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.
Highlights
Since its introduction by Euler in eighteen century, graph theory has proven its important applications in many different scientific fields
In this work we examined the centrality measures such as closeness, degree, eigenvector, Katz and subgraph
We showed the relationship between the Katz centrality and eigenvector as well as degree centrality
Summary
Since its introduction by Euler in eighteen century, graph theory has proven its important applications in many different scientific fields. 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. 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]. We will use the Kendall correlation coefficient [2] in the experimental work to determine the correlations
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