Egocentric Effective Conductance Centrality and General Degree

We study the popular centrality measure known as effective conductance or in some circles as information centrality. After reinterpreting this measure in terms of modulus (energy) of families of walks on the network, we introduce a new measure called shell modulus centrality, that relies on the egocentric structure of the graph. Egonetworks are networks formed around a focal node (ego) with a specific order of neighborhoods. We then propose efficient analytical and approximate methods for computing these measures on both undirected and directed networks. Finally, we describe a simple method inspired by shell modulus centrality, called general degree, which improves simple degree centrality and could prove to be a useful tool for network science.