  In a recent breakthrough STOC~2015 paper, a continuous diffusion process was
considered on hypergraphs (which has been refined in a recent JACM 2018 paper)
to define a Laplacian operator, whose spectral properties satisfy the
celebrated Cheeger's inequality. However, one peculiar aspect of this diffusion
process is that each hyperedge directs flow only from vertices with the maximum
density to those with the minimum density, while ignoring vertices having
strict in-beween densities. In this work, we consider a generalized diffusion
process, in which vertices in a hyperedge can act as mediators to receive flow
from vertices with maximum density and deliver flow to those with minimum
density. We show that the resulting Laplacian operator still has a second
eigenvalue satsifying the Cheeger's inequality. Our generalized diffusion model
shows that there is a family of operators whose spectral properties are related
to hypergraph conductance, and provides a powerful tool to enhance the
development of spectral hypergraph theory. Moreover, since every vertex can
participate in the new diffusion model at every instant, this can potentially
have wider practical applications.

Discrete Mathematics