Generalized Erdos Numbers for network analysis

TL;DR

This paper introduces Generalized Erdős Numbers (GENs), a novel topological measure of node closeness in weighted networks that is asymmetric, resource-aware, and correlates with PageRank, with applications in network structure and epidemic spread analysis.

Contribution

The paper proposes GENs as a new asymmetric, non-metric measure of closeness, linking it to centrality, PageRank, and network dynamics, including epidemic modeling.

Findings

GENs are effective in characterizing network structure.

GENs correlate highly with PageRank.

Application to epidemic spread demonstrates practical utility.

Abstract

In this paper we consider the concept of `closeness' between nodes in a weighted network that can be defined topologically even in the absence of a metric. The Generalized Erd\H{o}s Numbers (GENs) satisfy a number of desirable properties as a measure of topological closeness when nodes share a finite resource between nodes as they are real-valued and non-local, and can be used to create an asymmetric matrix of connectivities. We show that they can be used to define a personalized measure of the importance of nodes in a network with a natural interpretation that leads to a new global measure of centrality and is highly correlated with Page Rank. The relative asymmetry of the GENs (due to their non-metric definition) is linked also to the asymmetry in the mean first passage time between nodes in a random walk, and we use a linearized form of the GENs to develop a continuum model for `closeness' in spatial networks. As an example of their practicality, we deploy them to characterize the structure of static networks and show how it relates to dynamics on networks in such situations as the spread of an epidemic.