Vulnerability Importance Measures Toward Resilience-Based Network Design

Network resilience to a disruption is generally considered to be a function of the initial impact of the disruption (the network’s vulnerability) and the trajectory of recovery after the disruption (the network’s recoverability). In the context of network resilience, this work develops and compares several flow-based importance measures to prioritize network edges for the implementation of preparedness options. For a particular preparedness option and particular geographically located disruption, we compare the different importance measures in their resulting network vulnerability, as well as network resilience for a general recovery strategy. Results suggest that a weighted flow capacity rate, which accounts for both (i) the contribution of an edge to maximum network flow and (ii) the extent to which the edge is a bottleneck in the network, shows most promise across four instances of varying network sizes and densities. Resilience, broadly defined as the ability to stave off the effects of a disruption and subsequently return to a desired state, has been studied across a number of fields, including engineering (Hollnagel et al. 2006, Ouyang and DuenasOsorio 2012) and risk contexts (Haimes 2009, Aven 2011), to name a few. Resilience has increasingly been seen in the literature (Park et al. 2013), recognizing the need to prepare for the inevitability of disruptions. Figure 1: Graphical depiction of network performance, φ(t), over time (adapted from Henry and Ramirez-Marquez (2012)). Figure 1 illustrates three dimensions of resilience: reliability, vulnerability, and recoverability. The network service function φ(t) describes the behavior or performance of the network at time t (e.g., φ(t) could describe traffic flow or delay for a highway network). Prior to disruption e, the ability of the network to meet performance expectations is described by its reliability, often considered to the likelihood of connectivity of a network. Research in the area of recoverability is related to understanding the ability and speed of networks to recover after a disruptive event, similar in concept to rapidity in the “resilience triangle” literature in civil infrastructure (Bruneau et al. 2003). Emphasis in this paper is placed on the vulnerability dimension, or the ability of e to impact network performance in an adverse manner is a function of the network’s vulnerability (Nagurney and Qiang 2008, Zio et al. 2008, Zhang et al. 2011), similar in concept to 12 th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 2 robustness in the “resilience triangle” literature. Haimes (2006) broadly offers that that the states of the system are described by a state vector, suggesting that vulnerability is multifaceted (i.e., certain aspects of a system may be adversely affected by certain events and not others). Our work adopts this qualitative perspective, though we assume that the vulnerabilities found in the different aspects of the network can still be measured by changes in a single network service function φ(t) . As such, Jonsson et al. (2008) define vulnerability appropriately for our work as the magnitude of damage given the occurrence of a particular disruptive event. Networks have been characterized in two broad categories with respect to how their vulnerability is analyzed (Mishkovski et al. 2011): (i) those that involve “structural robustness,” or how networks behave after the removal of a set of nodes or links based only on topological features, and (ii) those that involve “dynamic robustness,” or how networks behave after the removal of a set of nodes or links given load redistribution leading to potential cascading failures. With respect to Figure 1, in networks that are primarily described by structural robustness (e.g., inland waterway, railway), te and td would coincide with each other such that network performance drops immediately as disruption e occurs. The performance of networks exhibiting dynamic robustness would dissipate over time after a disruption due to cascading effects (e.g., electric power networks), such that td is subsequent to te . This paper focuses on networks described by structural robustness. Emphasis is placed on vulnerability in the larger context of network resilience. Resilience is defined here as the time dependent ratio of recovery over loss, noting the notation for resilience, Я (Whitson and Ramirez-Marquez 2009) as R is commonly reserved for reliability. Similar in concept to the resilience triangle, we make use of the resilience paradigm provided in Figure 1, and we quantify resilience with Eq. (1) (Pant et al. 2014, Baroud et al. 2014, Barker et al. 2013, Henry and Ramirez-Marquez 2012). φ(t0) is the “as-planned” performance level of the network, td is the point in time after the disruption where network performance is at its most disrupted level, and recovery of the network occurs between times ts and tf. Яφ(t|e ) = φ(t|e)−φ(td|e j) φ(t0)−φ(td|e j) (1) 1. QUANTIFYING NETWORK VULNERABILITY A common approach to quantifying network vulnerability is with graph invariants (e.g., connectivity, diameter, betweenness centrality) as deterministic measures (Boesch et al. 2009). We focus on tangible metrics of network behavior in the form of a flow-based service function, φ(t) , rather than graph theoretic measures of performance. For this work, we choose all node pairs average maximum flow for φ, calculated by finding the maximum flow from a source node s to a sink node t, then exhausting all (s, t) pairs across the network and averaging the maximum flow for each (s, t) pair. This work considers geographic based physical networks with capacitated and symmetric arcs. Examples include transportation networks in which traffic per hour on a roadway or bridges with weight restrictions constrain traffic flow. We consider a class of disruptive events that impair the capacity of one or more edges in the network. To prioritize preemptive efforts to reduce network-wide vulnerability, we develop a variety of edge-specific, flow-based metrics to identify the most important edges. Edges deemed as the most important can be reinforced or otherwise protected prior to any event to reduce network vulnerability or can be candidates for expedited recovery (though we focus on the vulnerability, and not recoverability, aspect of network resilience in this work). In this section we provide details concerning various candidate edge importance measures relating to network vulnerability. 12 th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 3 1.1. Notation Let G = (V, E) denote a directed graph where V is a set of n vertices (also called nodes) and E ⊆ V × V is a set of m directed edges (also called arcs or links). For (i, j) ∈ E , the initial vertex i is called the tail and the terminal vertex j is called the head. Let cij and xij denote the capacity and flow on edge (i, j) ∈ E , respectively. A directed path P from a source node s to a target node t is a finite, alternating sequence of vertices and one or more edges starting at node s and ending at node t , P = {s, (s, v1), v1, (v1, v2), v2, ... , (vk, t), t} where all of the odd elements are distinct nodes in V and the even elements are directed edges in E. All nodes other than s and t are referred to as internal nodes. The length of path P is the number of edges it contains. The maximum capacity of a path is equal to the minimum capacity of all edges in the path. That is, the max capacity of path P equals min(i,j)∈P cij. The s-t max flow problem utilizes a subset of all possible paths between s and t to route a maximum amount of a commodity from s to t without exceeding the capacity of any edge. 1.2. Proposed Importance Measures Several importance measures for components of graphs have previously been offered. A frequent theme in these measures is the notion of centrality [Anthonisse 1971, Freeman 1977]. Edge betweenness, for example, of (i, j)εE is a function of the number of shortest paths between nodes s and t which include edge (i, j). The edge betweenness centrality of (i, j) is the sum of its edge betweenness for all s t pairs. Newman [2004] introduced a modified edge centrality that does not restrict the metric to only shortest paths between s and t but stochastically includes other paths. In our work we introduce or otherwise consider several flow-based and topological measures relating to max flow paths within a graph. 1.2.1. All Pairs Max Flow Edge Count The first importance measure is inspired by the basic edge betweenness centrality concept. However instead of shortest paths, we consider max flow paths. The all pairs max flow edge count is the total number of times a given edge is utilized in all s-t pairs max flow problems. The intuition is that if an edge is used more often than others in contributing to maximum flow, then a disruption that impacts its capacity is likely to have a significant impact on network performance φ. Let μst(i, j) = 1 if edge (i, j) is used in a given s t max flow problem and 0 otherwise. We define the first candidate for edge importance based on the raw max flow edge tally divided by the total number of s t pairs, as shown in Eq. (2). If multiple paths share a minimally capacitated edge, there will be multiple paths that contribute the same value to a given s t max flow problem. We arbitrarily choose among the shortest of these otherwise equally capacitated paths.