  In spectral graph theory, the Cheeger's inequality gives upper and lower
bounds of edge expansion in normal graphs in terms of the second eigenvalue of
the graph's Laplacian operator. Recently this inequality has been extended to
undirected hypergraphs and directed normal graphs via a non-linear operator
associated with a diffusion process in the underlying graph.
  In this work, we develop a unifying framework for defining a diffusion
operator on a directed hypergraph with stationary vertices, which is general
enough for the following two applications.
  1. Cheeger's inequality for directed hyperedge expansion.
  2. Quadratic optimization with stationary vertices in the context of
semi-supervised learning.
  Despite the crucial role of the diffusion process in spectral analysis,
previous works have not formally established the existence of the corresponding
diffusion processes. In this work, we give a proof framework that can indeed
show that such diffusion processes are well-defined. In the first application,
we use the spectral properties of the diffusion operator to achieve the
Cheeger's inequality for directed hyperedge expansion. In the second
application, the diffusion operator can be interpreted as giving a continuous
analog to the subgradient method, which moves the feasible solution in discrete
steps towards an optimal solution.
