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Traffic-driven model of the World-Wide-Web Graph A. Barrat, LPT, Orsay, France M. Barthélemy, CEA, France A. Vespignani, LPT, Orsay, France

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Outline The WebGraph Some empirical characteristics Various models Weights and strengths Our model: Definition Analysis: analytics+numerics Conclusions

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The Web as a directed graph i j l nodes i : web-pages directed links: hyperlinks in- and out- degrees:

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Small world : captured by Erdös-Renyi graphs Poisson distribution = p N With probability p an edge is established among couple of vertices Empirical facts

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Small world Large clustering: different neighbours of a node will likely know each other 1 2 3 n Higher probability to be connected =>graph models with large clustering, e.g. Watts-Strogatz 1998 Empirical facts

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Small world Large clustering Dynamical network Broad connectivity distributions also observed in many other contexts (from biological to social networks) huge activity of modeling Empirical facts (Barabasi-Albert 1999; Broder et al. 2000; Kumar et al. 2000; Adamic-Huberman 2001; Laura et al. 2003)

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Various growing networks models á Barab á si-Albert (1999): preferential attachment á Many variations on the BA model: rewiring (Tadic 2001, Krapivsky et al. 2001), addition of edges, directed model (Dorogovtsev-Mendes 2000, Cooper-Frieze 2001), fitness (Bianconi-Barab á si 2001),... Kumar et al. (2000): copying mechanism Pandurangan et al. (2002): PageRank+pref. attachment Laura et al. (2002): Multi-layer model Menczer (2002): textual content of web-pages

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The Web as a directed graph i j l nodes i : web-pages directed links: hyperlinks Broad P(k in ) ; cut-off for P(k out ) (Broder et al. 2000; Kumar et al. 2000; Adamic-Huberman 2001; Laura et al. 2003)

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Additional level of complexity: Weights and Strengths i j Links carry weights/traffic: w ij In- and out- strengths l Adamic-Huberman 2001: broad distribution of s in

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Model: directed network n i j (i) Growth (ii) Strength driven preferential attachment (n: k out =m outlinks) AND... “Busy gets busier”

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Weights reinforcement mechanism i j n The new traffic n-i increases the traffic i-j “Busy gets busier”

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Evolution equations (Continuous approximation) Coupling term

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Resolution Ansatz supported by numerics:

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Results

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Approximation Total in-weight i s in i : approximately proportional to the total number of in-links i k in i, times average weight h w i = 1+ Then: A=1+ s in 2 [2;2+1/m]

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Measure of A prediction of Numerical simulations Approx of

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Numerical simulations NB: broad P(s out ) even if k out =m

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Clustering spectrum i.e.: fraction of connected couples of neighbours of node i

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Clustering spectrum increases => clustering increases New pages: point to various well-known pages, often connected together => large clustering for small nodes Old, popular pages with large k: many in-links from many less popular pages which are not connected together => smaller clustering for large nodes

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Clustering and weighted clustering takes into account the relevance of triangles in the global traffic

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Clustering and weighted clustering Weighted Clustering larger than topological clustering: triangles carry a large part of the traffic

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Assortativity Average connectivity of nearest neighbours of i

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Assortativity k nn : disassortative behaviour, as usual in growing networks models, and typical in technological networks lack of correlations in popularity as measured by the in-degree

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Summary Web: heterogeneous topology and traffic Mechanism taking into account interplay between topology and traffic Simple mechanism=>complex behaviour, scale-free distributions for connectivity and traffic Analytical study possible Study of correlations: non-trivial hierarchical behaviour Possibility to add features (fitnesses, rewiring, addition of edges, etc...), to modify the redistribution rule... Empirical studies of traffic and correlations?

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