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Mining di dati web. Lezione n° 2 Il grafo del Web A.A 2006/2007. The Web Graph. The linkage structure of Web Pages forms a graph structure. The Web Graph (hereinafter called W ) is a directed graph W = (V,E) V is the vertex set and each vertex represents a page in the Web.

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Mining di dati web

Mining di dati web

Lezione n° 2

Il grafo del Web

A.A 2006/2007

The web graph
The Web Graph

  • The linkage structure of Web Pages forms a graph structure.

  • The Web Graph (hereinafter called W) is a directed graph W = (V,E)

    • V is the vertex set and each vertex represents a page in the Web.

    • E is the edge set and each directed edge (e1,e2) exists whenever a link appears in the page represented by e1 to the page represented by e2.

A toy example of w



















































A Toy Example of W

V= {1,2,3,4}

E= {(1,2), (1,4),

(2,3), (2,4),



The size of w
The size of W

  • What is being measured?

    • Number of hosts

    • Number of (static) html pages

      • Volume of data

  • Number of hosts - netcraft survey


    • Monthly report on how many web hosts & servers are out there!

  • Number of pages - numerous estimates

  • Recently Yahoo announced an index with 20B pages.

The real size of w
The “real” size of W

  • The web is really infinite

    • Dynamic content, e.g. calendars, online organizers, etc.


  • Static web contains syntactic duplication, mostly due to mirroring (~ 20-30%)

  • Some servers are seldom connected.

Recent measurement of w
Recent Measurement of W

  • [Gulli & Signorini, 2005]. Total web > 11.5B.

  • 2.3B the pages unknown to popular Search Engines.

  • 35-120B of pages are within the hidden web.

  • The index intersection between the largest available search engines -- namely Google, Yahoo!, MSN, AskJeeves -- is estimated to be 28.8%.

Evolution of w
Evolution of W

  • All of these numbers keep changing.

  • Relatively few scientific studies of the evolution of the web [Fetterly & al., 2003]


  • Sometimes possible to extrapolate from small samples (fractal models) [Dill & al., 2001]


Rate of change
Rate of change

  • There a number of different studies analyzing the rate of changes of pages in V.

  • [Cho & al., 2000] 720K pages from 270 popular sites sampled daily from Feb 17 - Jun 14, 1999

    • Any changes: 40% weekly, 23% daily

  • [Fetterly & al., 2003] Massive study 151M pages checked over few months

    • Significant changed -- 7% weekly

    • Slightly changed -- 25% weekly

  • [Ntoulas & al., 2004] 154 large sites re-crawled from scratch weekly

    • 8% new pages / week

    • 8% die

    • 5% new content

    • 25% new links/week

Rate of change fetterly al 2003
Rate of change [Fetterly & al., 2003]

Rate of change ntoulas al 2004
Rate of change [Ntoulas & al., 2004]

The power of power laws
The Power of Power Laws

  • A power law relationship between two scalar quantities x and y is one where the relationship can be written as

    y= axk

    where a (the constant of proportionality) and k (the exponent of the power law) are constants.

  • Power laws are observed in many subject areas, including physics, biology, geography, sociology, economics, and linguistics.

  • Power laws are among the most frequent scaling laws that describe the scale invariance found in many natural phenomena.

Power law probability distributions
Power Law Probability Distributions

  • Sometimes called heavy-tail or long-tail distributions.

  • Examples of power law probability distributions:

    • The Pareto distribution, for example, the distribution of wealth in capitalist economies

    • Zipf's law, for example, the frequency of unique words in large texts

    • Scale-free networks, where the distribution of links is given by a power law (in particular, the World Wide Web)

    • Frequency of events or effects of varying size in self-organized critical systems, e.g. Gutenberg-Richter Law of earthquake magnitudes and Horton's laws describing river systems

The in out degree
The in/out-degree

Power law trend:

Random graphs
Random Graphs

  • RGs are structures introduced by Paul Erdos and Alfred Reny.

  • There are several models of RGs. We are concerned with the model Gn,p.

  • A graph G = (V,E)  Gn,p is such that |V|=n and an edge (u,v)  E is selected uniformly at random with probability p.

