  Many natural, technological, and social systems incorporate multiway
interactions, yet are characterized and measured on the basis of weighted
pairwise interactions. In this article, I propose a family of models in which
pairwise interactions originate from multiway interactions, by starting from
ensembles of hypergraphs and applying projections that generate ensembles of
weighted projected networks. I calculate analytically the statistical
properties of weighted projected networks, and suggest ways these could be used
beyond theoretical studies. Weighted projected networks typically exhibit
weight disorder along links even for very simple generating hypergraph
ensembles. Also, as the size of a hypergraph changes, a signature of multiway
interaction emerges on the link weights of weighted projected networks that
distinguishes them from fundamentally weighted pairwise networks. This
signature could be used to search for hidden multiway interactions in weighted
network data. I find the percolation threshold and size of the largest
component for hypergraphs of arbitrary uniform rank, translate the results into
projected networks, and show that the transition is second order. This general
approach to network formation has the potential to shed new light on our
understanding of weighted networks.
