Random Walks on deterministic Scale-Free networks: Exact results

TL;DR

This paper provides exact analytical results for random walks on deterministic scale-free networks, revealing efficient navigation properties and scaling behaviors of first-passage times.

Contribution

It derives exact formulas for first-passage phenomena and mean hitting times on a class of deterministic scale-free networks, highlighting their efficiency.

Findings

Exact expression for the probability of first return to the main hub

Mean time to reach the main hub scales as V^{1-1/γ}

The process is notably efficient in navigating the network

Abstract

We study the random walk problem on a class of deterministic Scale-Free networks displaying a degree sequence for hubs scaling as a power law with an exponent $\gamma=\log 3/\log2$. We find exact results concerning different first-passage phenomena and, in particular, we calculate the probability of first return to the main hub. These results allow to derive the exact analytic expression for the mean time to first reach the main hub, whose leading behavior is given by $\tau \sim V^{1-1/\gamma}$, where $V$ denotes the size of the structure, and the mean is over a set of starting points distributed uniformly over all the other sites of the graph. Interestingly, the process turns out to be particularly efficient. We also discuss the thermodynamic limit of the structure and some local topological properties.