Integrating “Hard” and “Soft” Infrastructural Resilience Assessment for Water Distribution Systems

Table 2

GT metrics used for the analysis of the technical dimension of resilience. The number of WDS nodes and links is denoted by and , respectively.

Resilience dimension	Metric	Formula	Description
Robustness	Density of bridges		A bridge is a link whose removal isolates part of the network. It relates the number of bridges () to the edges [44].
	Central-point dominance		It is based on the betweenness centrality of each network node, , and of the most central node, . ranges from 0 (regular network) to 1 (star topology) [44, 45].
	Spectral gap		Difference between the first and second eigenvalues of the adjacency matrix. A small spectral gap would probably indicate the presence of bridges [44, 45].
	Algebraic connectivity		The second smallest eigenvalue of the normalized Laplacian matrix of the network. A larger value indicates enhanced fault tolerance against efforts to cut the network into isolated parts [44, 45].
Redundancy	Meshedness coefficient		Ratio between the total and the maximum number of independent loops in a planar graph. It ranges between 0 and 1 and is based on the existence of alternative supply paths [37, 38, 46].
	Clustering coefficient		Based on the ratio of the number of triangular loops to the number of connected triples . It is usually smaller in grid-like structures while higher values indicate a more clustered network [44].
Rapidity	Network efficiency		It is the harmonic mean physical distance between nodes. It ranges between 0 for least-efficient and 100% for most-efficient networks and may be used as proxy for average water travel time [38].