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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.

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A Toy Example of WV= {1,2,3,4}

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

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

(3,1),

(4,3)}

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.

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.

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

- 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

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

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

- 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

The in/out-degree

Power law trend:

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

- 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

- Let u, vV 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

- 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

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.

- [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.

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).

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.

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.

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.

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