2003/11/30 by Andreas Engel, Rémi Monasson, R�mi Monasson +1
Mathematics · Physics and Astronomy · #Complex Network Analysis Techniques #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.dis-nn #cond-mat.stat-mech
paper · pdf · doi:10.1007/s10955-004-2268-6
published as J. Stat. Phys. 117, 387 (2004) · 38 pages, 9 figures, Latex2e, corrected and extended version including numerical simulation results
arxiv created 2004/01/21 · openalex publication_date 2004/10/28 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/04
We show that large deviation properties of Erdös-Rényi random graphs can be derived from the free energy of the q-state Potts model of statistical mechanics. More precisely the Legendre transform of the Potts free energy with respect to ln q is related to the component generating function of the graph ensemble. This generalizes the well-known mapping between typical properties of random graphs and the q→ 1 limit of the Potts free energy. For exponentially rare graphs we explicitly calculate the number of components, the size of the giant component, the degree distributions inside and outside the giant component, and the distribution of small component sizes. We also perform numerical simulations which are in very good agreement with our analytical work. Finally we demonstrate how the same results can be derived by studying the evolution of random graphs under the insertion of new vertices and edges, without recourse to the thermodynamics of the Potts model.