Harmonic Centralization of Some Graph Families

TL;DR

This paper investigates the harmonic centralization of specific graph families, providing formulas that could aid in analyzing the centralization of more complex networks.

Contribution

It offers new formulas for harmonic centralization in important graph families, aiding future analysis of complex graphs.

Findings

Derived formulas for harmonic centralization of key graph families

Facilitates comparison of graph centralization across different networks

Provides tools for analyzing the importance distribution in graphs

Abstract

Centrality describes the importance of nodes in a graph and is modeled by various measures. Its global analogue, called centralization, is a general formula for calculating a graph-level centrality score based on the node-level centrality measure. The latter enables us to compare graphs based on the extent to which the connections of a given network are concentrated on a single vertex or group of vertices. One of the measures of centrality in social network analysis is harmonic centrality. It sums the inverse of the geodesic distances of each node to other nodes where it is 0 if there is no path from one node to another, with the sum normalized by dividing it by $m-1$, where $m$ is the number of nodes of the graph. In this paper, we present some results regarding the harmonic centralization of some important families of graphs with the hope that formulas generated herein will be of use when one determines the harmonic centralization of more complex graphs.