Random walks and diameter of finite scale-free networks

TL;DR

This paper investigates the dynamical scaling of random walks on finite scale-free networks, revealing how the network diameter influences the walk's end-to-end distance and proposing a method to efficiently measure network diameter through random walks.

Contribution

The study introduces a numerical analysis of random walk scaling behaviors on scale-free networks and links the network diameter to the random walk dynamics, providing a new approach for diameter measurement.

Findings

End-to-end distance scales with network parameters and time.

Network diameter relates to the degree exponent and network size.

Random walks can efficiently estimate network diameter.

Abstract

Dynamical scalings for the end-to-end distance $R_{ee}$ and the number of distinct visited nodes $N_v$ of random walks (RWs) on finite scale-free networks (SFNs) are studied numerically. $\left< R_{ee} \right>$ shows the dynamical scaling behavior $\left<R_{ee}({\bar \ell},t)\right>= \bar{\ell}^\alpha (\gamma, N) g(t/\bar{\ell}^z)$, where $\bar{\ell}$ is the average minimum distance between all possible pairs of nodes in the network, $N$ is the number of nodes, $\gamma$ is the degree exponent of the SFN and $t$ is the step number of RWs. Especially, $\left<R_{ee}({\bar \ell},t)\right>$ in the limit $t \to \infty$ satisfies the relation $\left< R_{ee} \right> \sim \bar{\ell}^\alpha \sim d^\alpha$, where $d$ is the diameter of network with $d ({\bar \ell}) \simeq \ln N$ for $\gamma \ge 3$ or $d ({\bar \ell}) \simeq \ln \ln N$ for $\gamma < 3$. Based on the scaling relation $\left< R_{ee} \right>$, we also find that the scaling behavior of the diameter of networks can be measured very efficiently by using RWs.