A Fuzzy Web Surfer Model

1. A Fuzzy Web Surfer Model Narayan L. Bhamidipati and Sankar K. Pal Indian Statistical Institute Kolkata

2. Contents • Web Surfer Models • Markov Chains • Fuzzy Markov Chains • Fuzzy Web Surfer Models • Advantages and Limitations

3. Web Surfer Models • Surfer visits pages randomly • Various assumptions, various models • Random Surfer Model (PageRank) • HITS (goes back and forth) • Intelligent/Directed Surfer Model • PHITS, WPSS

4. Markov Chains • MC is aperiodic if each of its states is • MC is irreducible if it is possible to reach any state from any other state • Regular if aperiodic and irreducible • A regular MC has a unique stationary distribution

5. Surfer Models as Markov Chains • Web pages are states • Moving between pages are the transitions • Clicking or typing URLs • Transition probabilities • Steady state distribution yields the page ranks

6. Fuzzy Markov Chains • As opposed to classical Markov Chains (which are based on Probability Theory) • Fuzzy Transition Matrix • Employ max-min algebra (instead of the usual addition and multiplication)

7. Fuzzy Web Surfer Model • Uncertainty involved in following links • Uncertainty involved in existence of links • Uncertainty involved in page(let) boundaries • Modeled in a fuzzy sense

8. Fuzziness in Link’s Existence ? • “A link either exists or it does not exist” • Pagelets: web pages split into sections • Link to a page, but which section ? • What if the sections are not made explicit ? • Have to guess if a link is intended for a particular pagelet

9. Pagelets • Coherent parts of web pages • Pagelets may differ widely in terms of content • Need not necessarily be split into explicit sections • Identification by considering the structure of the documents

10. Model Formulation • Each pagelet is referred to as a page • Every link in the web graph has an associated fuzzy number denoting the possibility of it being followed • Obtain fuzzy transition matrix Q • (i, j) element of Q denotes the belief of moving to page i, when the surfer is on page j

11. Model Formulation • Usual PageRank link computations are performed using the Fuzzy Transition Matrix on the max-min algebra • Concept of FuzzRank

12. Favorable Properties • Able to model fuzziness in links • Also can capture fuzziness in page contents • Finite Convergence • Analysis in terms of fuzzy eigen sets • Robust Computation

13. To be Explored… • Conditions for ergodicity (and hence regularity) of fuzzy Markov chains are not completely known (as yet) • The steady state fuzzy distribution depends on the initial state • What do the different convergent fuzzy distributions correspond to ?

14. Conjecture • The distinct steady state fuzzy distributions correspond to web communities • Probably helpful in identifying communities from a given set of pages

15. References • K. Avrachenkov and E. Sanchez. Fuzzy markov chains and decision-making. Fuzzy Optimization and Decision Making, 1(2):143–159, June 2002. • J. J. Buckley and E.Eslami. Fuzzy markov chains: Uncertain probabilities. Mathware and Soft Computing, 9(1):33–41, 2002. • M. Diligenti, M. Gori, and M. Maggini. A unified probabilistic framework for web page scoring systems. IEEE Transactions on Knowledge and Data Engineering, 16(1):4–16, January 2004.

17. Thank You