Characteristic times of biased random walks on complex networks

TL;DR

This paper investigates how degree-biased random walks on complex networks can be optimized by tuning a bias parameter to minimize characteristic times for various visitation tasks, with implications for real-world network navigation.

Contribution

It provides an analytical and numerical study of the optimal bias parameter for degree-biased random walks on real-world networks, linking it to degree correlations and network structure.

Findings

Optimal bias parameter varies with network assortativity.

Degree bias can reduce characteristic times compared to unbiased walks.

Analytical relation between degree correlation and optimal bias is established.

Abstract

We consider degree-biased random walkers whose probability to move from a node to one of its neighbors of degree $k$ is proportional to $k^{\alpha}$, where $\alpha$ is a tuning parameter. We study both numerically and analytically three types of characteristic times, namely: i) the time the walker needs to come back to the starting node, ii) the time it takes to visit a given node for the first time, and iii) the time it takes to visit all the nodes of the network. We consider a large data set of real-world networks and we show that the value of $\alpha$ which minimizes the three characteristic times is different from the value $\alpha_{\rm min}=-1$ analytically found for uncorrelated networks in the mean-field approximation. In addition to this, we found that assortative networks have preferentially a value of $\alpha_{\rm min}$ in the range $[-1,-0.5]$, while disassortative networks have $\alpha_{\rm min}$ in the range $[-0.5, 0]$. We derive an analytical relation between the degree correlation exponent $\nu$ and the optimal bias value $\alpha_{\rm min}$, which works well for real-world assortative networks. When only local information is available, degree-biased random walks can guarantee smaller characteristic times than the classical unbiased random walks, by means of an appropriate tuning of the motion bias.