Extreme events and event size fluctuations in biased random walks on networks

TL;DR

This paper investigates how biased random walks on networks influence the occurrence and size of extreme events, revealing that node connectivity and bias affect the likelihood of rare, large fluctuations.

Contribution

It introduces a generalized biased random walk model on networks, deriving analytical and simulation results for extreme event probabilities based on node 'generalized strength.'

Findings

Large fluctuations occur on small degree nodes when biased toward hubs.

Nodes with higher 'generalized strength' have lower probability of extreme events.

The model links node properties to the likelihood of extreme events.

Abstract

Random walk on discrete lattice models is important to understand various types of transport processes. The extreme events, defined as exceedences of the flux of walkers above a prescribed threshold, have been studied recently in the context of complex networks. This was motivated by the occurrence of rare events such as traffic jams, floods, and power black-outs which take place on networks. In this work, we study extreme events in a generalized random walk model in which the walk is preferentially biased by the network topology. The walkers preferentially choose to hop toward the hubs or small degree nodes. In this setting, we show that extremely large fluctuations in event-sizes are possible on small degree nodes when the walkers are biased toward the hubs. In particular, we obtain the distribution of event-sizes on the network. Further, the probability for the occurrence of extreme events on any node in the network depends on its 'generalized strength', a measure of the ability of a node to attract walkers. The 'generalized strength' is a function of the degree of the node and that of its nearest neighbors. We obtain analytical and simulation results for the probability of occurrence of extreme events on the nodes of a network using a generalized random walk model. The result reveals that the nodes with a larger value of 'generalized strength', on average, display lower probability for the occurrence of extreme events compared to the nodes with lower values of 'generalized strength'.