# Condensation of degrees emerging through a first-order phase transition in classical random graphs

## TL;DR

This paper demonstrates that classical random graphs exhibit a first-order phase transition in degree distribution, leading to a condensed phase with many nodes sharing similar degrees, supported by theoretical analysis and Monte Carlo simulations.

## Contribution

It reveals a previously unstudied degree condensation transition in Erdős-Rényi graphs, providing a detailed phase diagram and probabilistic analysis.

## Key findings

- Degree distribution undergoes a first-order phase transition.
- Condensed phase characterized by many nodes with similar degrees.
- Monte Carlo simulations confirm theoretical predictions.

## Abstract

Due to their conceptual and mathematical simplicity, Erd\"os-R\'enyi or classical random graphs remain as a fundamental paradigm to model complex interacting systems in several areas. Although condensation phenomena have been widely considered in complex network theory, the condensation of degrees has hitherto eluded a careful study. Here we show that the degree statistics of the classical random graph model undergoes a first-order phase transition between a Poisson-like distribution and a condensed phase, the latter characterized by a large fraction of nodes having degrees in a limited sector of their configuration space. The mechanism underlying the first-order transition is discussed in light of standard concepts in statistical physics. We uncover the phase diagram characterizing the ensemble space of the model and we evaluate the rate function governing the probability to observe a condensed state, which shows that condensation of degrees is a rare statistical event akin to similar condensation phenomena recently observed in several other systems. Monte Carlo simulations confirm the exactness of our theoretical results.