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

Giardinà, Cristian; Giberti, Claudio; Magnanini, Elena

doi:10.1007/s10955-021-02707-3

Cited by 7 publications

(4 citation statements)

References 31 publications

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“…This work also opens the perspective to explore systematically the role of short-range and long-range degree correlations in dynamical processes occurring on treelike networks, since in this case the equations for the dynamics are typically determined only by the degree distribution [50]. Finally, we point out that the free energy of ERG models can be mapped in the cumulant generating function of certain structural observables of Erdös-Rényi random graphs [51][52][53]. Therefore, the results of the present paper can be readily applied to study analytically the large deviations of higher-order topological properties of Erdös-Rényi random graphs in the limit N → ∞ [54].…”

Section: Final Remarksmentioning

confidence: 91%

Analytic solution of the two-star model with correlated degrees

et al. 2021

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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.

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Section: Final Remarksmentioning

confidence: 91%

Analytic solution of the two-star model with correlated degrees

et al. 2021

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“…We would like to mention that the characterization of the (infinite size) free energy out of the replica symmetric regime is still an open question. A numerical investigation of the free energy for the case α ≤ −2 has been performed in [14].…”

Section: Edge-triangle Modelmentioning

confidence: 99%

Limit theorems for exponential random graphs

Bianchi¹,

Collet²,

Magnanini³

2021

Preprint

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We consider the edge-triangle model, a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. In the so-called replica symmetric regime, the limiting free energy exists together with a complete characterization of the phase diagram of the model. We borrow tools from statistical mechanics to obtain limit theorems for the edge density. First, we determine the asymptotic distribution of this quantity, as the graph size tends to infinity, in the various phases. Then we study the fluctuations of the edge density around its average value off the critical curve and formulate conjectures about the behavior at criticality based on the analysis of a meanfield approximation of the model. Some of our results can be extended with no substantial changes to more general classes of exponential random graphs.

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“…Graphons are commonly used in combinatorics and computer science to analyze large dense graphs. For example, they have been used in extremal graph theory [13], mean-field games [12], analysis of large graphs [22], and to study the thermodynamic limit of statistical physics systems [14,28], to give a very non-exhaustive list.…”

Section: Introductionmentioning

confidence: 99%

The GHP scaling limit of uniform spanning trees of dense graphs

Archer,

Shalev

2024

Random Struct Algorithms

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We consider dense graph sequences that converge to a connected graphon and prove that the GHP scaling limit of their uniform spanning trees (USTs) is Aldous' Brownian CRT. Furthermore, we are able to extract the precise scaling constant from the limiting graphon. As an example, we can apply this to the scaling limit of the USTs of the Erdös–Rényi sequence for any fixed , and sequences of dense expanders. A consequence of GHP convergence is that several associated quantities of the spanning trees also converge, such as the height, diameter and law of a simple random walk.

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

Cited by 7 publications

References 31 publications

Analytic solution of the two-star model with correlated degrees

Analytic solution of the two-star model with correlated degrees

Limit theorems for exponential random graphs

The GHP scaling limit of uniform spanning trees of dense graphs

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