# Generalization of Effective Conductance Centrality for Egonetworks

## TL;DR

This paper extends effective conductance centrality to directed networks using a modulus framework, introduces efficient computation methods for egocentric networks, and proposes a new measure called shell degree for network analysis.

## Contribution

It generalizes effective conductance centrality to directed networks via modulus, and develops efficient algorithms for egocentric network measures, including the novel shell degree.

## Key findings

- Modulus centrality aligns with traditional effective conductance on simple networks.
- New methods enable efficient computation of centrality in directed and egocentric networks.
- Shell degree is a practical tool for local network analysis.

## 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.