On Balanced Clustering (Indices, Models, Examples)

Abstract

The paper describes and approach to balanced clustering problems. The list of balanced structures includes balanced partitioning of a set, balanced trees, balanced decomposition of a graph, and balanced multilevel structures. Balance indices (characteristics) for balanced structures (clustering solutions) are based on the difference between the cluster parameters: cardinality of a cluster, the total cluster weight, the total weight of edges/arcs of a cluster, and the structure of a cluster in terms of the types of its elements. The proposed balance indices are used as components for optimization models of balanced clustering: objective functions and constraints. Three numerical examples are presented: (1) calculating the balance indices for clustering based on the structure of a cluster in terms of the types of its elements; (2) calculating the balance indices for a clustering solution for a sample network; (3) balanced clustering for forming several student teams.