5 - Spectral Methods

Summary

In this chapter, we develop spectral techniques. We highlight some applications to Markov chain mixing and network analysis. The main tools are the spectral theorem and the variational characterization of eigenvalues, which we review together with some related results. We also give a brief introduction to spectral graph theory and detail an application to community recovery. Then we apply the spectral theorem to reversible Markov chains. In particular we define the spectral gap and establish its close relationship to the mixing time. We also show in that the spectral gap can be bounded using certain isoperimetric properties of the underlying network. We prove Cheeger’s inequality, which quantifies this relationship, and introduce expander graphs, an important family of graphs with good “expansion.” Applications to mixing times are also discussed. One specific technique is the “canonical paths method,” which bounds the spectral graph by formalizing a notion of congestion in the network.