Detectability of multiple link failures in multi-agent systems under the agreement protocol

The detectability of single link failures in a multi-agent network under the agreement protocol has been the subject of a previous study. This work expands the available results by considering the effect of multiple link losses. The analytical results offer sufficient graphical conditions for the detectability of a group of edges in the network information flow digraph. To this end, a powerful extension of the all-minors matrix tree theorem in algebraic graph theory is proved which relates the minors of the Laplace-transformed Laplacian of a directed graph to the number of shortest paths between its vertices. The results reveal an intricate relationship between the detectability of link failures and the inter-nodal paths in the network digraph.