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

@article{Metz2019CondensationOD, title={Condensation of degrees emerging through a first-order phase transition in classical random graphs.}, author={Fernando Lucas Metz and Isaac P'erez Castillo}, journal={Physical review. E}, year={2019}, volume={100 1-1}, pages={ 012305 } }

Due to their conceptual and mathematical simplicity, Erdös-Rényi 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…

## 4 Citations

### Phase transitions in atypical systems induced by a condensation transition on graphs.

- PhysicsPhysical review. E
- 2020

It is shown that this condensation transition induces distinct thermodynamic first-order transitions between the paramagnetic and the ferromagnetic phases of the Ising model, which leads to an abrupt change in the global eigenvalue statistics of the adjacency matrix, which renders the second moment of the eigen value distribution discontinuous.

### Approximating the Cumulant Generating Function of Triangles in the Erdös–Rényi Random Graph

- MathematicsJournal of Statistical Physics
- 2021

We study the pressure of the “edge-triangle model”, which is equivalent to the cumulant generating function of triangles in the Erdös–Rényi random graph. The investigation involves a population…

### Generalized optimal paths and weight distributions revealed through the large deviations of random walks on networks.

- Mathematics, Computer SciencePhysical review. E
- 2021

An exploration of optimal paths in the presence of obstacles, and networks that optimize flows under constraints on local observables are illustrated with an analysis of the large-deviation functions of random walks.

### Large deviation and anomalous fluctuations scaling in degree assortativity on configuration networks

- MathematicsJournal of Statistical Mechanics: Theory and Experiment
- 2021

By constructing a multicanonical Monte Carlo simulation, we obtain the full probability distribution ρ N (r) of the degree assortativity coefficient r on configuration networks of size N by using the…

## References

SHOWING 1-10 OF 38 REFERENCES

### The two-star model: exact solution in the sparse regime and condensation transition

- Physics, Mathematics
- 2015

The two-star model is the simplest exponential random graph model that displays complex behavior, such as degeneracy and phase transition. Despite its importance, this model has been solved only in…

### Condensation transition in joint large deviations of linear statistics

- Mathematics
- 2014

Real space condensation is known to occur in stochastic models of mass transport in the regime in which the globally conserved mass density is greater than a critical value. It has been shown within…

### Phase diagram and metastability of the Ising model on two coupled networks

- PhysicsJournal of Statistical Mechanics: Theory and Experiment
- 2018

We explore the cooperative behaviour and phase transitions of interacting networks by studying a simplified model consisting of Ising spins placed on the nodes of two coupled Erdös–Rényi random…

### Critical phenomena in complex networks

- PhysicsArXiv
- 2007

A wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, $k$-core percolations, phenomena near epidemic thresholds, condensation transitions,critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks are mentioned.

### Nonequilibrium statistical mechanics of the zero-range process and related models

- Mathematics
- 2005

We review recent progress on the zero-range process, a model of interacting particles which hop between the sites of a lattice with rates that depend on the occupancy of the departure site. We…

### Entropy distribution and condensation in random networks with a given degree distribution.

- Computer SciencePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2014

It is shown that the leading term of the entropy of scale-free network ensembles depends only on the network size and average degree and that entropy is self-averaging, meaning that its relative variance vanishes in the thermodynamic limit.

### Distribution of diameters for Erdős-Rényi random graphs.

- MathematicsPhysical review. E
- 2018

The distribution of diameters d of Erdős-Rényi random graphs with average connectivity c is studied numerically and the finite-size rate function Φ(d/N) is determined and extrapolate numerically to N→∞, indicating that the large-deviation principle holds.

### Bose-Einstein condensation in complex networks.

- PhysicsPhysical review letters
- 2001

The evolution of many complex systems, including the World Wide Web, business, and citation networks, is encoded in the dynamic web describing the interactions between the system's constituents, and addressing the dynamical properties of these nonequilibrium systems within the framework of equilibrium quantum gases predicts the "first-mover-advantage," "fit-get-rich," and "winner-takes-all" phenomena.

### Large deviations, condensation and giant response in a statistical system

- Mathematics
- 2015

We study the probability distribution P of the sum of a large number of non-identically distributed random variables nm. Condensation of fluctuations, the phenomenon whereby one of such variables…