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

TL;DR

This paper develops a unified framework linking projective limits of probability measures with symmetry groups, applying it to random graph limits and recovering known models like graphons and graphexes.

Contribution

It introduces a novel approach connecting projective limits and symmetry groups, providing a unified method to analyze various random graph limit models.

Findings

The direct limit group acts as the symmetry group of the projective limit measure.

Under certain conditions, the symmetry group extends to the entire infinite space.

The framework recovers classical graphon and graphex models as special cases.

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.