CMU Randomized Algorithms

Randomized Algorithms, Carnegie Mellon: Spring 2011

Lecture 21: Random walks on graphs

Today we talked about random walks on graphs, and the result that in any connected undirected graph G, for any given start vertex u, the expected time for a random walk to visit all the nodes of G (called the cover time of the graph) is at most 2m(n-1), where n is the number of vertices of G and m is the number of edges.

In the process, we proved that for any G, if we think of the walk as at any point in time being on some edge heading in some direction, then each edge/direction is equally likely at probability 1/(2m) at the stationary distribution. (Actually, since we didn’t need to, we didn’t prove it is unique. However, if G is connected, it is not hard to prove by contradiction that there is a unique stationary distribution). We then used that to prove that the expected gap between successive visits to any given (u,v) is 2m. See the notes.

We also gave a few examples to show this is existentially tight. For instance, on a line (n vertices, n-1 edges) we have an expected \Omega(n^2) time to reach the other end of the line. Also on a “lollipop graph” (a clique of n/2 vertices connected to a line of n/2 vertices), the expected time to get from the clique to the end of the line is \Omega(n^3). Since this is not in the notes, here is a quick argument. First of all, if you are in the clique, then each step has 2/n probability of taking you to the node connecting the clique and the handle (let’s call this “node 0”). When you are at node 0, your next step has probability 2/n to take you to the next point on the handle (let’s call this “node 1”). So, when you are in the clique, it takes \Omega(n^2) steps to get to node 1. Now, think about the following experiment. Say you go into a fair casino with 1 dollar and bet on fair games (betting a dollar each time) until either you lose all your money or you have n/2 dollars in your pocket. Whenever you lose all your money, you go back to your car to get another dollar, and if you get n/2 dollars, you go upstairs to the restaurant. What is the expected number of trips to your car before you go to the restaurant? The answer is n/2 because it’s a fair casino so in expectation, all the money in your pocket when you head to the restaurant came from your car. Now, think of node 0 as your car, node 1 as entering the casino, and the end of the lollipop stick (node n/2) as the restaurant.

We ended our discussion by talking about resistive networks, and using the connection to give another proof of the cover time of a graph. In particular, we have C_{uv} = 2m R_{uv} where C_{uv} is the commute-time between u and v, and R_{uv} is the effective resistance between u and v.

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