On the emergence of highly variable distributions in the autonomous system topology

Recent studies observe that vertex degree in the autonomous systems (AS) graph exhibits a highly variable distribution [14, 21]. The most prominent explanatory model for this phenomenon is the Barabasi-Albert (B-A) model [5, 2]. A central feature of the B-A model is preferential connectivity --- meaning that the likelihood a new node in a growing graph will connect to an existing node is proportional to the existing node's degree. In this paper we ask whether a more general explanation than the B-A model, and absent the assumption of preferential connectivity, is consistent with empirical data. We are motivated by two observations: first, AS degree and AS size are highly correlated [10]; and second, highly variable AS size can arise simply through exponential growth. We construct a model incorporating exponential growth in the size of the Internet and in the number of ASes, and show that it yields a size distribution exhibiting a power-law tail. In such a model, if an AS's link formation is roughly proportional to its size, then AS out-degree will also show high variability. Moreover, our approach is more flexible than previous work, since the choice of which AS to connect to does not impact high variability, thus can be freely specified. We instantiate such a model with empirically derived estimates of historical growth rates and show that the resulting degree distribution is in good agreement with that of real AS graphs.

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

[2]  Ibrahim Matta,et al.  BRITE: an approach to universal topology generation , 2001, MASCOTS 2001, Proceedings Ninth International Symposium on Modeling, Analysis and Simulation of Computer and Telecommunication Systems.

[3]  Ramesh Govindan,et al.  Heuristics for Internet map discovery , 2000, Proceedings IEEE INFOCOM 2000. Conference on Computer Communications. Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies (Cat. No.00CH37064).

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

[5]  Mischa Schwartz,et al.  ACM SIGCOMM computer communication review , 2001, CCRV.

[6]  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.

[7]  Stefano Mossa,et al.  Truncation of power law behavior in "scale-free" network models due to information filtering. , 2002, Physical review letters.

[8]  Mark Crovella,et al.  On the Size Distribution of Autonomous Systems , 2003 .

[9]  Walter Willinger,et al.  Network topology generators: degree-based vs. structural , 2002, SIGCOMM '02.

[10]  Fan Chung Graham,et al.  Random evolution in massive graphs , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[11]  Christos Gkantsidis,et al.  Spectral analysis of Internet topologies , 2003, IEEE INFOCOM 2003. Twenty-second Annual Joint Conference of the IEEE Computer and Communications Societies (IEEE Cat. No.03CH37428).

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

[13]  Derek de Solla Price,et al.  A general theory of bibliometric and other cumulative advantage processes , 1976, J. Am. Soc. Inf. Sci..

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

[15]  S. Redner,et al.  Connectivity of growing random networks. , 2000, Physical review letters.

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

[17]  Ellen W. Zegura,et al.  A quantitative comparison of graph-based models for Internet topology , 1997, TNET.

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

[19]  D J PRICE,et al.  NETWORKS OF SCIENTIFIC PAPERS. , 1965, Science.

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

[21]  Kenneth L. Calvert,et al.  Modeling Internet topology , 1997, IEEE Commun. Mag..

[22]  Walter Willinger,et al.  Inferring AS-level Internet topology from router-level path traces , 2001, SPIE ITCom.

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

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

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

[26]  Walter Willinger,et al.  Does AS size determine degree in as topology? , 2001, CCRV.

[27]  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.

[28]  Walter Willinger,et al.  Towards capturing representative AS-level Internet topologies , 2002, SIGMETRICS '02.

[29]  Walter Willinger,et al.  Toward an optimization-driven framework for designing and generating realistic Internet topologies , 2003, CCRV.