On the Distribution of the Maximal Clique Size for the Vertices in Real-World Network Graphs and Correlation Studies

Abstract

The high-level contributions of this paper are as follows: We modify an existing branch-andbound based exact algorithm (for maximum clique size of an entire graph) to determine the maximal clique size that the individual vertices in the graph are part of. We then run this algorithm on six real-world network graphs (ranging from random networks to scale-free networks) and analyze the distribution of the maximal clique size of the vertices in these graphs. We observe five of the six real-world network graphs to exhibit a Poisson-style distribution for the maximal clique size of the vertices. We analyze the correlation between the maximal clique size and the clustering coefficient of the vertices, and find these two metrics to be poorly correlated for the real-world network graphs. Finally, we analyze the Assortativity index of the vertices of the real-world network graphs and observe the graphs to exhibit positive assortativity with respect to maximal clique size and negative assortativity with respect to node degree; nevertheless, we observe the Assortativity index of the real-world network graphs with respect to both the maximal clique size and node degree to increase with decrease in the spectral radius ratio for node degree, indicating a positive correlation between the maximal clique size and node degree.