Distinct nodes visited by random walkers on scale-free networks

TL;DR

This paper investigates how multiple random walkers explore scale-free networks, revealing a universal stretched exponential pattern for unvisited nodes and simplifying the analysis to a single walker case.

Contribution

It introduces a universal description of node coverage by multiple walkers on scale-free networks, independent of network specifics and number of walkers.

Findings

Unvisited nodes follow a stretched exponential decay over time.

A power-law relation links unvisited nodes by multiple and single walkers.

Results are validated on four real-world scale-free networks.

Abstract

Random walks on discrete lattices are fundamental models that form the basis for our understanding of transport and diffusion processes. For a single random walker on complex networks, many properties such as the mean first passage time and cover time are known. However, many recent applications such as search engines and recommender systems involve multiple random walkers on complex networks. In this work, based on numerical simulations, we show that the fraction of nodes of scale-free network not visited by $W$ random walkers in time $t$ has a stretched exponential form independent of the details of the network and number of walkers. This leads to a power-law relation between nodes not visited by $W$ walkers and by one walker within time $t$. The problem of finding the distinct nodes visited by $W$ walkers, effectively, can be reduced to that of a single walker. The robustness of the results is demonstrated by verifying them on four different real-world networks that approximately display scale-free structure.