This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TFUZZ.2019.2953615, IEEE Transactions on Fuzzy Systems Modeling of Complex System Phenomena via Computing with Words in Fuzzy Cognitive Maps John T. Rickard, Senior Member, IEEE, Janet Aisbett, Member, IEEE, David G. Morgenthaler, Member, IEEE and Ronald R. Yager, Life Fellow, IEE Abstract--Fuzzy Cognitive Maps (FCMs) play an important role in high level reasoning but are limited in their ability to model complex systems with singularities. We are interested in systems which exhibit discontinuous behaviors as one or more of their internal node states approaches a threshold. In a new approach to FCM dynamics, we define general classes of aggregation functions which “jump” to a boundary value when any input crosses a threshold, or when all inputs do. The threshold value is a context- dependent parameter which can be readily understood by subject matter experts. Aggregation functions are applied separately to positively and negatively causal antecedents to each node then combined to form the nodal state. This modeling is applied in Computing with Words settings, in which link strengths and activation levels are elicited using vocabulary words represented by interval type-2 fuzzy membership functions. We illustrate the behaviors of these novel FCM systems in comparison with their non-singularity versions. Index terms—Aggregation functions, complex systems, computing with words, fuzzy cognitive maps. I. INTRODUCTION Fuzzy cognitive maps (FCM) [1]-[6] are fuzzy signed di- graphs in which the nodes represent high-level descriptive concepts and the links represent positively or negatively causal influences between concepts, along with the corresponding strengths of these influences. In Kosko’s original formulations of FCMs as qualitative models of social systems [7], the node activations and link strengths were described in linguistic terms, but in most subsequent work, they are described using scalar values. FCMs have been applied in numerous fields to model first-order feedback relationships in complex systems, with the general objective of predicting the activations of certain key nodes resulting from the fixed activations of one or more “exogenous” nodes [8]-[10]. These predictions involve iterating the FCM as a complex, nonlinear dynamical system, desirably until it Submitted July 18, 2019, revised October 24, 2019. John T. Rickard is with Meraglim Holdings Corporation, Larkspur, CO, USA (email: terry.rickard@hushmail.com) Janet Aisbett is with Meraglim Holdings Corporation, Newcastle, NSW, Australia (email: janet.aisbett@gmail.com) converges to a steady state, from which the corresponding activations of the key nodes are obtained. In almost all applications, the individual node structure of an FCM conforms to a neuronal model ubiquitous in artificial neural networks [11]-[13]. The functional form of the “squashing” transformation in the neuronal model, along with parameters such as scale, are generally specified by a fuzzy systems expert rather than by a subject matter expert (SME) (e.g. [4]). However, the concepts representing the nodes in an FCM and the signed directed links and their corresponding causal strengths are usually formulated via a consensus of SME opinion as to the important factors and the direction and approximate strength of influences (e.g. [4],[11]). While the link strengths may be expressed initially in terms of linguistic values, these are typically converted to numbers. Recent work in FCMs has revived the spirit of Kosko’s original formulation by modeling link strengths, concept values [14] and even the existence of the link itself as fuzzy sets on the unit interval (perhaps constrained to a particular form as intervals, intuitionistic fuzzy sets [15] or grey numbers [16]). Computing with Words (CWW) [17] approaches to FCMs in which vocabulary words are represented by type-2 fuzzy sets have also been taken [18]. Because the traditional neuronal model is hard to justify when modelling higher-level conceptual features, in [19] we investigated the class of mean operators [20]-[21] as potential aggregation operators. The subclass of weighted power means (WPMs) [22] was selected because it supports importance weighting of the antecedents and, through choice of the power exponent, can take a continuously variable perspective ranging from the most pessimistic (the minimum activation amongst the input antecedents) to the most optimistic (the maximum activation). WPMs are an example of a more general class of operators called quasi-means or Kolmogorov-Nagumo means [23] in which inputs are operated on by an invertible function prior to being aggregated as a weighted sum which is then transformed by the inverse of the initial function. With this form of aggregation, one can feasibly compute aggregations of fuzzy David G. Morgenthaler is with Leidos Holdings, Lakewood, CO, USA (email: david.g.morgenthaler@gmail.com) Ronald R. Yager is with the Machine Intelligence Institute, Iona College, New Rochelle, NY, USA (email: yager@panix.com) .