Resilient and Risk-Averse Network Systems

Abstract

Graphs are versatile modeling tools capable of effectively representing real-world systems by capturing individual components and their intricate interactions. The main objective of this research is to formulate and develop efficient solution methodologies for graph-theoretical problems involving topologically stochastic information appearing in various forms. Weutilize a stochastic programming framework grounded in coherent risk measures to identify minimum-risk graph structures under stochastic vertex and/or edge weights. First, we consider a network flow problem and propose an approach for developing resilience metrics, where resilience refers to the network’s capability to restore optimal or near-optimal operations following unforeseen (stochastic) disruptions in topology or operational parameters. We illustrate this approach through two examples: the resilient maximum network flow problem and the resilient minimum cost network flow problem. Specifically, these network flow problems aim to achieve resilience against unpredictable losses of network arc capacities by preallocating resources to restore, at least partially, the capacities of arcs. Additionally, we demonstrate that the proposed formulations of resilient network flow problems can be interpreted as ``network risk measures'', possessing properties analogous to convex risk measures. Efficient decomposition algorithms are developed for solving both the resilient maximum network flow problem and the resilient minimum cost network flow problem. Furthermore, we analyze network flow resilience in relation to network structure by conducting studies on three distinct types of network topology: uniform random graphs, scale-free graphs, and grid graphs. Next, we address the regulator's perspective on risk, which concerns regulators or large asset holders who seek to manage the risk of their assets through some investments. A central feature of this perspective is that losses can be “suppressed” by allocating additional resources, meaning that committing zero resources leaves losses unchanged. To illustrate this, we examine two well-known graph problems: the minimum vertex cover and the spanning tree problem. We present mathematical programming formulations for both and employ a stochastic programming framework grounded in coherent risk measures to identify minimum risk graph structures under stochastic vertex or edge weights. After establishing that the decision versions of these problems are NP-hard, we propose a combinatorial branch-and-bound algorithm and compare its performance with an equivalent mathematical programming approach on randomly generated networks. Lastly, we examine graph strength as a robust measure of connectivity. Graph strength quantifies a network's resilience by determining the number of edge-disjoint spanning trees that can be embedded within it. By leveraging this concept, we gain insights into designing networks capable of maintaining functionality despite edge failures, ensuring multiple disjoint paths between any two vertices. We propose a cutting-plane mathematical programming method for computing graph strength, where constraints are iteratively generated by solving a minimum spanning tree problem. Additionally, we investigate the problem of identifying a subgraph with at least a prescribed strength and the largest possible cardinality. We show that while deleting edges from a graph cannot increase its strength, removing vertices can potentially enhance the strength of the remaining subgraph, thereby making the problem well-defined. We prove that finding the maximum subgraph of a given strength is solvable in polynomial time and present the corresponding algorithm. Finally, we provide numerical experiments conducted on a diverse set of real-world graphs to demonstrate the computational performance of the proposed algorithms.