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

摘要

The notion of complex-valued information entropy measure is presented. Itapplies in particular to directed networks (digraphs). The correspondingstatistical physics notions are outlined. The studied network, serving as acase study, in view of illustrating the discussion, concerns citations byagents belonging to two distinct communities which have markedly differentopinions: the Neocreationist and Intelligent Design Proponents, on one hand,and the Darwinian Evolution Defenders, on the other hand. The whole, intra- andinter-community adjacency matrices, resulting from quotations of published workby the community agents, are elaborated and eigenvalues calculated. Sinceeigenvalues can be complex numbers, the information entropy may become alsocomplex-valued. It is calculated for the illustrating case. The role of theimaginary part finiteness is discussed in particular and given some physicalsense interpretation through local interaction range consideration. It isconcluded that such generalizations are not only interesting and necessary fordiscussing directed networks, but also may give new insight into conceptualideas about directed or other networks. Notes on extending the above to Tsallisentropy measure are found in an Appendix.