Complex-valued information entropy measure for networks with directed links (digraphs). Application to citations by community agents with opposite opinions

TL;DR

This paper introduces a complex-valued information entropy measure for directed networks, exemplified by citation networks between communities with opposing views, revealing new insights into network structure and dynamics.

Contribution

It presents a novel complex-valued entropy measure for directed networks and applies it to community citation networks, offering new physical interpretations and potential extensions.

Findings

Eigenvalues of adjacency matrices can be complex, leading to complex entropy values.

The imaginary part of entropy provides insights into local interaction ranges.

The measure offers new perspectives on directed network analysis.

Abstract

The notion of complex-valued information entropy measure is presented. It applies in particular to directed networks (digraphs). The corresponding statistical physics notions are outlined. The studied network, serving as a case study, in view of illustrating the discussion, concerns citations by agents belonging to two distinct communities which have markedly different opinions: the Neocreationist and Intelligent Design Proponents, on one hand, and the Darwinian Evolution Defenders, on the other hand. The whole, intra- and inter-community adjacency matrices, resulting from quotations of published work by the community agents, are elaborated and eigenvalues calculated. Since eigenvalues can be complex numbers, the information entropy may become also complex-valued. It is calculated for the illustrating case. The role of the imaginary part finiteness is discussed in particular and given some physical sense interpretation through local interaction range consideration. It is concluded that such generalizations are not only interesting and necessary for discussing directed networks, but also may give new insight into conceptual ideas about directed or other networks. Notes on extending the above to Tsallis entropy measure are found in an Appendix.