Time-Dependent Variation of the Centrality Measures of the Nodes during the Evolution of a Scale-Free Network

Scale-free networks are a type of complex networks in which the degree distribution of the nodes is according to the power-law. Centrality of the nodes is a quantitative measure of the importance of the nodes according to the topological structure of the network. The commonly used centrality measures are the degree-based degree centrality and eigenvector centrality and the shortest path-based closeness centrality and betweenness centrality. We use the widely studied Barabasi-Albert (BA) model to simulate the evolution of scale-free networks. The model works by adding new nodes to the network, one at a time, with the new node connected to m of the currently existing nodes. Accordingly, nodes that have been in the network for a longer time have greater chances of acquiring more links and hence a larger degree centrality. While the degree centrality of the nodes has been observed to show a concave down pattern of increase with time; but the time-dependent variation of the other centrality measures has not been analyzed until now. In this paper, we study the time-dependent variation of degree centrality, eigenvector centrality, closeness centrality and betweenness centrality of the nodes during the evolution of a scale-free network according to the BA model