The Quantum Theil Index: Characterizing Graph Centralization using von Neumann Entropy

TL;DR

This paper introduces a quantum-inspired measure called the von Neumann Theil index to quantify graph centralization, providing a novel, information-theoretic approach that captures macro-structural changes more effectively than traditional metrics.

Contribution

It establishes a new graph centralization measure based on von Neumann entropy and Theil index, linking quantum information theory with graph analysis for the first time.

Findings

The von Neumann index bounds the graph's Theil index, characterizing centralization.

The von Neumann Theil index offers a more comprehensive measure than classical centralization metrics.

Future work includes generalizing the measure using Renyi entropy and bounding the Jain fairness index.

Abstract

We show that the von Neumann entropy (from herein referred to as the von Neumann index) of a graph's trace normalized combinatorial Laplacian provides structural information about the level of centralization across a graph. This is done by considering the Theil index, which is an established statistical measure used to determine levels of inequality across a system of `agents', e.g., income levels across a population. Here, we establish a Theil index for graphs, which provides us with a macroscopic measure of graph centralization. Concretely, we show that the von Neumann index can be used to bound the graph's Theil index, and thus we provide a direct characterization of graph centralization via the von Neumann index. Because of the algebraic similarities between the bound and the Theil index, we call the bound the von Neumann Theil index. %From an information theoretic perspective, the von Neumann Theil index describes the mutual information between the quantum state corresponding to a graph and the maximally mixed state (which corresponds to the complete graph). We elucidate our ideas by providing examples and a discussion of different $n=7$ vertex graphs. We also discuss how the von Neumann Theil index provides a more comprehensive measure of centralization when compared to traditional centralization measures, and when compared to the graph's classical Theil index. This is because it more accurately accounts for macro-structural changes that occur from micro-structural changes in the graph (e.g., the removal of a vertex). Finally, we provide future direction, showing that the von Neumann Theil index can be generalized by considering the R\'enyi entropy. We then show that this generalization can be used to bound the negative logarithm of the graph's Jain fairness index.