  We propose a dynamical model for cascading failures in single-commodity
network flows. In the proposed model, the network state consists of flows and
activation status of the links. Network dynamics is determined by a, possibly
state-dependent and adversarial, disturbance process that reduces flow capacity
on the links, and routing policies at the nodes that have access to the network
state, but are oblivious to the presence of disturbance. Under the proposed
dynamics, a link becomes irreversibly inactive either due to overload condition
on itself or on all of its immediate downstream links. The coupling between
link activation and flow dynamics implies that links to become inactive
successively are not necessarily adjacent to each other, and hence the pattern
of cascading failure under our model is qualitatively different than standard
cascade models. The magnitude of a disturbance process is defined as the sum of
cumulative capacity reductions across time and links of the network, and the
margin of resilience of the network is defined as the infimum over the
magnitude of all disturbance processes under which the links at the origin node
become inactive. We propose an algorithm to compute an upper bound on the
margin of resilience for the setting where the routing policy only has access
to information about the local state of the network. For the limiting case when
the routing policies update their action as fast as network dynamics, we
identify sufficient conditions on network parameters under which the upper
bound is tight under an appropriate routing policy. Our analysis relies on
making connections between network parameters and monotonicity in network state
evolution under proposed dynamics.

Systems and Control