Lecture 08 . RandomWalk on Graphs : Fiedler Vector , Cheeger inequality

In this class, we introduced the random walk on graphs. The last lecture shows Perron-Frobenius theory to the analysis of primary eigenvectors which is the stationary distribution. In this lecture we will study the second eigenvector. To analyze the properties of the graph, we construct two matrices: one is (unnormalized) graph Laplacian and the other is normalized graph Laplacian. In the first part, we introduce Fiedler Theory for the unnormalized graph Laplacian, which shows the second eigenvector can be used to bipartite the graph into two connected components. In the second part, we study the eigenvalues and eigenvectors of normalized Laplacian matrix to show its relations with random walks or Markov chains on graphs. In the third part, we will introduce the Cheeger Inequality for second eigenvector of normalized Laplacian, which leads to an approximate algorithm for Normalized graph cut (NCut) problem, an NP-hard problem itself.