  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.
