  There has been recent work [Louis STOC 2015] to analyze the spectral
properties of hypergraphs with respect to edge expansion. In particular, a
diffusion process is defined on a hypergraph such that within each hyperedge,
measure flows from nodes having maximum weighted measure to those having
minimum. The diffusion process determines a Laplacian, whose spectral
properties are related to the edge expansion properties of the hypergraph.
  It is suggested that in the above diffusion process, within each hyperedge,
measure should flow uniformly in the complete bipartite graph from nodes with
maximum weighted measure to those with minimum. However, we discover that this
method has some technical issues. First, the diffusion process would not be
well-defined. Second, the resulting Laplacian would not have the claimed
spectral properties.
  In this paper, we show that the measure flow between the above two sets of
nodes must be coordinated carefully over different hyperedges in order for the
diffusion process to be well-defined, from which a Laplacian can be uniquely
determined. Since the Laplacian is non-linear, we have to exploit other
properties of the diffusion process to recover a spectral property concerning
the "second eigenvalue" of the resulting Laplacian. Moreover, we show that
higher order spectral properties cannot hold in general using the current
framework.
