1 / 13

PERTEMUAN 26

PERTEMUAN 26. Markov Chains and Random Walks. Fundamental Theorem of Markov Chains. If M g is an irreducible, aperiodic Markov Chain: All states are positive reccurent. P k converges to W , where each row of W is the same (and equal to  , say)

alamea
Télécharger la présentation

PERTEMUAN 26

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. PERTEMUAN 26

  2. Markov Chains and Random Walks

  3. Fundamental Theorem of Markov Chains If Mg is an irreducible, aperiodic Markov Chain: • All states are positive reccurent. • Pk converges to W, where each row of W is the same (and equal to , say) •  is the unique vector for which P = 

  4. Fundamental Theorem of Markov Chains • Let Mg be a Markov Chain with states S0…Sn • The Fundamental Theoremtells us that after a sufficiently large number of time steps, the probability of being in state Si+1 is the same as being in state Si. • This steady-state condition is known as a stationary distribution • The rate at which a Markov Chain converges to a stationary distribution is called the mixing rate.

  5. Random Walks • A Random Walk on connected, undirected, non-bipartite Graph G can be modeled as a Markov Chain Mg, where the vertices of the Graph, V(G), are represented by the states of the the Markov Chain and the transition matrix is as follows If (u,v) is a member of E otherwise

  6. Random Walks • Mg is irreducible because G is connected • Mg is aperiodic • Periodicity is the GCD of the length of all closed walks on G • Since G is undirected, there exist closed walks of length 2 (u,v  E, exists walk u-v-u) • Since G is non-bipartite it contains odd cycles • Therefore GCD of all closed walks is 1 • Mg is aperiodic

  7. Random Walks • Given that Mg is aperiodic and irreducible, we can apply the Fundamental Theorem of Markov Chains and deduce that Mg converges to a stationary distribution. • Lemma: • For all v  V, v = d(v) / 2 |E| ( d(v) = the degree of v) • Proof denote the component corresponding to vertex v in the probability vector

  8. Hitting time (huv) – expected number of steps in a Random Walk that starts at u and ends upon its first visit to v • Commute time (cuv) -- expected number of steps in a Random Walk that starts at u, visitsv once and returns to u. (cuv = huv + hvu)

  9. The Lollipop Graph • Lollipop Graph consists of n vertices • A clique on n/2 vertices • A path on n/2 vertices • Let u,v  V, u is in the clique, v is at the far end of the path. • Surprisingly, huv != hvu (huv is (n3) hvu is  (n2)

  10. Markov Chains: an Application • Link Prediction and Path Analysis using Markov Chains • Use Markov Chains to perform probabilistic analysis and modeling of weblink sequences; ie. If a user requests page n, what will be her most likely next choice • Possible Applications • Web Server Request Prediction • Adaptive Web Navigation • Tour Generation • Personalized Hub • Model can be used in adaptive mode; transition matrix can be updated as new data (example: Web Server Request) arrives

  11. Markov Chains: an Application Link Prediction and Path Analysis using Markov Chains System Overview

  12. Markov Chains: an Application Experimental Results HTTP Server Request Prediction • 6572 URIs (including html documents, directories, gifs, and cgi requests) • 40,000 Requests • Over 50% of the web server requests can be predicted to be the state with the highest probability

  13. References • L. Lovasz. Random Walks on Graphs: A Survey. Combinatorics: Paul Erdos is Eighty (vol. 2), 1996, pp. 353-398. (http://www.cs.yale.edu/HTML/YALE/CS/HyPlans/lovasz/erdos.ps) • R. Sarukkai, "Link Prediction and Path Analysis Using Markov Chains: 9th World Wide Wide Conference, May, 2000. (http://www9.org/w9cdrom/68/68.html) • Introduction to Markov chains : with special emphasis on rapid mixing / Ehrhard Behrends. Germany [1990-onward] Vieweg & Sohn, GW Am Math Soc 2000.

More Related