Cheeger Inequalities for General Edge-Weighted Directed Graphs

We consider Cheeger Inequalities for general edge-weighted directed graphs. Previously the directed case was considered by Chung for a probability transition matrix corresponding to a strongly connected graph with weights induced by a stationary distribution. An Eulerian property of these special weights reduces these instances to the undirected case, for which recent results on multi-way spectral partitioning and higher-order Cheeger Inequalities can be applied. We extend Chung's approach to general directed graphs. In particular, we obtain higher-order Cheeger Inequalities for the following scenarios: (1) The underlying graph needs not be strongly connected. (2) The weights can deviate (slightly) from a stationary distribution.