First passage times of transport on planar spatial networks and their connections to off-network planar diffusion

TL;DR

This paper investigates how transport on planar spatial networks transitions from one-dimensional to two-dimensional behavior over different length scales, providing a numerical method to analyze first passage times and their relation to planar diffusion.

Contribution

It introduces a numerical linear algebra approach to compute exact moments of first passage times on planar networks and connects these to planar diffusion behavior.

Findings

Quantifies the length scale where networked diffusion appears planar.

Provides exact moments of first passage times for given networks.

Develops an analytical approximation to the entire distribution of first passage times.

Abstract

Consider a network embedded in the 2D plane, where a particle diffuses along the edges of the network. It is clear that over short length scales a particle moves along a single edge and thus undergoes one-dimensional diffusion. However, on larger length scales it is no longer immediately clear how the transport will behave. One could intuit that as the network is embedded in two dimensions for "large enough" length scales the transport will also appear two-dimensional. Is this true for all networks? Can we quantify the length scales upon which this transition occurs? What is the transport behaviour on intermediate spatial scales? In this paper, we answer these question by presenting a numerical linear algebra approach that provides the exact moments of first passage times for a given network. Comparing these networked first-passage times to first-passage times for planar diffusion reveals several interesting properties of networked transport. In particular we can directly quantify the length scale upon which networked diffusion will appear planar if it does at all. Finally, we introduce an adaptation of the method of maximum entropy to use the moments of first-passage times to construct an analytical approximation to the first-passage times entire probability distribution.