Approximating the cumulant generating function of triangles in the Erdös-Rényi random graph

Giardinà, Cristian; Giberti, Claudio; Magnanini, Elena

doi:10.48550/arxiv.2007.12971

Cited by 2 publications

(2 citation statements)

References 25 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 tree-like networks, since in this case the equations for the dynamics are typically determined only by the degree distribution [49]. 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 [50][51][52] . 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 → ∞ [53].…”

Section: Final Remarksmentioning

confidence: 92%

Analytic solution of the two-star model with correlated degrees

Bolfe,

Metz,

Guzmán-González

et al. 2021

Preprint

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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 non-monotonic 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: 92%

Analytic solution of the two-star model with correlated degrees

Bolfe,

Metz,

Guzmán-González

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Various other processes have been explored with related methologies in publications dealing with, e.g., percolation transitions in single or multilayer networks subject to rare initial configurations, [36,37], paths leading to epidemic extinction [38,39], the connection between the rate of rare events and heterogeneity in population networks [40], or large-fluctutation-induced phase switch in majority-vote models [41]. Additionally, large-deviation and rare-event techniques have been employed in the exploration of structural properties, such as the assortativity in configuration-model networks [42], the study of ensembles of random graphs satisfying structural contraints [43,44], and the existence of a first-order condensation transition in the node degrees [45].…”

Section: Introductionmentioning

confidence: 99%

Generalized optimal paths and weight distributions revealed through the large deviations of random walks on networks

Gutiérrez,

Pérez-Espigares

2020

Preprint

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Numerous problems of both theoretical and practical interest are related to finding shortest (or otherwise optimal) paths in networks, frequently in the presence of some obstacles or constraints. A somewhat related class of problems focus on finding optimal distributions of weights which, for a given connection topology, maximize some kind of flow or minimize a given cost function. We show that both sets of problems can be approached through an analysis of the large-deviation functions of random walks. Specifically, a study of ensembles of trajectories allows us to find optimal paths, or design optimal weighted networks, by means of an auxiliary stochastic process (the generalized Doob transform). In contrast to standard approaches, the paths are not limited to shortest paths nor the weights need to optimize a given function. Paths and weights can in fact be tailored to a given statistics of a time-integrated observable, which may be an activity or current, or local functions marking the passing of the random walker through a given node or link. We illustrate this idea with an exploration of optimal paths in the presence of obstacles and optimal networks for flows in the presence of forbidden regions or upper bounds in the local activity.

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Approximating the cumulant generating function of triangles in the Erdös-Rényi random graph

Cited by 2 publications

References 25 publications

Analytic solution of the two-star model with correlated degrees

Analytic solution of the two-star model with correlated degrees

Generalized optimal paths and weight distributions revealed through the large deviations of random walks on networks

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