Global mean first-passage times of random walks on complex networks

TL;DR

This paper introduces a framework for bounding the global mean first-passage time (GMFPT) of random walks on complex networks, revealing how network size and target connectivity influence walk efficiency and challenging the representativeness of GMFPT.

Contribution

It provides a rigorous lower bound for GMFPT applicable to various networks and identifies conditions under which minimal scaling occurs, advancing understanding of random walk dynamics.

Findings

GMFPT has a minimal scaling related to the target's connectivity.

Transient target sites lead to arbitrarily slow GMFPT scaling.

Target-averaged GMFPT obeys stricter bounds, preventing sublinear scaling.

Abstract

We present a general framework, applicable to a broad class of random walks on complex networks, which provides a rigorous lower bound for the mean first-passage time of a random walker to a target site averaged over its starting position, the so-called global mean first-passage time (GMFPT). This bound is simply expressed in terms of the equilibrium distribution at the target, and implies a minimal scaling of the GMFPT with the network size. We show that this minimal scaling, which can be arbitrarily slow for a proper choice of highly connected target, is realized under the simple condition that the random walk is transient at the target site, and independently of the small-world, scale free or fractal properties of the network. Last, we put forward that the GMFPT to a specific target is not a representative property of the network, since the target averaged GMFPT satisfies much more restrictive bounds, which forbid any sublinear scaling with the network size.