Generalized network measures based on modulus of families of walks

The modulus of a family of walks quantifies the richness of the family by favoring many short walks over fewer longer ones. In this paper we investigate various families of walks in order to introduce new measures for quantifying network properties using modulus. The proposed new measures are compared to other known quantities such as current-flow closeness centrality, out-degree centrality, and current-flow betweenness centrality. Our proposed method is based on walks on a network, and therefore will work in great generality. For instance, the networks we consider can be directed, multi-edged, weighted, and even contain disconnected parts. Examples are provided to show the effectiveness of our measures.