On the formation of degree and cluster-degree correlations in scale-free networks

The cluster-degree of a vertex is the number of connections among the neighbors of this vertex. In this paper we study the cluster-degree of the generalized Barabasi–Albert model (GBA model) whose exponent of degree distribution ranges from 2 to ∞. We present the mean-field rate equation for clustering and obtain analytically the degree-dependence of the cluster-degree. We study the distribution of the cluster-degree, which is size dependent but the tail is kept invariant for different degree exponents in the GBA model. In addition, for the degree dependence of the clustering coefficient, very different behaviors arise for different cases of the GBA model. The physical sense of the invariance property of cluster-degree is explained and more general cases are discussed. All the above theoretical results are verified by simulation.

[1]  R. Albert,et al.  The large-scale organization of metabolic networks , 2000, Nature.

[2]  S. Redner,et al.  Organization of growing random networks. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  A. Barabasi,et al.  Lethality and centrality in protein networks , 2001, Nature.

[4]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[5]  Ph. Blanchard,et al.  The “Cameo Principle” and the Origin of Scale-Free Graphs in Social Networks , 2004 .

[6]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[7]  S. N. Dorogovtsev,et al.  Structure of growing networks with preferential linking. , 2000, Physical review letters.

[8]  S. Redner.  How popular is your paper? An empirical study of the citation distribution , 1998, cond-mat/9804163.

[9]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[10]  Walter Willinger,et al.  The origin of power laws in Internet topologies revisited , 2002, Proceedings.Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies.

[11]  Reka Albert,et al.  Mean-field theory for scale-free random networks , 1999 .

[12]  S. Strogatz.  Exploring complex networks , 2001, Nature.

[13]  Alessandro Vespignani,et al.  Large-scale topological and dynamical properties of the Internet. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  S. N. Dorogovtsev,et al.  Scaling properties of scale-free evolving networks: continuous approach. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Albert-László Barabási,et al.  Internet: Diameter of the World-Wide Web , 1999, Nature.

[16]  S. N. Dorogovtsev,et al.  Evolution of networks , 2001, cond-mat/0106144.

[17]  Michalis Faloutsos,et al.  On power-law relationships of the Internet topology , 1999, SIGCOMM '99.

[18]  Albert-László Barabási,et al.  Hierarchical organization in complex networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  J. Kertész,et al.  Structural transitions in scale-free networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Chi K. Tse,et al.  Effect of clustering in a complex user network on the telephone traffic , 2006 .

[21]  A. Barabasi,et al.  Scale-free characteristics of random networks: the topology of the world-wide web , 2000 .

[22]  Per Ottar Seglen,et al.  The skewness of science , 1992 .

[23]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[24]  János Kertész,et al.  Clustering in complex networks , 2004 .