W cannot be a rg
W cannot be a RG

  • Let Xk be a discrete value indicating the number of nodes having degree equal to k.

  • Obviously in Gn,p the expected value of XpE(Xp) is .

  • Xk is asintotically distributed as a Poisson variable with mean k.

The avg distance of a graph g
The avg distance of a graph G

  • Let u, vV be two nodes of G.

  • Let d(u,v) be the distance from u to v expressed as the length of the shortest path connecting u to v. If u and v are not connected then the distance is set to .

  • Definewhere S is the set of pairs of distinct nodes u, v of W with the property that d(u,v) is finite.

The avg distance of w
The avg distance of W

  • A small world graph is a graph whose avg distance is much smaller that the order of the graph.

  • For instance L(G)  O(log(|V(G)|)).

  • L(W) is about 7.

  • Ld(W) is about 18

What s the best model for w

It is still an open problem

to find a web graph model

that produces graphs which

provably has all four properties.

What’s the best model for W?

  • A graph model for the web should have (at least) the following features:

    • On-line property. The number of nodes and edges changes with time.

    • Power law degree distribution. The degree distribution follows a power law, with an exponent >2.

    • Small world property. The average distance is much smaller that the order of the graph.

    • Many dense bipartite subgraphs. The number of distinct bipartite cliques or cores is large when compared to a random graph with the same number of nodes and edges.

W models proposed so far
W Models proposed so far.

  • [Bollobas & al., 2001]. Linearized Chord Diagram (LCD).

  • [Aiello & al., 2001]. ACL.

  • [Chung & al., 2003]. CL.

  • [Kumar & al., 1999]. Copying model.

  • [Chung & al., 2004]. CL-del growth-deletion model.

  • [Cooper & al., 2004]. CFV.


  • [Gulli & Signorini, 2005]. Antonio Gulli and Alessio Signorini. The indexable web is more than 11.5 billion pages. WWW (Special interest tracks and posters) 2005: 902-903.

  • [Fetterly & al., 2003]. Dennis Fetterly, Mark Manasse, Marc Najork, and Janet Wiener. A Large-Scale Study of the Evolution of Web Pages. 12th International World Wide Web Conference (May 2003), pages 669-678.

  • [Dill & al., 2001]. Stephen Dill, Ravi Kumar, Kevin S. McCurley, Sridhar Rajagopalan, D. Sivakumar, Andrew Tomkins: Self-similarity in the web. ACM Trans. Internet Techn. 2(3): 205-223 (2002).


  • [Cho & al., 2000]. Junghoo Cho, Hector Garcia-Molina. The Evolution of the Web and Implications for an Incremental Crawler. VLDB 2000: 200-209.

  • [Ntoulas & al., 2004]. Alexandros Ntoulas, Junghoo Cho, Christopher Olston. What's new on the web?: the evolution of the web from a search engine perspective. WWW 2004: 1-12.

  • [Bollobas & al., 2001]. Bela Bollobas, Oliver Riordan, G. Tusnary and Joel Spencer. The degree sequence of a scale-free random graph process. Random Structures and Algorithms, vol 18, 2001, 279-290.


  • [Aiello & al., 2001]. William Aiello, Fan R. K. Chung, Linyuan Lul. Random Evolution in Massive Graphs. FOCS 2001: 510-519.

  • [Chung & al., 2003]. Fan R. K. Chung, L. Lu. The average distances in random graphs with given expected degrees. Internet Mathematics. 1(2003): 91-114.

  • [Kumar & al., 1999]. R. Kumar, P. Raghavan, S. Rajagopalan, D. Sivakumar, A. Tomkins, and Eli Upfal. Stochastic models for the Web graph. Proceedings of the 41th FOCS. 2000, pp. 57-65.


  • [Chung & al., 2004]. F. Chung, L. Lu. Coupling Online and Offline Analyses for Random Power Law Graphs. Internet Mathematics. Vol 1 (2003). 409-461.

  • [Cooper & al., 2004]. C. Cooper, A. Frieze, J. Vera. Random Deletions in a Scale Free Random Graph Process. Internet Mathematics. Vol 1 (2003). 463 - 483.