The Library
The von Neumann Theil index : characterizing graph centralization using the von Neumann index
Tools
Tools
Tools
Simmons, David E., Coon, Justin P. and Datta, Animesh (2018) The von Neumann Theil index : characterizing graph centralization using the von Neumann index. Journal of Complex Networks, 6 (6). pp. 859-876. doi:10.1093/comnet/cnx061
|
PDF
WRAP-von-Neumann-Theil-index-characterizing-graph-centralization-Datta-2018.pdf - Accepted Version - Requires a PDF viewer. Download (953Kb) | Preview |
Official URL: https://doi.org/10.1093/comnet/cnx061
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’, for example, 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. 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 Rényi entropy. We then show that this generalization can be used to bound the negative logarithm of the graph’s Jain fairness index.
| Item Type: | Journal Article | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | |||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Physics | |||||||||
| Library of Congress Subject Headings (LCSH): | Graph theory, Matrices | |||||||||
| Journal or Publication Title: | Journal of Complex Networks | |||||||||
| Publisher: | OUP | |||||||||
| Official Date: | 1 December 2018 | |||||||||
| Dates: |
|
|||||||||
| Volume: | 6 | |||||||||
| Number: | 6 | |||||||||
| Page Range: | pp. 859-876 | |||||||||
| DOI: | 10.1093/comnet/cnx061 | |||||||||
| Status: | Peer Reviewed | |||||||||
| Publication Status: | Published | |||||||||
| Reuse Statement (publisher, data, author rights): | David E Simmons, Justin P Coon, Animesh Datta; The von Neumann Theil index: characterizing graph centralization using the von Neumann index, Journal of Complex Networks, Volume 6, Issue 6, 1 December 2018, Pages 859–876, https://doi.org/10.1093/comnet/cnx061 | |||||||||
| Access rights to Published version: | Restricted or Subscription Access | |||||||||
| Date of first compliant deposit: | 11 January 2019 | |||||||||
| Date of first compliant Open Access: | 24 January 2019 | |||||||||
| RIOXX Funder/Project Grant: |
|
|||||||||
| Open Access Version: |
Request changes or add full text files to a record
Repository staff actions (login required)
 |
View Item |
Downloads
Downloads per month over past year
