06109946

j

06109946 Niesink, Patrick Poulin, Keven \v{S}ajna, Mateja Computing transitive closure of bipolar weighted digraphs. Discrete Appl. Math. 161, No. 1-2, 217-243 (2013). 2013 Elsevier Science B.V. (North-Holland), Amsterdam EN fuzzy cognitive map bipolar weighted digraph bipolar fuzzy digraph bipolar random digraph transitive closure

doi:10.1016/j.dam.2012.06.013

Summary: We define a bipolar weighted digraph as a weighted digraph together with the sign function on the arcs such that the weight of each arc lies between 0 and 1, and no two parallel arcs have the same sign. Bipolar weighted digraphs are utilized to model so-called fuzzy cognitive maps, which are used in science, engineering, and the social sciences to represent the causal structure of a body of knowledge. It has been noted in the literature that a transitive closure of a bipolar weighted digraph contains useful new information for the fuzzy cognitive map it models. In this paper we ask two questions: what is a sensible and useful definition of transitive closure of a bipolar weighted digraph, and how do we compute it? We give two answers to each of these questions, that is, we present two distinct models. First, we give a review of the fuzzy digraph model, which has been, in a different form and less rigorously, studied previously in the fuzzy systems literature. Second, we carefully develop a probabilistic model, which is related to the notion of network reliability. This paper is intended for a mathematical audience.