An undecidability result on limits of sparse graphs

TL;DR

This paper proves that it is undecidable to determine whether a sequence of bounded degree graphs can approximate a given set of local neighborhoods with high probability, highlighting fundamental limits in graph theory.

Contribution

It establishes an undecidability result for the existence of graph sequences with prescribed local neighborhood distributions, extending to unimodular random graphs.

Findings

Undecidability of local neighborhood approximation in bounded degree graphs

Extension of the result to unimodular random graphs

Implications for limits of sparse graphs

Abstract

Given a set B of finite rooted graphs and a radius r as an input, we prove that it is undecidable to determine whether there exists a sequence (G_i) of finite bounded degree graphs such that the rooted r-radius neighbourhood of a random node of G_i is isomorphic to a rooted graph in B with probability tending to 1. Our proof implies a similar result for the case where the sequence (G_i) is replaced by a unimodular random graph.