On resilience of distributed routing in networks under cascade dynamics

We consider network flow over graphs between a single origin-destination pair, where the network state consists of flows and activation status of the links. The evolution of the activation status of a link is given by an irreversible transition that depends on the saturation status of that link and the activation status of the downstream links. The flow dynamics is determined by activation status of the links and node-wise routing policies under the flow balance constraints at the nodes. We formulate a deterministic discrete time dynamics for the network state, where the time epochs correspond to a change in the activation status of the links, and study network resilience towards disturbances that reduce link-wise flow capacities, under distributed routing policies. The margin of resilience is defined as the minimum, among all possible disturbances, of the link-wise sum of reductions in flow capacities, under which the links outgoing from the origin node become inactive in finite time. We propose a backward propagation algorithm to compute an upper bound on the margin of resilience for tree-like network topologies with breadth at most 2, and show that this bound is tight for trees with the additional property of having depth at most 2.