Sometimes There Is No ''Most-Vital'' Arc: Assessing and Improving the Operational Resilience of Systems

Abstract

T his paper shows that no simple, common-sense rule of thumb can be used to identify a most-vital arc, even in a simple maximum-flow problem. The correct answer requires analysis equivalent in difficulty to completely solving the maximum-flow problem, perhaps repeatedly. This insight generalizes to finding a most-vital component, or set of components, in a system whose operation is described by a more general model. Our paper shows how to evaluate the criticality of sets of components, how to assess the worst-case set of components that might be lost to a given number of simultaneous hostile attacks (or engineering failures, or losses to Mother Nature), and how to allocate limited defensive resources to minimize the maximum damage from a subsequent attack. Collateral insights include the fact that there is no way to prioritize individual components by criticality, and that the analysis that determines critical component sets also yields objective assessments of operational system resilience and can provide constructive advice on how to increase it.

Assessing and Improving Operational Resilience of Critical Infrastructures and Other Systems

“…The dashed line connects the worst-case disruptions for this system is our resilience curve, where the term "curve" refers here to a discrete frontier of points. (Figure from Alderson et al [7], Figure 5. )…”

“…We illustrate the basic steps in assessing and improving operational resilience using a simple example, adapted from a historical case study on the Soviet rail network (Harris and Ross [49], [7]). Notes.…”

“…Alderson et al [7] analyze the resilience of the Soviet rail example from Harris and Ross [49] by adding bidirectional flows on track sections and modifying the defensive options and attacks to influence both antiparallel arcs representing flow on a track section, as described above. The assumption is that following an attack, each targeted track section has zero capacity.…”

“…Instead, the focus will be on delivering as much of that function to a particular location (or from a particular location, or both) as the system will bear, and a maximum-flow formulation is the standard way to estimate this capacity of an infrastructure system. The Soviet rail system (Harris and Ross [49], Alderson et al [7]) is a prime example of this; the operator wanted to know how much military equipment and personnel could be moved into Europe (a single "demand node") from various starting locations that can be supplied by a single "source node," limited only by the tonnage restrictions on tracks connecting adjacent locations. Another maximum-flow application is figuring out how much fuel the United States can extract in a fixed time period from the strategic petroleum reserve (Brown et al [22]) during a fuel crisis.…”

“…Schrijver [78], Alderson et al [7]), in which the operator's goal is to move as much cargo as possible from one particular station to another (see Figure 1). In this figure, each line represents a section of track having a finite tonnage limit, and each circle represents a station where adjacent sections of track meet.…”