPRW), while the MFPT ⟨Ts(hz)⟩ of MSRW is much greater than MPRW; see Figures 6(b) and 3(b)

Summary

Introduction

A random walk on a network [1] is precisely what its name says: a walk X0X1 ⋅ ⋅ ⋅ operated in a certain random fashion. Random walks on complex networks are a fundamental dynamic process, which can be used to model various dynamic stochastic systems in physical, biological, or social contexts such as diffusive motion [2], traffic flow [3], synchronization [4], animals movement [5], and epidemic spread [6] It could be a mechanism of transport and search in real-world networks [7,8,9], when no global information of the underlying networks is available. FPT and CT of a single walker or multiple walkers in complex networks have been studied in a number of articles [16,17,18,19,20,21,22] These results are frequently based on the spectral properties of the adjacency matrix of the graph and of the transition matrix of the random walk [13, 21, 22]. The MRW search strategy could be used to fast detect the unknown shortest path and avoid the worst routing between a pair of nodes

Single Random Walks on Networks

Multiple Random Walks on Networks

Simulations and Discussions

MSRW on a Simple Random Network

Conclusions