Critical Flow Centrality Measures on Interdependent Networks with Time-Varying Demands

This paper presents a method that allows urban planners and municipal engineers to identify critical components of interdependent infrastructure systems. The intent of the method is to provide a means of modeling the impact of capacity-related changes (e.g., population growth, component degradation) on a city’s ability to deliver resources to critical locations. Infrastructure systems are modeled as ﬂow networks in which capacities, demands, and supply constraints vary over time; demand nodes also have criticality ratings that allow a user to model levels of importance. Interconnections between infrastructure systems are represented by physical and geospatial dependencies at a component level. A ﬂow-based centrality measure is used to rank components according to their role in the delivery of resources to critical locations. A simple instantiation of the method is presented and evaluated on a district-scale model of a city that contains interconnected water and electricity networks. Finally, two forms of reliability analysis are demonstrated: a composite measure incorporating edge reliability, and a variation on standard component failure/degradation analysis.


Introduction
This paper presents a method for identifying critical components in interderatio. Flows of water or electricity are not modeled. 180 Buldyrev et al. [10] examine the impact of electricity system disruptions on  provide a method for computing higher order risks from such a graph. 215 Stergiopoulos et al [55] use an RDG in combination with centrality measures 216 (e.g., betweenness, eigenvector centrality, node degree) to identify critical systems.

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The centrality measures are computed in order to identify nodes that affect critical 218 risk paths in the RDG. A decision-tree algorithm is then used to select a subset of these nodes for risk mitigation. Testing their approach on empirical data, the 220 authors make a number of observations about the relationship between centrality 221 and risk (e.g., that the most critical paths in the RDG tend to involve nodes with 222 high centrality). 223 Shahraeini and Kotzanikolaou [56] provide a method to aid in the design of wide

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Of all the aforementioned works, the method in this paper is closest to Apos-242 tolakis and Lemon [45] in overall intent. In both cases the computational model 243 allows for co-location. However, the present work also allows for physical depen-244 dencies, as well as dynamic behavior through the use of time series for key system 245 variables. The temporal aspects of the current work also distinguish it from many 246 of its predecessors.

Methodology
The goal of this work is to explore means by which urban Computation of the centrality measure, critical flow centrality ("CFC"), can 261 be accomplished in several ways (see [57]). In the current paper, a discrete-valued 262 approach is taken in which: (1) an infrastructure system is represented as a flow-263 network; (2) demands, capacities, and supply limits are given as integers, and;

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(3) each demand node in the network is assigned a real-valued criticality rating.  Since infrastructure networks are not independent of each other, physical and 269 geospatial dependencies may be introduced between individual infrastructures.

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The most important of these for the present paper are physical dependencies in 271 which resources provided by one system (e.g., electricity) are used by another 272 system (e.g., water pumps). One of the main contributions of the paper is to show 273 how CFC values can be propagated from one infrastructure system to another.

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The method is demonstrated by applying it to a district-level model of a city. that focuses on dependencies, and; (2) a traditional view of each individual system 311 that is familiar to managers/specialists.

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One means of providing infrastructure models that support multiple perspec-313 tives is through the use of geographical information systems ("GIS") software.

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In fact, the critical information protection community has begun to use GIS as a 315 platform for resilience and vulnerability analysis [62]. For this reason, the method 316 described in this paper was explicitly designed for integration within GIS software. techniques for inferring asset locations from proxy data sources (e.g., [63,64]).

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The model used in this paper is a mixture of synthetic and empirical com-

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This section discusses the building blocks of the simplified model, including:

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(1) the network representation; (2) the use of time series for key system variables;

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(3) criticality ratings, and; (4) interdependencies.  Each network is a multi-graph in which multiple edges may connect a given pair of nodes, allowing for redundant (fallback) connections. Bi-directional rela-352 tionships, cycles, and self-loops are all permitted.

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As a flow network, contains both source (supply) and sink (demand) nodes.

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Supply constraints and resource demands are represented as discrete, integer-

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A criticality function : → R maps demand nodes ∈ to a criticality 385 rating ( ). Although it is possible to use binary (e.g., critical, non-critical) or 386 categorical (e.g., low, medium, high) representations, this paper focuses on the 387 continuous variant in which criticality ratings take on values between 0 and 1.  Dependencies between network elements imply dependencies between sys-411 tems. If an interconnection record exists that maps elements of 1 to elements 412 of 2 , we say that 1 is physically dependent on 2 , represented as 1 → 2 .

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Mutual dependency between systems makes the computational task more diffi-  In this paper, the set of physical (resource) dependencies between systems in 418 S is taken to form a directed, acyclic graph ("DAG") G that can be ordered with a 419 topological sort (see [68]). In contrast, geospatial dependencies are not restricted 420 in such a fashion.

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The Critical Flow Centrality ("CFC") of a component is a measure of its role in supplying resources to critical locations [57]. Recall that the flow in network (given assignment ) is the aggregate of all flows reaching the demand nodes: The critical flow in network (given assignment ) is the set of flows reaching the demand nodes, weighted by criticality: A component (i.e., node or edge) is deemed to be important to the extent that it carries critical flow. Let ( , ) be the flow that reaches ∈ from given assignment , and let [ ( , )] be its expectation. Then the critical flow centrality ("CFC") of component under assignment is: This quantity may be normalized by the critical flow ( ): Computing the CFC thus reduces to computing the probability ( | ) that 423 a unit of commodity passing through component ends up in demand node .

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While there are numerous ways to accomplish this task (e.g., Markov chains), this 425 paper uses a discrete, search-based approach.                   To recap, Algorithm 5 results in a set of CFC values ( ), where is a timestep and is a component. For instance, the output for the water system edges can be represented as a matrix in which rows are timesteps and columns are edges: One major issue not addressed by classical works on network centrality (e.g., [9]) 569 is the choice of ranking method for component measures taken at different times.

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The most intuitive approach to ranking the components is to take the sample mean 571 of each column and to subsequently rank columns in descending order. This would 572 be an appropriate strategy if each row of the matrix was a sample from the space of 573 assignments (i.e., in a Monte Carlo approach) at a given time . However, the rows 574 in the matrix are assessments of the system at different points in time. The use of descriptive statistical measures (e.g., average, variance) elides system dynamics.

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The same is true of various other methods (e.g., spectral analysis, information 577 theory) that might be employed to analyze the matrix.

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The choice of ranking approach is dependent upon the purpose of analysis.

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Consider a long-term (e.g., multi-year) analysis that attempts to study the distri-  An integral is calculated from the cubic spline (as shown in Figure 11

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This section demonstrates that the CFC measure may be combined with stan-604 dard approaches to network reliability-namely, (1) edge reliability measures, and;

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Algorithm 6 shows a high-level view of a procedure in which capacities of edges in an infrastructure system are degraded one-at-a-time. For each time < , appropriate demands and criticality values are loaded into the graph. Then each edge ∈ is considered in order, degrading its capacity and performing the CFC  The edge failure mechanism was tested on the network from