Abstract
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.
Highlights
As critical infrastructure networks, such as transport and energy, are being utilized closer and closer to their capacity limits, the complex interaction between physical systems, cyber layers, and human decision makers has created new challenges in simultaneously achieving efficiency and reliability
We introduce a class of monotone dynamical flow networks, characterized by structural properties of the dependence of the flow variables on the network state
After reviewing some structural stability properties of linear dynamical flow networks, we have focused on an important class of nonlinear dynamical flow networks characterized by an additional monotonicity property
Summary
As critical infrastructure networks, such as transport and energy, are being utilized closer and closer to their capacity limits, the complex interaction between physical systems, cyber layers, and human decision makers has created new challenges in simultaneously achieving efficiency and reliability. This paper: (i) presents results relating the (differential) stability of (nonlinear) monotone dynamical flow networks to graph-theoretical properties; (ii) introduces the notion of margin of resilience as a measure of their robustness against exogenous perturbations; and (iii) studies a class of locally responsive feedback routing and flow control policies that are able to achieve the maximum possible margin of resilience for a given network topology in spite of relying on local information only and requiring no global knowledge of the network. We introduce the notion of margin of resilience as a quantitative measure of robustness and compute the margin of resilience of different classes of distributed routing and flow control policies We end this introductory section by gathering some notational conventions to be adopted throughout the paper.
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