TheoryFest Day 2: Directed Spectral methods

There was an interesting talk on developing spectral tools for directed graphs. The magic of spectral methods for undirected graphs comes from the interpretation of the graph as a Markov chain, and then using spectral properties of the Laplacian (which is symmetric and positive semidefinite) to reason about a host of properties like conductance, mixing times, sparsification, and others.

But for directed graphs none of this works as stated. The Laplacian is no longer symmetric (goodbye real eigenvalues) and even if we symmetrize it it may not be positive definite (goodbye nonnegative eignenvalues). One of the key tricks that this paper exploits (and relates to work by Fan Chung on Cheeger inequalities for directed graphs) is the following:

Suppose our directed graph is Eulerian, which means that each vertex has the same incoming and outgoing degree. Then if we construct the associated Laplacian $L$ and symmetrize it as $L' = (L + L^T)/2$, then the resulting symmetric matrix $L'$ is positive definite!

This turns out to be a key ingredient of designing efficient algorithms for estimating spectral quantities on directed graphs, and is a trick worth remembering.