Optimal and suboptimal networks for efficient navigation measured by mean-first passage time of random walks

TL;DR

This paper investigates the efficiency of network navigation through mean first-passage time, proving complete graphs are optimal and analyzing scale-free fractal networks to understand their navigability.

Contribution

It provides a new proof that complete graphs minimize Kemeny constant and derives exact solutions for scale-free fractal networks' navigation efficiency.

Findings

Complete graphs have the minimum Kemeny constant, growing linearly with network size.

Certain scale-free fractal networks also exhibit linear Kemeny constant growth.

Scale-free and fractal properties enhance network navigability.

Abstract

For a random walk on a network, the mean first-passage time from a node $i$ to another node $j$ chosen stochastically according to the equilibrium distribution of Markov chain representing the random walk is called Kemeny constant, which is closely related to the navigability on the network. Thus, the configuration of a network that provides optimal or suboptimal navigation efficiency is a question of interest. It has been proved that complete graphs have the exact minimum Kemeny constant over all graphs. In this paper, by using another method we first prove that complete graphs are the optimal networks with a minimum Kemeny constant, which grows linearly with the network size. Then, we study the Kemeny constant of a class of sparse networks that exhibit remarkable scale-free and fractal features as observed in many real-life networks, which cannot be described by complete graphs. To this end, we determine the closed-form solutions to all eigenvalues and their degeneracies of the networks. Employing these eigenvalues, we derive the exact solution to the Kemeny constant, which also behaves linearly with the network size for some particular cases of networks. We further use the eigenvalue spectra to determine the number of spanning trees in the networks under consideration, which is in complete agreement with previously reported results. Our work demonstrates that scale-free and fractal properties are favorable for efficient navigation, which could be considered when designing networks with high navigation efficiency.