# Harmonic Analysis of Symmetric Random Graphs

## TL;DR

This paper explores the harmonic analysis of symmetric random graphs, representing their probability distributions as mixtures of characters on a semigroup, offering an alternative view of de Finetti's theorem for exchangeable graphs.

## Contribution

It introduces a harmonic analysis framework on semigroups to understand graph limits and provides a new derivation of de Finetti's theorem for exchangeable graphs.

## Key findings

- Representation of random graph distributions as mixtures of characters
- Alternative derivation of de Finetti's theorem for graphs
- Insight into graph limits via harmonic analysis

## Abstract

This note attempts to understand graph limits as defined by Lovasz and Szegedy (2006)} in terms of harmonic analysis on semigroups. This is done by representing probability distributions of random exchangeable graphs as mixtures of characters on the semigroup of unlabeled graphs with node-disjoint union, thereby providing an alternative derivation of de Finetti's theorem for random exchangeable graphs.