Generalization of effective conductance centrality for egonetworks

Abstract

We study the popular centrality measure known as effective conductance or in some circles as information centrality. This is an important notion of centrality for undirected networks, with many applications, e.g., for random walks, electrical resistor networks, epidemic spreading, etc. In this paper, we first reinterpret this measure in terms of modulus (energy) of families of walks on the network. This modulus centrality measure coincides with the effective conductance measure on simple undirected networks, and extends it to much more general situations, e.g., directed networks as well. Secondly, we study a variation of this modulus approach in the egocentric network paradigm. Egonetworks are networks formed around a focal node (ego) with a specific order of neighborhoods. We propose efficient analytical and approximate methods for computing these measures on both undirected and directed networks. Finally, we describe a simple method inspired by the modulus point-of-view, called shell degree, which proved to be a useful tool for network science.

Modulus of time-respecting paths

“…Proof. The limit (11) follows directly from the lemma. To obtain the monotonicity, let λ 2 ≥ λ 1 > 0 and let ρ 1 be an optimal density for Mod λ 1 p,σ (Γ).…”

“…Discrete p-modulus (see, e.g., [1,2,6,7]) has proven to be a powerful and versatile tool for exploring the structure of graphs and networks. Applications discovered thus far include clustering and community detection [12], the construction of a large class of graph metrics [4], measures of centrality [11], hierarchical graph decomposition [3], and the solution to game-theoretic models of secure network broadcast [5]. The flexibility of p-modulus allows it to be applied to a wide variety of graph types; the essential definitions are easily adapted for directed or undirected graphs, for weighted or unweighted graphs, for simple graphs or multigraphs, or even for hypergraphs.…”