Diversity of graphs with highly variable connectivity.

A popular approach for describing the structure of many complex networks focuses on graph theoretic properties that characterize their large-scale connectivity. While it is generally recognized that such descriptions based on aggregate statistics do not uniquely characterize a particular graph and also that many such statistical features are interdependent, the relationship between competing descriptions is not entirely understood. This paper lends perspective on this problem by showing how the degree sequence and other constraints (e.g., connectedness, no self-loops or parallel edges) on a particular graph play a primary role in dictating many features, including its correlation structure. Building on recent work, we show how a simple structural metric characterizes key differences between graphs having the same degree sequence. More broadly, we show how the (often implicit) choice of a background set against which to measure graph features has serious implications for the interpretation and comparability of graph theoretic descriptions.

[1]  Edward A. Bender,et al.  The Asymptotic Number of Labeled Graphs with Given Degree Sequences , 1978, J. Comb. Theory A.

[2]  Steven Skiena,et al.  The Algorithm Design Manual , 2020, Texts in Computer Science.

[3]  P. Gács,et al.  Algorithms , 1992 .

[4]  Bruce A. Reed,et al.  A Critical Point for Random Graphs with a Given Degree Sequence , 1995, Random Struct. Algorithms.

[5]  Benoit B. Mandelbrot,et al.  Fractals and Scaling in Finance , 1997 .

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

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

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

[9]  A. Barabasi,et al.  Spectra of "real-world" graphs: beyond the semicircle law. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

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

[12]  K. Sneppen,et al.  Specificity and Stability in Topology of Protein Networks , 2002, Science.

[13]  M E J Newman.  Assortative mixing in networks. , 2002, Physical review letters.

[14]  M. Newman,et al.  Mixing patterns in networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  M. Newman,et al.  Why social networks are different from other types of networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Stefan Bornholdt,et al.  Handbook of Graphs and Networks: From the Genome to the Internet , 2003 .

[17]  R. Pastor-Satorras,et al.  Class of correlated random networks with hidden variables. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Fan Chung Graham,et al.  The Average Distance in a Random Graph with Given Expected Degrees , 2004, Internet Math..

[19]  Uncorrelated random networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[21]  Amitabha Tripathi,et al.  A note on a theorem of Erdös & Gallai , 2003, Discret. Math..

[22]  Alessandro Vespignani,et al.  Cut-offs and finite size effects in scale-free networks , 2003, cond-mat/0311650.

[23]  S. N. Dorogovtsev.  Clustering of correlated networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  K. Sneppen,et al.  Detection of topological patterns in complex networks: correlation profile of the internet , 2002, cond-mat/0205379.

[25]  I. Sokolov,et al.  Reshuffling scale-free networks: from random to assortative. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Jennifer Rexford,et al.  Proceedings of the 2004 conference on Applications, technologies, architectures, and protocols for computer communications , 2004, SIGCOMM 2004.

[27]  Albert-László Barabási,et al.  Evolution of Networks: From Biological Nets to the Internet and WWW , 2004 .

[28]  M. Newman,et al.  Statistical mechanics of networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  Sergey N. Dorogovtsev,et al.  Organization of complex networks without multiple connections , 2005, Physical review letters.

[30]  Marián Boguñá,et al.  Tuning clustering in random networks with arbitrary degree distributions. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  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 .

[32]  R. Tanaka,et al.  Scale-rich metabolic networks. , 2005, Physical review letters.

[33]  R. Pastor-Satorras,et al.  Generation of uncorrelated random scale-free networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  Walter Willinger,et al.  Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications , 2005, Internet Math..

[35]  S. Low,et al.  The "robust yet fragile" nature of the Internet. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[36]  Walter Willinger,et al.  Understanding Internet topology: principles, models, and validation , 2005, IEEE/ACM Transactions on Networking.

[37]  Alessandro Vespignani,et al.  Detecting rich-club ordering in complex networks , 2006, physics/0602134.