Quotient FCMs - A decomposition theory for fuzzy cognitive maps

In this paper, we introduce a decomposition theory for fuzzy cognitive maps (FCMs). First, we partition the set of vertices of an FCM into blocks according to an equivalence relation, and by regarding these blocks as vertices we construct a quotient FCM. Second, each block induces a natural sectional FCM of the original FCM, which inherits the topological structure as well as the inference from the original FCM. In this way, we decompose the original FCM into a quotient FCM and some sectional FCMs. As a result, the analysis of the original FCM is reduced to the analysis of the quotient and sectional FCMs, which are often much smaller in size and complexity. Such a reduction is important in analyzing large-scale FCMs. We also propose a causal algebra in the quotient FCM, which indicates that the effect that one vertex influences another in the quotient depends on the weights and states of the vertices along directed paths from the former to the latter. To illustrate the process involved, we apply our decomposition theory to university management networks. Finally, we discuss possible approaches to partitioning an FCM and major concerns in constructing quotient FCMs. The results represented in this paper provide an effective framework for calculating and simplifying causal inference patterns in complicated real-world applications.