Mining di dati web

1. Mining di dati web Lezione n° 2 Il grafo del Web A.A 2006/2007

2. 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.

3. 2 1 Link21 Link11 Link22 Link12 4 Link31 3 Link41 1 2 3 4 1 1 2,4 2,2 = 1 0 1 1 2 2 2 3,1 3,4 0 = 1 1 3 3 3 1 1 1 0 = = 4 4 4 3 3 0 0 1 = A Toy Example of W V= {1,2,3,4} E= {(1,2), (1,4), (2,3), (2,4), (3,1), (4,3)}

4. A More Realistic W

5. The size of W • 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.

6. The “real” size of W • 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.

7. 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%.

8. Evolution of W • 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

9. 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

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

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

12. The Bow-Tie Structure

13. 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.

14. 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 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

15. The in/out-degree Power law trend:

16. 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.

17. 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.

18. 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.

19. 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

20. 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.

21. 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.

22. General Characteristics

23. References • [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).

24. References • [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.

25. References • [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.

26. References • [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.