Joint Optimization of Resources and Routes for Minimum Resistance: From Communication Networks to Power Grids

In this chapter, we are concerned with robustness design in complex communication networks and power grids. We define robustness as the ability of a network to adapt to environmental variations such as traffic fluctuations, topology modifications, and changes in the source (sink) of external traffic. We present a network theory approach to the joint optimization of resources and routes in a communication network to provide robust network operation. Our main metrics are the well-known point-to-point resistance distance and network criticality (average resistance distance) of a graph. We show that some of the key performance metrics in a communication network, such as average link betweenness sensitivity or average network utilization, can be expressed as a weighted combination of point-to-point resistance distances. A case of particular interest is when the external demand is specified by a traffic matrix. We extend the notion of network criticality to be a traffic-aware metric. Traffic-aware network criticality is then a weighted linear combination of point-to-point resistance distances of the graph. For this reason, in this chapter, we focus on a weighted linear sum of resistance distances (which is a convex function of link weights) as the main metric and we discuss a variety of optimization problems to jointly assign routes and flows in a network. We provide a complete mathematical analysis of the network planning problem (optimal weight assignment), where we assume that a routing algorithm is already in place and governs the distribution of network flows. Then, we extend the analysis to a more general case involving the simultaneous optimization of resources and flows (routes) in a network. Furthermore, we briefly discuss the problems of finding the best set of demands that can be matched to a given network topology (joint optimization of resources, flows, and demands) subject to the condition that the weighted linear sum of all point-to-point resistance distances of the network should remain below a certain threshold. We discuss applications of the proposed optimization methods to the design of virtual networks. Moreover, we show how our techniques can be used in the design of robust communication networks and robust sparse power grids.