Flow on sweeping networks

TL;DR

This paper introduces a cellular automaton coupled with a transport equation to model flow dynamics on networks, capturing complex behaviors like hysteresis and multiple steady states.

Contribution

It develops a novel coupled cellular automaton and transport model with mean-field analysis for flow on graphs, including pedestrian dynamics.

Findings

Derived a master equation and mean-field models

Identified multiple meta-stable states and hysteresis

Provided analytical steady-state solutions

Abstract

We introduce a cellular automaton model coupled with a transport equation for flows on graphs. The direction of the flow is described by a switching process where the switching probability dynamically changes according to the value of the transported quantity in the neighboring cells. A motivation is pedestrian dynamics in a small corridor where the propagation of people in a part of the corridor can be either left or rightgoing. Under the assumptions of propagation of chaos and mean-field limit, we derive a master equation and the corresponding meanfield kinetic and macroscopic models. Steady--states are computed and analyzed analytically and exhibit the possibility of multiple meta-stable states and hysteresis.