Extended Lorenz majorization and frequencies of distances in an undirected network

TL;DR

This paper introduces an extension of Lorenz majorization to analyze the distribution of distances in undirected networks, revealing that such networks have smaller average and median distances compared to chains, with implications for small-world phenomena.

Contribution

The paper extends Lorenz majorization to undirected networks and establishes that their distance distribution is majorized by that of a chain, providing new insights into network structure.

Findings

Distance distribution in undirected networks Lorenz majorizes that of a chain

Average and median distances in undirected networks are smaller than or equal to those in a chain

Results are relevant for understanding small-world properties and six degrees of separation

Abstract

Findings: We show that the distance distribution in an undirected network Lorenz majorizes the one of a chain. As a consequence, the average and median distances in any such network are smaller than or equal to those of a chain. Research limitations: We restricted our investigations to undirected, unweighted networks. Practical implications: We are convinced that these results are useful in the study of small worlds and the so-called six degrees of separation property.