2021/02/28 by Maíra Bolfe, Maíra Angélica Bolfe, Fernando L. Metz +2 · 7 citations
Mathematics · Physics and Astronomy · Psychology · #Artificial intelligence #Astrophysics #Combinatorics #Complex Network Analysis Techniques #Complex network #Computer science #Critical point (mathematics) #Degree (music) #Degree distribution #Mathematical analysis #Mathematics #Mental Health Research Topics #Opinion Dynamics and Social Influence #Order (exchange) #Phase (matter) #Phase diagram #Phase transition #Physics #Quantum mechanics #Star (game theory) #Stars #Statistical physics #cond-mat.dis-nn #cond-mat.stat-mech #k-nearest neighbors algorithm #physics.soc-ph
paper · pdf · doi:10.1103/physreve.104.014147
published in Physical review. E 104(1), 014147 (American Physical Society) · 13 pages, 11 figures
openalex publication_date 2021/07/29 · arxiv created 2021/08/03 · arxiv updated 2021/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Exponential random graphs are important to model the structure of real-world complex networks. Here we solve the two-star model with degree-degree correlations in the sparse regime. The model constraints the average correlation between the degrees of adjacent nodes (nearest neighbors) and between the degrees at the end-points of two-stars (next nearest neighbors). We compute exactly the network free energy and show that this model undergoes a first-order transition to a condensed phase. For non-negative degree correlations between next nearest neighbors, the degree distribution inside the condensed phase has a single peak at the largest degree, while for negative degree correlations between next nearest neighbors the condensed phase is characterized by a bimodal degree distribution. We calculate the degree assortativities and show they are nonmonotonic functions of the model parameters, with a discontinuous behavior at the first-order transition. The first-order critical line terminates at a second-order critical point, whose location in the phase diagram can be accurately determined. Our results can help to develop more detailed models of complex networks with correlated degrees.