The Sum-Over-Paths Covariance: A novel covariance measure between nodes of a graph

This work introduces a link-based covariance measure between the nodes of a weighted, directed, graph where a cost is associated to each arc. To this end, a probability distribution on the (usually infinite) set of paths through the network is defined by minimizing the sum of the expected costs between all pairs of nodes while fixing the total relative entropy spread in the network. This results in a probability distribution on the set of paths such that long paths (with a high cost) occur with a low probability while short paths (with a low cost) occur with a high probability. The sum-over-paths (SoP) covariance measure is then computed according to this probability distribution: two nodes will be highly correlated if they often co-occur together on the same – preferably short – paths. The resulting covariance matrix between nodes (say n in total) is a Gram matrix and therefore defines a valid kernel matrix on the graph; it is obtained by inverting a n × n matrix. The proposed model could be used for various graph mining tasks such as computing betweenness centrality, semisupervised classification, visualization, etc.