Understanding and modeling the internet topology: economics and evolution perspective

In this paper, we seek to understand the intrinsic reasons for the well-known phenomenon of heavy-tailed degree in the Internet AS graph and argue that in contrast to traditional models based on preferential attachment and centralized optimization, the Pareto degree of the Internet can be explained by the evolution of wealth associated with each ISP. The proposed topology model utilizes a simple multiplicative stochastic process that determines each ISP's wealth at different points in time and several "maintenance" rules that keep the degree of each node proportional to its wealth. Actual link formation is determined in a decentralized fashion based on random walks, where each ISP individually decides when and how to increase its degree. Simulations show that the proposed model, which we call Wealth-based Internet Topology (WIT), produces scale-free random graphs with tunable exponent α and high clustering coefficients (between 0.35 and 0.5) that stay invariant as the size of the graph increases. This evolution closely mimics that of the Internet observed since 1997.

[1]  K-I Goh,et al.  Fluctuation-driven dynamics of the internet topology. , 2002, Physical review letters.

[2]  Arnold L. Rosenberg,et al.  Comparing the structure of power-law graphs and the Internet AS graph , 2004, Proceedings of the 12th IEEE International Conference on Network Protocols, 2004. ICNP 2004..

[3]  Fan Chung,et al.  Spectral Graph Theory , 1996 .

[4]  L. Sander,et al.  Geography in a scale-free network model. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  B. Mohar.  THE LAPLACIAN SPECTRUM OF GRAPHS y , 1991 .

[6]  Michalis Faloutsos,et al.  A Systematic Framework for Unearthing the Missing Links: Measurements and Impact , 2007, NSDI.

[7]  Hawoong Jeong,et al.  Modeling the Internet's large-scale topology , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[8]  Ronald W. Wolff,et al.  Stochastic Modeling and the Theory of Queues , 1989 .

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

[10]  Christos H. Papadimitriou,et al.  Heuristically Optimized Trade-Offs: A New Paradigm for Power Laws in the Internet , 2002, ICALP.

[11]  F. Chung,et al.  Connected Components in Random Graphs with Given Expected Degree Sequences , 2002 .

[12]  Fan Chung Graham,et al.  A random graph model for massive graphs , 2000, STOC '00.

[13]  Rajendra Kulkarni,et al.  Spatial Small Worlds: New Geographic Patterns for an Information Economy , 2003 .

[14]  Donald F. Towsley,et al.  On distinguishing between Internet power law topology generators , 2002, Proceedings.Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies.

[15]  D. Sornette.  Multiplicative processes and power laws , 1997, cond-mat/9708231.

[16]  Reuven Cohen,et al.  Geographical embedding of scale-free networks , 2003 .

[17]  T. Erlebach,et al.  A Spectral Analysis of the Internet Topology , 2001 .

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

[19]  Damien Magoni,et al.  Analysis of the autonomous system network topology , 2001, CCRV.

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

[21]  Arun Venkataramani,et al.  iPlane: an information plane for distributed services , 2006, OSDI '06.

[22]  Walter Willinger,et al.  To Peer or Not to Peer: Modeling the Evolution of the Internet's AS-Level Topology , 2006, Proceedings IEEE INFOCOM 2006. 25TH IEEE International Conference on Computer Communications.

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

[24]  Shi Zhou,et al.  Accurately modeling the Internet topology , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Ibrahim Matta,et al.  On the origin of power laws in Internet topologies , 2000, CCRV.

[26]  Walter Willinger,et al.  Network topology generators: degree-based vs. structural , 2002, SIGCOMM 2002.

[27]  Bernardo A. Huberman,et al.  Intentional Walks on Scale Free Small Worlds , 2001, ArXiv.

[28]  Jie Wu,et al.  Small Worlds: The Dynamics of Networks between Order and Randomness , 2003 .

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

[30]  S Redner,et al.  Degree distributions of growing networks. , 2001, Physical review letters.

[31]  J M Carlson,et al.  Highly optimized tolerance: a mechanism for power laws in designed systems. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[32]  Walter Willinger,et al.  In search of the elusive ground truth: the internet's as-level connectivity structure , 2008, SIGMETRICS '08.

[33]  Lixin Gao.  On inferring autonomous system relationships in the internet , 2001, TNET.

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

[35]  Walter Willinger,et al.  Internet connectivity at the AS-level: an optimization-driven modeling approach , 2003, MoMeTools '03.

[36]  S. Solomon,et al.  Dynamical Explanation For The Emergence Of Power Law In A Stock Market Model , 1996 .

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

[38]  Azer Bestavros,et al.  Small-World Internet Topologies: Possible Causes and Implications on Scalability of End-System Multicast , 2002 .

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

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

[41]  Abhijit Kar Gupta.  Models of wealth distributions: a perspective , 2006 .

[42]  Xiang Li,et al.  A local-world evolving network model , 2003 .

[43]  Yuval Shavitt,et al.  DIMES: let the internet measure itself , 2005, CCRV.

[44]  Christos H. Papadimitriou,et al.  On the Eigenvalue Power Law , 2002, RANDOM.

[45]  Herbert A. Simon,et al.  Some Further Notes on a Class of Skew Distribution Functions , 1960, Inf. Control..

[46]  M. Levy,et al.  POWER LAWS ARE LOGARITHMIC BOLTZMANN LAWS , 1996, adap-org/9607001.

[47]  S. Redner,et al.  Finiteness and fluctuations in growing networks , 2002, cond-mat/0207107.

[48]  BERNARD M. WAXMAN,et al.  Routing of multipoint connections , 1988, IEEE J. Sel. Areas Commun..

[49]  H. Simon,et al.  ON A CLASS OF SKEW DISTRIBUTION FUNCTIONS , 1955 .

[50]  S. N. Dorogovtsev.  Networks with given correlations , 2003 .

[51]  S. Bornholdt,et al.  World Wide Web scaling exponent from Simon's 1955 model. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[52]  Ibrahim Matta,et al.  On the geographic location of Internet resources , 2003, IEEE J. Sel. Areas Commun..

[53]  I M Sokolov,et al.  Evolving networks with disadvantaged long-range connections. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.