Robust Distributed Routing in Dynamical Flow Networks - Part II: Strong Resilience, Equilibrium Selection and Cascaded Failures

TL;DR

This paper investigates the strong resilience of dynamical flow networks under distributed routing policies, identifying maximum resilience conditions, proposing optimization for delay minimization, and exploring cascading failures through simulations.

Contribution

It introduces a class of locally responsive routing policies that maximize strong resilience and links this resilience to node residual capacities, with optimization and simulation analyses.

Findings

Maximum strong resilience equals the minimum node residual capacity.

Distributed policies can achieve optimal resilience under local information constraints.

Simulations illustrate the relationship between cascaded failures and network resilience.

Abstract

Strong resilience properties of dynamical flow networks are analyzed for distributed routing policies. The latter are characterized by the property that the way the inflow at a non-destination node gets split among its outgoing links is allowed to depend only on local information about the current particle densities on the outgoing links. The strong resilience of the network is defined as the infimum sum of link-wise flow capacity reductions under which the network cannot maintain the asymptotic total inflow to the destination node to be equal to the inflow at the origin. A class of distributed routing policies that are locally responsive to local information is shown to yield the maximum possible strong resilience under such local information constraints for an acyclic dynamical flow network with a single origin-destination pair. The maximal strong resilience achievable is shown to be equal to the minimum node residual capacity of the network. The latter depends on the limit flow of the unperturbed network and is defined as the minimum, among all the non-destination nodes, of the sum, over all the links outgoing from the node, of the differences between the maximum flow capacity and the limit flow of the unperturbed network. We propose a simple convex optimization problem to solve for equilibrium limit flows of the unperturbed network that minimize average delay subject to strong resilience guarantees, and discuss the use of tolls to induce such an equilibrium limit flow in transportation networks. Finally, we present illustrative simulations to discuss the connection between cascaded failures and the resilience properties of the network.