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The Fuzzy Cognitive Map (FCM) provides a robust model for knowledge representation. FCM is a fuzzy signed weighted directed graph that depicts the knowledge of the domain as nodes representing the factors of the domain and arcs representing the connections among these factors. The centrality of a node in FCM, also called the importance of a node in this paper, is considered the most important index of all the graph theory indices applying to FCM which helps decision makers in analysing their FCM models. By finding the centrality values of the nodes in FCM, the important (central) nodes, which are the focal point for decision makers, are determined. The highest centrality value of a node in FCM is the most important one. Little research has addressed the centrality of the nodes in an FCM using only the degree centrality measure. The degree centrality measure only accounts for the direct connections of the node. Although the degree centrality index is considered an important measure in determining the centrality of a node in an FCM, it is not sufficient and has significant shortcomings; it ignores the importance of the indirect connections, the role of the node's position and flow of information through that node, i.e., how a node is close to other nodes and how the node contributes to the flow of information (communication control) through that node. In the literature, there are other centrality measures that can handle direct and indirect connections to determine the central nodes in a directed graph. This paper presents a new method for identifying the central nodes in an FCM. In order to achieve that, we provide, in addition to the degree measure, new important measures to overcome the above drawbacks. These new centrality measures are: betweenness and closeness measures. In this paper, we calculate and normalize the three centrality measures values for each node in the FCM. These values are then transformed into linguistic terms using 2-tuple fuzzy linguistic representation model. We use the 2-tuple model because it describes the granularity of uncertainty of the fuzzy sets and avoids the loss of information resulted from the imprecision and normalization of the measures. The calculated centrality measures values for each node in the FCM are then aggregated using a 2-tuple fuzzy fusion approach to obtain consensus centrality measure. The resulting aggregated values are then ranked in descending order to identify the most central nodes in the FCM, and this would improve the decision-making and help in simplify the FCM by removing the least important nodes from it. Finally, a list of future works related to this paper is suggested.

A new method for identifying the central nodes in Fuzzy Cognitive Maps using Consensus Centrality Measure