Traffic-driven model of the World-Wide-Web Graph

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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 analyticsnumerics
• Conclusions

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

• Small world captured by Erdös-Renyi graphs

With probability p an edge is established among
couple of vertices
ltkgt p N
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Empirical facts

• Small world
• Large clustering different neighbours of a node
• will
  likely know each other

gtgraph models with large clustering, e.g.
Watts-Strogatz 1998
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Empirical facts

• Small world
• Large clustering
• Dynamical network
• Broad connectivity distributions
• also observed in many other contexts
• (from biological to social networks)
• huge activity of modeling

(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

• Barabási-Albert (1999) preferential attachment
• 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-Barabási
  2001), ...
• Kumar et al. (2000) copying mechanism
• Pandurangan et al. (2002) PageRankpref.
  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
nodes i web-pages directed links hyperlinks
l
j
i
Broad P(kin) cut-off for P(kout)
(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
l
j
Links carry weights/traffic wij
i
In- and out- strengths
Adamic-Huberman 2001 broad distribution of sin
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Model directed network
(i) Growth
j
(ii) Strength driven preferential
attachment (n koutm outlinks)
i
Busy gets busier
AND...
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Weights reinforcement mechanism
j
i
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 sini approximately
proportional to the total number of in-links ?i
kini , times average weight hwi 1?
Then A1?
gsin 2 221/m
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Numerical simulations
Measure of A prediction of ?
Approx of g
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Numerical simulations
NB broad P(sout) even if koutm
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Clustering spectrum
i.e. fraction of connected couples of neighbours
of node i
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Clustering spectrum

• d increases gt clustering increases
• New pages point to various well-known pages,
  often connected
• together gt large clustering for small nodes
• Old, popular pages with large k many in-links
  from many less popular pages which are not
  connected together
• gt 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

• knn 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 mechanismgtcomplex 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?