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 timedependent 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.