Hypernode Graphs for Learning from Binary Relations between Groups in Networks

The aim of this paper is to propose methods for learning from interactions between groups in networks. We propose a proper extension of graphs, called hypernode graphs as a formal tool able to model group interactions. A hypernode graph is a collection of weighted relations between two disjoint groups of nodes. Weights quantify the individual participation of nodes to a given relation. We define Laplacians and kernels for hypernode graphs and prove that they strictly generalize over graph kernels and hypergraph kernels. We then proceed to prove that hypernode graphs correspond to signed graphs such that the matrix D − W is positive semi-definite. As a consequence, homophilic relations between groups may lead to non homophilic relations between individuals. We also define the notion of connected hypernode graphs and a resistance distance for connected hypernode graphs. Then, we propose spectral learning algorithms on hypernode graphs allowing to infer node ratings or node labelings. As a proof of concept, we model multiple players games with hypernode graphs and we define skill rating algorithms competitive with specialized algorithms.