Stability and phase transitions of dynamical flow networks with finite capacities

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 analyse using tools from monotone systems and contraction theory. Specifically, we prove that there exists a set of equilibrium points that is globally asymptotically stable. Such set of equilibrium points reduces to a single globally asymptotically stable equilibrium point for generic exogenous demand vectors. Moreover, we show that the critical exogenous demand vectors giving rise to non-unique equilibrium points correspond to phase transitions in the asymptotic behavior of the dynamical flow network.