Convergence law for hyper-graphs with prescribed degree sequences

TL;DR

This paper establishes a convergence law for the first-order logic properties of random hyper-graphs with prescribed degree sequences, extending to classical models like Erdős-Rényi graphs.

Contribution

It introduces a logical framework for analyzing hyper-graphs with specified degree distributions and characterizes their limit theories, generalizing known results for simpler models.

Findings

Proves a convergence law for first-order logic in hyper-graphs.

Characterizes the limit theories for a wide class of degree distributions.

Extends convergence results to classical models like Erdős-Rényi graphs.

Abstract

We view hyper-graphs as incidence graphs, i.e. bipartite graphs with a set of nodes representing vertices and a set of nodes representing hyper-edges, with two nodes being adjacent if the corresponding vertex belongs to the corresponding hyper-edge. It defines a random hyper-multigraph specified by two distributions, one for the degrees of the vertices, and one for the sizes of the hyper-edges. We develop the logical analysis of this framework and first prove a convergence law for first-order logic, then characterise the limit first-order theories defined by a wide class of degree distributions. Convergence laws of other models follow, and in particular for the classical Erd\H{o}s-R\'enyi graphs and $k$-uniform hyper-graphs.