Predicting First-Passage Dynamics in Disordered Systems Exactly: Application to Sparse Networks

TL;DR

This paper introduces an exact analytical method to study first-passage dynamics of diffusing particles on disordered networks, revealing new transport behaviors and challenging previous assumptions about search processes.

Contribution

The authors develop a general analytical framework for first-passage times on sparse graphs, enabling exact calculations and uncovering novel bi-modal regimes in small-world networks.

Findings

Existence of bi-modal first-passage time distribution in small-world networks

Network features influence the shape of first-passage probability distributions

Temporal features can reveal new transport paradigms in disordered environments

Abstract

Quantifying how spatial disorder affects the movement of a diffusing particle or agent is fundamental to target search studies. When diffusion occurs on a network, that is on a highly disordered environment, we lack the mathematical tools to calculate exactly the temporal characteristics of search processes, instead relying on estimates provided by stochastic simulations. To close this knowledge gap we devise a general methodology to represent analytically the movement and search dynamics of a diffusing random walk on sparse graphs. We show its utility by uncovering the existence of a bi-modality regime in the time-dependence of the first-passage probability to hit a target node in a small-world network. By identifying the network features that give rise to the bi-modal regime, we challenge long-held beliefs on how the statistics of the so-called direct, intermediate, and indirect trajectories influence the shape of the resulting first-passage and first-absorption probabilities and the interpretation of their mean values. Overall these findings show that temporal features in first-passage studies can be utilised to unearth novel transport paradigms in spatially heterogeneous environments.