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

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.