Great Ideas of Computer Science: Random Walks and Searching the Web

Transcription

1 CS/MA 109 Fall 2016 Wayne Snyder Department Boston University Great Ideas of : Random Walks and Searching the Web

2 Internet Search Engines Earlier search engines were fairly useless, because they could not figure out how to ORDER the pages that fit the search phrase you typed in.

3 How does Google figure out which pages you might be most likely to want to see? Why am I the top Wayne Snyder on the web?

4 Remember our dumb algorithm for solving the traveling salesman problem? Another way to write this algorithm would be to RANDOMLY walk around the graph and try to find the minimal path. After a very long time, you would happen upon it (with very high probability). Let s explore the idea of Random Walks with a computer simulation.

5 Remember our dumb algorithm for solving the traveling salesman problem? Another way to write this algorithm would be to RANDOMLY walk around the graph and try to find the minimal path. After a very long time, you would happen upon it (with very high probability). Even dumber, you say? But random walks have lots of uses.

6 The Mathematical Model of Random Walks have many uses as a simulation tool: Brownian Motion of Molecules Stock prices and behavior of investors Modeling of cascades of neuron firings in brain A tourist wandering around a city, or a customer exploring a new store A web surfer aimlessly clicking from a web page to another And they have important practical uses in the Internet: Twitter uses random walks to suggest who to follow Sculpture in London designed using a random 3D walk Google uses random walks to order pages which match a search phrase

7 Google: What pages are most important? Consider a tourist wandering around in Boston

8 Google: What pages are most important? To put it in mathematical/computer science terms: What are the relative frequencies (or probabilities) of visiting the various nodes in the graph during a random walk? Examples: 1. What is the likelihood of finding a wandering tourist at a given subway station? 2. What is the likelihood that a wandering tourist will stumble upon the Museum of Science? 3. What is the likelihood that a web surfer will end up visiting Google? How about the BU web site? This will give us an estimate of how important a page is, and how interested you might be in that page when you search Google!

9 Suppose we just consider the shuttle bus between two BU campuses? What is the probability that a random student will be in either place?

10 Suppose we just consider the shuttle bus between two BU campuses? What is the probability that a random student will be in either place? We can model this with a graph/network which has probabilities attached to each path, showing the probability that that path is taken: Probability = 1.0 Probability = 1.0 What is the probability that a random student, traveling back and forth between the two locations is in either of the two places?

11 Suppose we just consider the shuttle bus between two BU campuses? What is the probability that a random student will be in either place? We can model this with a graph/network which has probabilities attached to each path, showing the probability that that path is taken: Probability = % Probability = % What is the probability that a random student, traveling back and forth between the two locations is in either of the two places?

12 What about this graph? When a visitor leaves a vertex, she travels the given link with some probability (say, by flipping a coin). What is the probability of a random walker being in A or in B now?

13 Here is a probabilistic algorithm ( Monte Carlo Method ) for finding out: 1. Start with equal percentage at each vertex; 2. At each iteration, split the visitors according to the percentages and send them to the next vertex (as if they walked on a link). 3. Eventually this will stabilize with the probability of a random visitor being on a vertex at some random time!

14 Here is a probabilistic algorithm ( Monte Carlo Method ) for finding out: 1. Start with equal percentage at each vertex; 2. At each iteration, split the visitors according to the percentages and send them to the next vertex (as if they walked on a link). 3. Eventually this will stabilize with the probability of a random visitor being on a vertex at some random time! Let s make this concrete: suppose we have 1000 walkers, and they start out evenly divided at each vertex.. A B Population at A Population at B

15 Here is a probabilistic algorithm ( Monte Carlo Method ) for finding out: 1. Start with equal percentage at each vertex; 2. At each iteration, split the visitors according to the percentages and send them to the next vertex (as if they walked on a link). 3. Eventually this will stabilize with the probability of a random visitor being on a vertex at some random time! 500 A B Now, all of A s visitors go to B; But 50% of B s visitors go to A, and 50% walk in a circle and stay! Population at A Population at B

16 Here is a probabilistic algorithm ( Monte Carlo Method ) for finding out: 1. Start with equal percentage at each vertex; 2. At each iteration, split the visitors according to the percentages and send them to the next vertex (as if they walked on a link). 3. Eventually this will stabilize with the probability of a random visitor being on a vertex at some random time! Now, all of A s visitors go to B; But 50% of B s visitors go to A, and 50% walk in a circle and stay! A 250 B 750 Population at A Population at B

17 Here is a probabilistic algorithm ( Monte Carlo Method ) for finding out: 1. Start with equal percentage at each page; 2. At each iteration, split the visitors according to the percentages and send them to the next page (as if they walked on a link). 3. Eventually this will stabilize with the probability of a random visitor being on a page at some random time! Do it again! All of A s visitors go to B; But 50% of B s visitors go to A, and 50% walk in a circle and stay! A 375 B 625 Population at A Population at B = 625

18 Here is a probabilistic algorithm ( Monte Carlo Method ) for finding out: 1. Start with equal percentage at each vertex; 2. At each iteration, split the visitors according to the percentages and send them to the next vertex (as if they walked on a link). 3. Eventually this will stabilize with the probability of a random visitor being on a vertex at some random time! A B And again! All of A s visitors go to B; But 50% of B s visitors go to A, and 50% walk in a circle and stay! Population at A Population at B Eventually it stabilizes at the correct values!

19 Ok what does this have to do with Google and what web pages are most popular? Vertices = Web Pages Visitors = people viewing the page

20 Ok what does this have to do with Google and what web pages are most popular? Vertices = Web Pages Visitors = people viewing the page Links = Links in the web page to another web page Homework 1 syllabus. MCS 109 Syllabus See below for. Homework 1.. Assumptions: Viewer only moves by clicking links, and chooses randomly!

21 So for any collection of pages, we can calculate the relative importance of a page, by using the Monte Carlo method to figure out which are more popular (based on the structure of the links).

22 Now we can calculate which pages are most likely to have viewers: 1. Start with equal percentage at each page; 2. At each iteration, split the viewers according to the percentages and send them to the next page (as if they clicked on a link). 3. Eventually this will stabilize with the probability of a random viewer being on a page at some random time!

23 Is this all there is? NO! 1. People are not robots clicking randomly on links 2. People are not synchronized in their clicks 3. People often just type the URL they want 4. Links on a page may change over time 5. Many web pages are dynamic 6. Etc., etc., etc.. But despite all that 1. Random walks on a web graph tell us something about how the structure of the web influences surfers. 2. It gives Google a lot of information about the structure of the web. 3. That s what models are for! 4. And it explains why I m the top Wayne Snyder on the web: A lot of pages link to my page, so Google thinks I m important!