Modular hierarchical and power-law small-world networks bear structural optima for minimal first passage times and cover time

TL;DR

This paper investigates how modular hierarchical and power-law small-world networks optimize random walk efficiency, revealing structural features that minimize first passage and cover times, with implications for understanding natural network organization.

Contribution

It demonstrates that certain modular hierarchical network structures inherently optimize random walk dynamics, highlighting their potential evolutionary and functional advantages.

Findings

Optimal network structures minimize first passage and cover times.

Power-law and hierarchical networks exhibit small-world effects similar to Kleinberg models.

Analytic predictions fail to capture the minima, indicating structural effects.

Abstract

Networks that are organized as a hierarchy of modules have been the subject of much research, mainly focusing on algorithms that can extract this community structure from data. The question of why modular hierarchical organizations are so ubiquitous in nature, however, has received less attention. One hypothesis is that modular hierarchical topologies may provide an optimal structure for certain dynamical processes. We revisit a modular hierarchical network model that interpolates, using a single parameter, between two known network topologies: from strong hierarchical modularity to an Erd\H{o}s-R\'enyi random connectivity structure. We show that this model displays a similar small-world effect as the Kleinberg model, where the connection probability between nodes decays algebraically with distance. We find that there is an optimal structure, in both models, for which the pair-averaged first passage time (FPT) and mean cover time of a discrete-time random walk are minimal, and provide a heuristic explanation for this effect. Finally, we show that analytic predictions for the pair-averaged FPT based on an effective medium approximation fail to reproduce these minima, which implies that their presence is due to a network structure effect.