# Phase Transitions in Edge-Weighted Exponential Random Graphs: Near-Degeneracy and Universality

## TL;DR

This paper extends exponential random graph models to weighted networks by introducing a common distribution for edge weights, addressing limitations of traditional models that only handle simple graphs, and explores properties like near-degeneracy and universality.

## Contribution

It proposes a new framework for weighted exponential random graphs with minimal assumptions on edge weight distribution, enabling modeling of more realistic weighted networks.

## Key findings

- Identifies conditions for near-degeneracy in weighted models
- Demonstrates universality properties in the extended framework
- Provides theoretical insights into weighted network phase transitions

## Abstract

Conventionally used exponential random graphs cannot directly model weighted networks as the underlying probability space consists of simple graphs only. Since many substantively important networks are weighted, this limitation is especially problematic. We extend the existing exponential framework by proposing a generic common distribution for the edge weights. Minimal assumptions are placed on the distribution, that is, it is non-degenerate and supported on the unit interval. By doing so, we recognize the essential properties associated with near-degeneracy and universality in edge-weighted exponential random graphs.