Sparsity Measure of a Network Graph: Gini Index

TL;DR

This paper introduces a generalized Gini Index-based sparsity measure for network graphs, addressing limitations of edge density by satisfying key properties and correlating with degree distributions.

Contribution

The paper formulates a new sparsity index based on Gini Index, demonstrating its properties, advantages over edge density, and its relation to power law degree distributions.

Findings

The sparsity index satisfies key properties like Robin Hood and Scaling.

It correlates inversely with edge density in network graphs.

The index relates analytically to power law degree distributions.

Abstract

This article examines the application of a popular measure of sparsity, Gini Index, on network graphs. A wide variety of network graphs happen to be sparse. But the index with which sparsity is commonly measured in network graphs is edge density, reflecting the proportion of the sum of the degrees of all nodes in the graph compared to the total possible degrees in the corresponding fully connected graph. Thus edge density is a simple ratio and carries limitations, primarily in terms of the amount of information it takes into account in its definition. In this paper, we have provided a formulation for defining sparsity of a network graph by generalizing the concept of Gini Index and call it sparsity index. A majority of the six properties (viz., Robin Hood, Scaling, Rising Tide, Cloning, Bill Gates and Babies) with which sparsity measures are commonly compared are seen to be satisfied by the proposed index. A comparison between edge density and the sparsity index has been drawn with appropriate examples to highlight the efficacy of the proposed index. It has also been shown theoretically that the two measures follow similar trend for a changing graph, i.e., as the edge density of a graph increases its sparsity index decreases. Additionally, the paper draws a relationship, analytically, between the sparsity index and the exponent term of a power law distribution, a distribution which is known to approximate the degree distribution of a wide variety of network graphs. Finally, the article highlights how the proposed index together with Gini index can reveal important properties of a network graph.