# Displacement of transport processes on networked topologies

## TL;DR

This paper introduces the winding distance to measure particle displacement on networks, revealing a topology-induced transition from diffusive to ballistic behavior for Brownian particles and extending analysis to anomalous diffusion.

## Contribution

It proposes the winding distance as a new measure and derives a macroscopic model showing topology-driven displacement transitions in networked transport.

## Key findings

- Confinement can induce a diffusive to ballistic transition in displacement.
- A topological condition predicts when the transition occurs.
- Network topology influences long-time behavior of anomalous diffusion.

## Abstract

Consider a particle whose position evolves along the edges of a network. One definition for the displacement of a particle is the length of the shortest path on the network between the current and initial positions of the particle. Such a definition fails to incorporate information of the actual path the particle traversed. In this work we consider another definition for the displacement of a particle on networked topologies. Using this definition, which we term the winding distance, we demonstrate that for Brownian particles, confinement to a network can induce a transition in the mean squared displacement from diffusive to ballistic behaviour, $\langle x^2(t) \rangle \propto t^2$ for long times. A multiple scales approach is used to derive a macroscopic evolution equation for the displacement of a particle and uncover a topological condition for whether this transition in the mean squared displacement will occur. Furthermore, for networks satisfying this topological condition, we identify a prediction of the timescale upon which the displacement transitions to long-time behaviour. Finally, we extend the investigation of displacement on networks to a class of anomalously diffusive transport processes, where we find that the mean squared displacement at long times is affected by both network topology and the character of the transport process.