  Networks are important structures used to model complex systems where
interactions take place. In a basic network model, entities are represented as
nodes, and interaction and relations among them are represented as edges.
However, in a complex system, we cannot describe all relations as pairwise
interactions, rather should describe as higher-order interactions. Hypergraphs
are successfully used to model higher-order interactions in complex systems. In
this paper, we present two new hypergraph Laplacians based on diffusion
framework. Our Laplacians take the relations between higher-order interactions
into consideration, hence can be used to model diffusion on hypergraphs not
only between vertices but also higher-order structures. These Laplacians can be
employed in different network mining problems on hypergraphs, such as social
contagion models on hypergraphs, influence study on hypergraphs, and hypergraph
classification, to list a few.
