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

On the Connectivity of Unions of Random Graphs

Graph-theoretic tools and techniques have seen wide use in the multi-agent systems literature, and the unpredictable nature of some multi-agent communications has been successfully modeled using random communication graphs. Across both network control and network optimization, a common assumption is that the union of agents' communication graphs is connected across any finite interval of some prescribed length, and some convergence results explicitly depend upon this length. Despite the prevalence of this assumption and the prevalence of random graphs in studying multi-agent systems, to the best of our knowledge, there has not been a study dedicated to determining how many random graphs must be in a union before it is connected. To address this point, this paper solves two related problems. The first bounds the number of random graphs required in a union before its expected algebraic connectivity exceeds the minimum needed for connectedness. The second bounds the probability that a union of random graphs is connected. The random graph model used is the Erd\H{o}s-R\'enyi model, and, in solving these problems, we also bound the expectation and variance of the algebraic connectivity of unions of such graphs. Numerical results for several use cases are given to supplement the theoretical developments made.Comment: 16 pages, 3 tables; accepted to 2017 IEEE Conference on Decision and Control (CDC