Robust Network Routing under Cascading Failures

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

Resilience of Locally Routed Network Flows: More Capacity is Not Always Better

In this paper, we are concerned with the resilience of locally routed network flows with finite link capacities. In this setting, an external inflow is injected to the so-called origin nodes. The total inflow arriving at each node is routed locally such that none of the outgoing links are overloaded unless the node receives an inflow greater than its total outgoing capacity. A link irreversibly fails if it is overloaded or if there is no operational link in its immediate downstream to carry its flow. For such systems, resilience is defined as the minimum amount of reduction in the link capacities that would result in the failure of all the outgoing links of an origin node. We show that such networks do not necessarily become more resilient as additional capacity is built in the network. Moreover, when the external inflow does not exceed the network capacity, selective reductions of capacity at certain links can actually help averting the cascading failures, without requiring any change in the local routing policies. This is an attractive feature as it is often easier in practice to reduce the available capacity of some critical links than to add physical capacity or to alter routing policies, e.g., when such policies are determined by social behavior, as in the case of road traffic networks. The results can thus be used for real-time monitoring of distance-to-failure in such networks and devising a feasible course of actions to avert systemic failures.Comment: Accepted to the IEEE Conference on Decision and Control (CDC), 201

Efficient robust routing for single commodity network flows

We study single commodity network flows with suitable robustness and efficiency specs. An original use of a maximum entropy problem for distributions on the paths of the graph turns this problem into a steering problem for Markov chains with prescribed initial and final marginals. From a computational standpoint, viewing scheduling this way is especially attractive in light of the existence of an iterative algorithm to compute the solution. The present paper builds on [13] by introducing an index of efficiency of a transportation plan and points, accordingly, to efficient-robust transport policies. In developing the theory, we establish two new invariance properties of the solution (called bridge) \u2013 an iterated bridge invariance property and the invariance of the most probable paths. These properties, which were tangentially mentioned in our previous work, are fully developed here. We also show that the distribution on paths of the optimal transport policy, which depends on a \u201ctemperature\u201d parameter, tends to the solution of the \u201cmost economical\u201d but possibly less robust optimal mass transport problem as the temperature goes to zero. The relevance of all of these properties for transport over networks is illustrated in an example

On resilient control of dynamical flow networks

Resilience has become a key aspect in the design of contemporary infrastructure networks. This comes as a result of ever-increasing loads, limited physical capacity, and fast-growing levels of interconnectedness and complexity due to the recent technological advancements. The problem has motivated a considerable amount of research within the last few years, particularly focused on the dynamical aspects of network flows, complementing more classical static network flow optimization approaches. In this tutorial paper, a class of single-commodity first-order models of dynamical flow networks is considered. A few results recently appeared in the literature and dealing with stability and robustness of dynamical flow networks are gathered and originally presented in a unified framework. In particular, (differential) stability properties of monotone dynamical flow networks are treated in some detail, and the notion of margin of resilience is introduced as a quantitative measure of their robustness. While emphasizing methodological aspects -- including structural properties, such as monotonicity, that enable tractability and scalability -- over the specific applications, connections to well-established road traffic flow models are made.Comment: accepted for publication in Annual Reviews in Control, 201

Equilibria and Systemic Risk in Saturated Networks

We undertake a fundamental study of network equilibria modeled as solutions of fixed point equations for monotone linear functions with saturation nonlinearities. The considered model extends one originally proposed to study systemic risk in networks of financial institutions interconnected by mutual obligations and is one of the simplest continuous models accounting for shock propagation phenomena and cascading failure effects. It also characterizes Nash equilibria of constrained quadratic network games with strategic complementarities. We first derive explicit expressions for network equilibria and prove necessary and sufficient conditions for their uniqueness encompassing and generalizing results available in the literature. Then, we study jump discontinuities of the network equilibria when the exogenous flows cross certain regions of measure 0 representable as graphs of continuous functions. Finally, we discuss some implications of our results in the two main motivating applications. In financial networks, this bifurcation phenomenon is responsible for how small shocks in the assets of a few nodes can trigger major aggregate losses to the system and cause the default of several agents. In constrained quadratic network games, it induces a blow-up behavior of the sensitivity of Nash equilibria with respect to the individual benefits.Comment: 26 page

On resilient control of dynamical flow networks

Resilience has become a key aspect in the design of contemporary infrastructure networks. This comes as a result of ever-increasing loads, limited physical capacity, and fast-growing levels of interconnectedness and complexity due to the recent technological advancements. The problem has motivated a considerable amount of research within the last few years, particularly focused on the dynamical aspects of network flows, complementing more classical static network flow optimization approaches.In this tutorial paper, a class of single-commodity first-order models of dynamical flow networks is considered. A few results recently appeared in the literature and dealing with stability and robustness of dynamical flow networks are gathered and originally presented in a unified framework. In particular, (differential) stability properties of monotone dynamical flow networks are treated in some detail, and the notion of margin of resilience is introduced as a quantitative measure of their robustness. While emphasizing methodological aspects -including structural properties, such as monotonicity, that enable tractability and scalability- over the specific applications, connections to well-established road traffic flow models are made

Generalized Proportional Allocation Policies for Robust Control of Dynamical Flow Networks

We study a robust control problem for dynamical flow networks. In the considered dynamical models, traffic flows along the links of a transportation network --modeled as a capacited multigraph-- and queues up at the nodes, whereby control policies determine which incoming queues at a node are to be allocated service simultaneously, within some predetermined scheduling constraints. We first prove a fundamental performance limitation by showing that for a dynamical flow network to be stabilizable by some control policy it is necessary that the exogenous inflows belong to a certain stability region, that is determined by the network topology, link capacities, and scheduling constraints. Then, we introduce a family of distributed controls, referred to as Generalized Proportional Allocation (GPA) policies, and prove that they stabilize a dynamical transportation network whenever the exogenous inflows belong to such stability region. The proposed GPA control policies are decentralized and fully scalable as they rely on local feedback information only. Differently from previously studied maximally stabilizing control strategies, the GPA control policies do not require any global information about the network topology, the exogenous inflows, or the routing, which makes them robust to demand variations and unpredicted changes in the link capacities or the routing decisions. Moreover, the proposed GPA control policies also take into account the overhead time while switching between services. Our theoretical results find one application in the control of urban traffic networks with signalized intersections, where vehicles have to queue up at junctions and the traffic signal controls determine the green light allocation to the different incoming lanes