Random walks in weighted networks with a perfect trap: An application of Laplacian spectra

TL;DR

This paper develops a spectral graph theory framework to analyze trapping times in weighted networks, deriving explicit formulas and bounds for mean first-passage times and applying them to specific network models.

Contribution

It introduces a general spectral approach for calculating trapping times in weighted networks and derives bounds based on local information, with applications to Levy walks and weighted uncorrelated networks.

Findings

Network weights significantly influence trapping times.

Modifying weight parameters can drastically change the average trapping time.

Optimal Levy walk exponents minimize average trapping times.

Abstract

In this paper, we propose a general framework for the trapping problem on a weighted network with a perfect trap fixed at an arbitrary node. By utilizing the spectral graph theory, we provide an exact formula for mean first-passage time (MFPT) from one node to another, based on which we deduce an explicit expression for average trapping time (ATT) in terms of the eigenvalues and eigenvectors of the Laplacian matrix associated with the weighted graph, where ATT is the average of MFPTs to the trap over all source nodes. We then further derive a sharp lower bound for the ATT in terms of only the local information of the trap node, which can be obtained in some graphs. Moreover, we deduce the ATT when the trap is distributed uniformly in the whole network. Our results show that network weights play a significant role in the trapping process. To apply our framework, we use the obtained formulas to study random walks on two specific networks: trapping in weighted uncorrelated networks with a deep trap, the weights of which are characterized by a parameter, and L\'evy random walks in a connected binary network with a trap distributed uniformly, which can be looked on as random walks on a weighted network. For weighted uncorrelated networks we show that the ATT to any target node depends on the weight parameter, that is, the ATT to any node can change drastically by modifying the parameter, a phenomenon that is in contrast to that for trapping in binary networks. For L\'evy random walks in any connected network, by using their equivalence to random walks on a weighted complete network, we obtain the optimal exponent characterizing L\'evy random walks, which have the minimal average of ATTs taken over all target nodes.