Using Spectral Radius Ratio for Node Degree to Analyze the Evolution of Complex Networks

In this paper, we show that the spectral radius ratio for node degree could be used to analyze the variation of node degree during the evolution of complex networks. We focus on three commonly studied models of complex networks: random networks, scale-free networks and small-world networks. The spectral radius ratio for node degree is defined as the ratio of the principal (largest) eigenvalue of the adjacency matrix of a network graph to that of the average node degree. During the evolution of each of the above three categories of networks (using the appropriate evolution model for each category), we observe the spectral radius ratio for node degree to exhibit high-very high positive correlation (0.75 or above) to that of the coefficient of variation of node degree (ratio of the standard deviation of node degree and average node degree). We show that the spectral radius ratio for node degree could be used as the basis to tune the operating parameters of the evolution models for each of the three categories of complex networks as well as analyze the impact of specific operating parameters for each model.