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: analytics+numerics



Conclusions

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The Web as a directed graph

i

l j

nodes i: web-pages

directed links: hyperlinks

in- and out- degrees:

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• Small world : captured by Erdös-Renyi graphs

Poisson distribution

<k> = 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

 Barabá_ási-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-Barabá_ási 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

l j

nodes i: web-pages

directed links: hyperlinks

Broad P(k_in) ; cut-of 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 sⁱⁿ

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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 sⁱⁿ_i : approximately proportional to the

total number of in-links _i kⁱⁿ_i , times average weight hwi = 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?