Centrality Measures Based on Matrix Functions

Abstract

Network is considered naturally as a wide range of different contexts, such as biological systems, social relationships as well as various technological scenarios. Investigation of the dynamic phenomena taking place in the network, determination of the structure of the network and community and description of the interactions between various elements of the network are the key issues in network analysis. One of the huge network structure challenges is the identification of the node(s) with an outstanding structural position within the network. The popular method for doing this is to calculate a measure of centrality. We examine node centrality measures such as degree, closeness, eigenvector, Katz and subgraph centrality for undirected networks. We show how the Katz centrality can be turned into degree and eigenvector centrality by considering limiting cases. Some existing centrality measures are linked to matrix functions. We extend this idea and examine the centrality measures based on general matrix functions and in particular, the logarithmic, cosine, sine, and hyperbolic functions. We also explore the concept of generalised Katz centrality. Various experiments are conducted for different networks generated by using random graph models. The results show that the logarithmic function in particular has potential as a centrality measure. Similar results were obtained for real-world networks. 80Open Journal of Discrete Mathematics cal, social and biological scenarios. Networks are used to model a variety of highly interconnected systems, both in nature and man-made world of technology. These networks include protein-protein interaction networks, social networks, food webs, scientific collaboration networks, metabolic networks, lexical or semantic networks, neural networks, the World Wide Web and others. The use of network analysis is in various situations: from determining network structure and communities, describing the interactions between various elements of the network and investigating the dynamics phenomena taking place in the network [1]. One of the ground laying questions analysis of network is how to determine the "most important" nodes in a given network. Many centrality measures have been proposed, starting with the simplest of all, node degree centrality. This measure has being considered too "local", as it does not take into account the connectivity of the immediate neighbours of the node under consideration. A number of centrality measures have been introduced that take into account the global connectivity properties of the network. These include various types of eigenvector centrality for both directed and undirected networks, Katz centrality, subgraph centrality and PageRank centrality [1]. The use of centrality scoresprovides rankings of the nodes in the networks. The higher the ranking of a node, the more important the node is believed to be within the network. There are many different ranking methods in use, and many algorithms have been developed to compute these rankings.The purpose of this p...