Projective limits of probabilistic symmetries and their applications to random graph limits

TL;DR

The paper develops a general framework that couples projective limits of probability measures with direct limits of symmetry groups, showing that probabilistic symmetries persist in the limit under natural compatibility conditions. It proves existence and invariance of projective-limit point processes and extends invariance from finite-stage groups to the full limit group under a finite-characteristic condition. Applying the theory to random graphs, the authors recover graphons (dense graphs) and graphexes (sparse graphs) as limit objects and present a new ultrasparse, rotation-invariant graph limit in Euclidean space. This unified approach provides a versatile toolkit for analyzing limits across diverse random-graph regimes, enabling a principled path from finite constructions to canonical limit objects. The results offer both conceptual clarity and practical pathways for representing and sampling limits in dense, sparse, and ultrasparse networks.

Abstract

We couple projective limits of probability measures to direct limits of their symmetry groups. We show that the direct limit group is the group of symmetries of the projective limit probability measure. If projective systems of probability measures represent point processes in increasingly larger finite regions of the same infinite space, then we show that under some additional niceness and consistency assumptions, an extension of the direct limit group is the symmetry group of the projective limit point process in the whole infinite space. The application of these results to random graph limits provides ``shortest paths'' to graphons and graphexes as it recovers these random graph limits as trivial corollaries. Another application example encompasses a broad class of limits of random graphs with bounded average degrees. This class includes a representative collection of paradigmatic random graph models that have attracted significant research attention in diverse areas of science. Our approach thus provides a general unified framework to study limits of very different types of random graphs.

Figures (1)

• Figure 1: Projective limits of random graphs as point processes.(a) A random graph of size $4$ with vertices labeled by $\{1,2,3,4\}$. Each point represents an edge. The graph is undirected, so the point process is symmetric, with each edge represented by two symmetric points. The shown graph consists of four edges $a=\{1,2\}$, $b=\{2,3\}$, $c=\{3,4\}$, and $d=\{4,2\}$. Since the label space is $\mathbb{N}$ in the limit, the point process is confined to locations on the $\mathbb{N}^2$ lattice. (b) The same graph, shown in the inset, but with vertices labeled by real numbers in $[0,4]$. Points can now be anywhere on the $[0,4]^2$ square, or on $\mathbb{R}_+^2$ in the limit. (c) A random geometric graph on a disk. The visualization is different compared to (a) and (b) in that only a single copy of the label space is shown, the label space is two-dimensional, a disk in $\mathbb{R}^2$, points represent graph vertices versus edges, while edges are shown explicitly. The arrows in all the panels indicate the projective expansion of the label space as the graph grows.

Theorems & Definitions (30)

• Definition 2.1: Projective system of topological spaces bourbaki1995topological
• Definition 2.2: Projective limit of topological spaces bourbaki1995topological
• Theorem 2.1: Projective limit of topological spaces bourbaki1995topological
• Definition 2.3: Projective system of probability measures bochner2005harmonic
• Definition 2.4: Projective limit of probability measures bochner2005harmonic
• Theorem 2.2: Projective limit of probability measures bochner2005harmonic
• Definition 2.5: Direct system of groups bourbaki2004algebraI
• Definition 2.6: Direct limit of groups bourbaki2004algebraI
• Theorem 2.3: Direct limit of groups bourbaki2004algebraI
• Definition 2.7: Direct pre-limit of groups