As noted in Nagurney and Qiang (2007), in order to be able to evaluate the vulnerability and the reliability of a network, a measure that can quantifiably capture the efficiency/performance of a network must be developed. For example, in a series of papers, beginning
in 2001, Latora and Marchiori discussed the network performance issue by measuring
the “global efficiency” in a weighted network as compared to that of the simple non-weighted
small-world network. In a weighted network, the network is not only characterized by the
edges that connect different nodes, but also by the weights associated with different edges
in order to capture the relationships between different nodes. The network efficiency E of a
network G is defined in the paper of Latora and Marchiori (2001) as