Stability and phase transitions of dynamical flow networks with finite capacities

TL;DR

This paper analyzes the stability and phase transitions in deterministic dynamical flow networks with finite capacities, revealing conditions for unique equilibrium and the occurrence of phase transitions under varying demands.

Contribution

It introduces a linear saturated system model for flow networks and proves global stability and conditions for phase transitions using monotone systems and contraction theory.

Findings

Existence of a globally asymptotically stable equilibrium set.

Unique equilibrium for generic exogenous demands.

Identification of phase transitions linked to non-unique equilibria.

Abstract

We study deterministic continuous-time lossy dynamical flow networks with constant exogenous demands, fixed routing, and finite flow and buffer capacities. In the considered model, when the total net flow in a cell ---consisting of the difference between the total flow directed towards it minus the outflow from it--- exceeds a certain capacity constraint, then the exceeding part of it leaks out of the system. The ensuing network flow dynamics is a linear saturated system with compact state space that we analyze using tools from monotone systems and contraction theory. Specifically, we prove that there exists a set of equilibria that is globally asymptotically stable. Such equilibrium set reduces to a single globally asymptotically stable equilibrium for generic exogenous demand vectors. Moreover, we show that the critical exogenous demand vectors giving rise to non-unique equilibria correspond to phase transitions in the asymptotic behavior of the dynamical flow network.