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

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 the regime of dense connectivity. In this work we solve the model in the finite connectivity regime, f...

Full description

Saved in:
Bibliographic Details
Published in:Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2015-09, Vol.48 (36), p.365001-21
Main Authors: Annibale, A, Courtney, O T
Format: Article
Language:English
Subjects:
complex networks
Condensation
Condensing
Exact solutions
Graphs
Heterogeneity
large deviation
Mathematical analysis
Mathematical models
Phase transformations
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 the regime of dense connectivity. In this work we solve the model in the finite connectivity regime, far more prevalent in real world networks. We show that the model undergoes a condensation transition from a liquid to a condensate phase along the critical line corresponding, in the ensemble parameters space, to the Erdös-Rényi (ER) graphs. In the fluid phase the model can produce graphs with a narrow degree statistics, ranging from regular to ER graphs, while in the condensed phase, the 'excess' degree heterogeneity condenses on a single site with degree . This shows the unsuitability of the two-star model, in its standard definition, to produce arbitrary finitely connected graphs with degree heterogeneity higher than ER graphs and suggests that non-pathological variants of this model may be attained by softening the global constraint on the two-stars, while keeping the number of links hardly constrained.
ISSN:1751-8113
1751-8121
DOI:10.1088/1751-8113/48/36/365001