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Our primary objective in this paper is to study the distribution of themaximal clique size of the vertices in complex networks. We define the maximalclique size for a vertex as the maximum size of the clique that the vertex ispart of and such a clique need not be the maximum size clique for the entirenetwork. We determine the maximal clique size of the vertices using a modifiedversion of a branch-and-bound based exact algorithm that has been originallyproposed to determine the maximum size clique for an entire network graph. Wethen run this algorithm on two categories of complex networks: One category ofnetworks capture the evolution of small-world networks from regular network(according to the wellknown Watts-Strogatz model) and their subsequentevolution to random networks; we show that the distribution of the maximalclique size of the vertices follows a Poisson-style distribution at differentstages of the evolution of the small-world network to a random network; on theother hand, the maximal clique size of the vertices is observed to bein-variant and to be very close to that of the maximum clique size for theentire network graph as the regular network is transformed to a small-worldnetwork. The second category of complex networks studied are real-worldnetworks (ranging from random networks to scale-free networks) and we observethe maximal clique size of the vertices in five of the six real-world networksto follow a Poisson-style distribution. In addition to the above case studies,we also analyze the correlation between the maximal clique size and clusteringcoefficient as well as analyze the assortativity index of the vertices withrespect to maximal clique size and node degree.

Distribution of maximal clique size of the vertices for theoretical small-world networks and real-world networks