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

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.

On resilience of distributed routing in networks under cascade dynamics

“…We also plan to extend the formulation here to continuous time framework in the context of dynamical flow networks as pursued in our previous and current work [1], [2], [16].…”

“…The objective of this paper is to provide a tight characterization of margin of resilience under such a setting in terms of network parameters. The setting of this paper is to be contrasted with the dynamical flow network in our previous work [1], [2] where the state of a link is described by density, and whose dynamics is driven by the difference between the inflow and outflow on that link in such a way that the flow always remains within its capacity.…”

Density flow over networks: A mean-field game theoretic approach

“…The first step is to show that the problem results in the following mean-field game system for the unknown scalar functions v(x, t) and ρ(t) when each player behaves according to (5):…”

“…The current paper finds inspiration in the distributed routing problem presented in [5], [6]. We provide a detailed analysis of a similar problem via mean-field games theory.…”

Game Theoretic Decentralized Feedback Controls in Markov Jump Processes

“…Differently from [4], we consider a decentralized control (as in [5][6][7]), in which the density of each node is controlled locally. We highlight next three distinct approaches relating to routing/jump problems.…”