Hierarchical organization in complex networks.

Many real networks in nature and society share two generic properties: they are scale-free and they display a high degree of clustering. We show that these two features are the consequence of a hierarchical organization, implying that small groups of nodes organize in a hierarchical manner into increasingly large groups, while maintaining a scale-free topology. In hierarchical networks, the degree of clustering characterizing the different groups follows a strict scaling law, which can be used to identify the presence of a hierarchical organization in real networks. We find that several real networks, such as the Worldwideweb, actor network, the Internet at the domain level, and the semantic web obey this scaling law, indicating that hierarchy is a fundamental characteristic of many complex systems.

[1]  Ginestra Bianconi,et al.  Competition and multiscaling in evolving networks , 2001 .

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

[3]  J. Hopfield,et al.  From molecular to modular cell biology , 1999, Nature.

[4]  Alessandro Vespignani,et al.  Internet topology at the router and autonomous system level , 2002, ArXiv.

[5]  Alessandro Vespignani,et al.  Epidemic spreading in scale-free networks. , 2000, Physical review letters.

[6]  Albert,et al.  Topology of evolving networks: local events and universality , 2000, Physical review letters.

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

[8]  A. Wagner.  The yeast protein interaction network evolves rapidly and contains few redundant duplicate genes. , 2001, Molecular biology and evolution.

[9]  Albert-Laszlo Barabasi,et al.  Deterministic scale-free networks , 2001 .

[10]  Mariano Sigman,et al.  Global organization of the Wordnet lexicon , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[11]  S N Dorogovtsev,et al.  Language as an evolving word web , 2001, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[12]  A. Barabasi,et al.  Bose-Einstein condensation in complex networks. , 2000, Physical review letters.

[13]  M E Newman,et al.  Scientific collaboration networks. I. Network construction and fundamental results. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  Jean-Pierre Eckmann,et al.  Curvature of co-links uncovers hidden thematic layers in the World Wide Web , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[15]  C. Lee Giles,et al.  Accessibility of information on the web , 1999, Nature.

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

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

[18]  V. Eguíluz,et al.  Highly clustered scale-free networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  Mark S. Granovetter.  The Strength of Weak Ties , 1973, American Journal of Sociology.

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

[21]  Alessandro Vespignani,et al.  Topology and correlations in structured scale-free networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Ramon Ferrer i Cancho,et al.  The small world of human language , 2001, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[23]  Mark Newman,et al.  Models of the Small World , 2000 .

[24]  S. Havlin,et al.  Breakdown of the internet under intentional attack. , 2000, Physical review letters.

[25]  Beom Jun Kim,et al.  Growing scale-free networks with tunable clustering. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[27]  M E J Newman,et al.  Identity and Search in Social Networks , 2002, Science.

[28]  A. Barabasi,et al.  Evolution of the social network of scientific collaborations , 2001, cond-mat/0104162.

[29]  Albert-László Barabási,et al.  Error and attack tolerance of complex networks , 2000, Nature.

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

[31]  D. Fell,et al.  The small world inside large metabolic networks , 2000, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[32]  B. Kahng,et al.  Geometric fractal growth model for scale-free networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[33]  宁北芳,et al.  疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A , 2005 .

[34]  L. Amaral,et al.  The web of human sexual contacts , 2001, Nature.

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

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

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

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

[39]  M. Newman,et al.  The structure of scientific collaboration networks. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[40]  A. Barabasi,et al.  Hierarchical Organization of Modularity in Metabolic Networks , 2002, Science.