Bayesian Importance Measures for Network Edges under Saturated Lagrangian Poisson Failures

Bayesian importance measures (BIMs) are useful tools for quantifying the contribution of an edge to the up or down state of the network. This article investigates BIMs for the K-terminal networks under the assumption that the failures of edges occur according to a branching process in which the total number of the failed edges follows a saturated Lagrangian Poisson distribution (SLPD). First, we derive two types of BIM equations when the total number of the failed edges follows a certain probability distribution. Both BIMs are represented in terms of network spectra that depend only on the network structure. It is also found that both BIMs are equivalent as they lead to the identical ranking order of network edges. Next, when the total number of the failed edges has an SLPD, several unique properties of BIMs rankings are explicitly derived. We further prove that under certain conditions, the rankings based on the BIMs belong to the structural ranking, namely, the spectra-based rankings solely depend on the network structure. Finally, the numerical analyses of a transportation network show that the BIMs can effectively measure the edge importance for the medium and relatively large networks.