A set of master variables for the two-star random graph
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915562692542464 |
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| author | Akara-pipattana, Pawat Evnin, Oleg |
| author_facet | Akara-pipattana, Pawat Evnin, Oleg |
| contents | The two-star random graph is the simplest exponential random graph model with nontrivial interactions between the graph edges. We propose a set of auxiliary variables that control the thermodynamic limit where the number of vertices N tends to infinity. Such `master variables' are usually highly desirable in treatments of `large N' statistical field theory problems. For the dense regime when a finite fraction of all possible edges are filled, this construction recovers the mean-field solution of Park and Newman, but with an explicit control over the 1/N corrections. We use this advantage to compute the first subleading correction to the Park-Newman result, which encodes the finite, nonextensive contribution to the free energy. For the sparse regime with a finite mean degree, we obtain a very compact derivation of the Annibale-Courtney solution, originally developed with the use of functional integrals, which is comfortably bypassed in our treatment. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07423 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A set of master variables for the two-star random graph Akara-pipattana, Pawat Evnin, Oleg Statistical Mechanics Disordered Systems and Neural Networks High Energy Physics - Theory Mathematical Physics Probability The two-star random graph is the simplest exponential random graph model with nontrivial interactions between the graph edges. We propose a set of auxiliary variables that control the thermodynamic limit where the number of vertices N tends to infinity. Such `master variables' are usually highly desirable in treatments of `large N' statistical field theory problems. For the dense regime when a finite fraction of all possible edges are filled, this construction recovers the mean-field solution of Park and Newman, but with an explicit control over the 1/N corrections. We use this advantage to compute the first subleading correction to the Park-Newman result, which encodes the finite, nonextensive contribution to the free energy. For the sparse regime with a finite mean degree, we obtain a very compact derivation of the Annibale-Courtney solution, originally developed with the use of functional integrals, which is comfortably bypassed in our treatment. |
| title | A set of master variables for the two-star random graph |
| topic | Statistical Mechanics Disordered Systems and Neural Networks High Energy Physics - Theory Mathematical Physics Probability |
| url | https://arxiv.org/abs/2509.07423 |