Minimum spanning tree hierarchical clustering algorithm: A new Pythagorean fuzzy similarity measure for the analysis of functional brain networks

Clustering structures are one of the most important aspects of complex networks. Minimum spanning tree (MST), the tree that connects all vertices with minimum total weight, can be considered as a fundamental unit of original weighted graphs. There are different types of algorithms that identify clusters in a network, but the existing theories and algorithms for searching trees have not been investigated for uncertain scenarios. This paper tackles the situations where network parameters may be uncertain. Rather, we permit the parameters to take the form of Pythagorean fuzzy numbers (PFNs). Moreover, to represent qualitative aspects of uncertainty, the use of linguistic variables (LVs) has effective means for experts in expressing their views. The current study proposes a graph theory-based agglomerative hierarchical clustering technique for Pythagorean fuzzy sets. We first define generalized Pythagorean fuzzy numbers (GPFNs) and LR-type PFNs. Then we compute the PF distance between two GPFNs and also LR-type PFNs, and formulate the expressions for PF similarity measure. This paper mainly examines the use of PF distance and similarity measures in a minimum spanning tree agglomerative hierarchical clustering method by considering PFLVs. Since the structural and functional systems of brain are characterized by complex networks, we apply the proposed algorithm on a functional brain network to prove its practicality and efficiency. We discuss the hierarchical clustering consequences of the proposed algorithm in the shape of a dendrogram. Finally, we compare the clustering results obtained by different similarity measures.