2 Local and Almost Linear-time Clustering and Partitioning 2.1 Review of Local Clustering 2.2 General Strategy

1 Administrivia You should probably know that • the first problem set (due October 15) is posted on the class website, and • its hints are also posted there. Also, today in class there was a majority vote for posting problem sets earlier. Professor Kelner will post the problem sets from two years ago, but he reserves the right to add new problems once a problem set has already been posted. Questions from last time. • What is a level set? The level set of a function corresponding to a (fixed) constant c is the set of points in the function's domain whose image equals c. • What is a good reference on applications of expander graphs? A course taught by Nathan Linial and Avi Wigderson [3]. Plan for today. We use what we proved last time to obtain a local clustering algorithm from a random walk scheme. Then, noting that similar results to the ones proved last time also hold for PageRank, we obtain a second scheme that yields a second, better local clustering algorithm. Finally, we briefly motivate the technique of sparsification, which we will discuss next time. Let us briefly review local clustering, which we introduced last time. Given a vertex v in some graph G, we would like know if v is contained in a cluster, i.e. a subset of vertices that defines a cut with low conductance. However, we want the running time of our algorithm to depend on the cluster size, and not on the size of the graph. Last time we mentioned that a good example of a problem of this sort is trying to find a cluster of web pages around mit.edu; we surely do not want the running time of this task to depend on the number of sites created on the other side of the world. Let us make our goal a little bit more precise: in this lecture we will describe an algorithm that, after running for time almost linear in K, outputs a cluster of size at least K/2 around the starting vertex, if such a cluster exists. We observe that if we run a random walk starting from some vertex v contained in a cluster, then low-conductance cuts will be an obstacle to mixing; i.e., the random walk has trouble leaving the cluster. Hence, a good guess for the cluster is the set of vertices with the …