An extended clique degree distribution and its heterogeneity in cooperation–competition networks

After Xiao et al. [W.-K. Xiao, J. Ren, F. Qi, Z.W. Song, M.X. Zhu, H.F. Yang, H.Y. Jin, B.-H. Wang, Tao Zhou, Empirical study on clique-degree distribution of networks, Phys. Rev. E 76 (2007) 037102], in this article we present an investigation on so-called k -cliques, which are defined as complete subgraphs of k ( k > 1 ) nodes, in the cooperation–competition networks described by bipartite graphs. In the networks, the nodes named actors are taking part in events, organizations or activities, named acts. We mainly examine a property of a k -clique called “ k -clique act degree”, q , defined as the number of acts, in which the k -clique takes part. Our analytic treatment on a cooperation–competition network evolution model demonstrates that the distribution of k -clique act degrees obeys Mandelbrot distribution, P ( q ) ∝ ( q + α ) − γ . To validate the analytical model, we have further studied 13 different empirical cooperation–competition networks with the clique numbers k = 2 and k = 3 . Empirical investigation results show an agreement with the analytic derivations. We propose a new “heterogeneity index”, H , to describe the heterogeneous degree distributions of k -clique and heuristically derive the correlation between H and α and γ . We argue that the cliques, which take part in the largest number of acts, are the most important subgraphs, which can provide a new criterion to distinguish important cliques in the real world networks.