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# Mining di dati web - PowerPoint PPT Presentation

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

Lezione n° 2

Il grafo del Web

A.A 2006/2007

• 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.

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A Toy Example of W

V= {1,2,3,4}

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

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

(3,1),

(4,3)}

• What is being measured?

• Number of hosts

• Number of (static) html pages

• Volume of data

• Number of hosts - netcraft survey

• http://news.netcraft.com/archives/web_server_survey.html

• 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 web is really infinite

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

• http://www.raingod.com/raingod/resources/Programming/JavaScript/Software/RandomStrings/index.html

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

• Some servers are seldom connected.

• [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%.

• All of these numbers keep changing.

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

• http://research.microsoft.com/research/sv/sv-pubs/p97-fetterly/p97-fetterly.pdf

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

• http://www.vldb.org/conf/2001/P069.pdf

• 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

Rate of change [Fetterly & al., 2003]

Rate of change [Ntoulas & al., 2004]

• 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.

• 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 http://wordcount.org/main.php

• 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

Power law trend:

• 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

• 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.

• 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.

• 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)|)).

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.

• [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.