Spectral Radius as a Measure of Variation in Node Degree for Complex Network Graphs

Abstract

The spectral radius of a network graph is the largest Eigen value of the adjacency matrix of the graph. We hypothesize the spectral radius to be a measure of the variation in the degrees of the nodes. In this pursuit, we define a metric called the spectral radius ratio for node degree as the ratio of the spectral radius to the average node degree. We validate our hypothesis by determining this metric on some of the commonly studied classical large real-world complex network graphs (undirected) for network analysis. Based on the results collected, we observe the spectral radius ratio for node degree to be positively correlated (correlation coefficient: 0.75) to the coefficient of variation in node degree (the ratio of the average node degree to the standard deviation in node degree), thus confirming our hypothesis.