Weighted projected networks: mapping hypergraphs to networks

TL;DR

This paper introduces models linking hypergraphs to weighted networks, analytically studying their properties and revealing signatures of multiway interactions that can be used to detect hidden structures in weighted network data.

Contribution

It develops a framework for deriving weighted projected networks from hypergraphs, analyzing their statistical properties and identifying signatures of multiway interactions.

Findings

Weighted projected networks exhibit weight disorder even from simple hypergraph ensembles.

A signature of multiway interactions appears in link weights as hypergraph size varies.

The percolation transition in hypergraphs is second order and can be analyzed through projected networks.

Abstract

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.