Spatial Gibbs random graphs

Mourrat, Jean-Christophe; Valesin, Daniel

doi:10.1214/17-aap1316

Cited by 6 publications

(29 citation statements)

References 47 publications

(49 reference statements)

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“…This changes drastically by the introduction of the Gibbs weight (at least if the parameter b is large enough). The main result of [MV16], reproduced as Theorem 1 below, is the convergence in probability of the random variable N −1 log H p (G N ) under P b,p N,γ , when γ, b, p are fixed and N is taken to infinity. The limit is deterministic and given explicitly as a function of the parameters.…”

Section: Introductionmentioning

confidence: 99%

Local limits of spatial Gibbs random graphs

Endo¹,

Valesin²

2019

ALEA

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We study the spatial Gibbs random graphs introduced in [MV16] from the point of view of local convergence. These are random graphs embedded in an ambient space consisting of a line segment, defined through a probability measure that favors graphs of small (graph-theoretic) diameter but penalizes the presence of edges whose extremities are distant in the geometry of the ambient space. In [MV16] these graphs were shown to exhibit threshold behavior with respect to the various parameters that define them; this behavior was related to the formation of hierarchical structures of edges organized so as to produce a small diameter. Here we prove that, for certain values of the underlying parameters, the spatial Gibbs graphs may or may not converge locally, in a manner that is compatible with the aforementioned hierarchical structures.

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Section: Introductionmentioning

confidence: 99%

Local limits of spatial Gibbs random graphs

Endo¹,

Valesin²

2019

ALEA

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Section: Introductionmentioning

confidence: 99%

“…In [132], the authors introduced and studied a class of random graphs which they called spatial Gibbs random graphs. These are random graphs embedded in an ambient space, which in [132], was a finite line segment. They are distributed according to a measure that penalizes the presence of edges whose extremities are distant (in terms of the ambient space geometry), but also penalizes graphs with large graph-theoretic diameter.…”

Section: Introductionmentioning

confidence: 99%

“…Graphs sampled from this measure may thus be thought of as answering to a compromise between the conflicting requirements of using few long edges and having vertices close to each other in graph distance. The main result of [132] describes the typical aspect of these graphs depending on the various parameters that define them. Here, we continue the study of spatial Gibbs random graphs on line segments by considering their local convergence properties.…”

Section: Introductionmentioning

confidence: 99%

“…Let us explain the definition of spatial Gibbs random graphs and briefly present the results of [132]. Define the set of graphs on Z as G = {g = (V, E) : V ⊂ Z and g is locally finite}.…”

Section: Introductionmentioning

confidence: 99%

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Gibbs Measures for Models on Lines and Trees

Endo¹

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Esta versão da tese contém as correções e alterações sugeridas pela Comissão Julgadora durante a defesa da versão original do trabalho, realizada em 31/07/2018. Uma cópia da versão original está disponível no AgradecimentosThe best part of the thesis is here, since I can write about the people who I can thank a lot, the good moments of my Ph. D., and add anything that, in the future, when someone opens this part, he or she will see that I am grateful for these wonderful four years.First I thank my supervisors, Aernout and Rodrigo, and my cosupervisor, Daniel, to open the door of statistical mechanics. Thanks to you, I found the area that I will hopefully follow for the rest of my life. I started studying this area when I was in the master degree, and Rodrigo gave a lecture on the introduction of models for ferromagnetism. He showed for the students that the statistical mechanics is not a rocket science, and it is actually a nice area. Thanks Rodrigo to give this amazing lecture. Aernout, who showed me that it is possible to think in the qualitative ideas avoiding heavy maths to understand how a problem could be solved, opened my mind. Thanks for many discussions that we had in Groningen. Daniel, I would like to thank you to have nice discussions about the spatial Gibbs random graphs problem, because those discussions were fruitful to understand how an idea and a proof can be simplified. Thanks to many advices and to help me how to "survive" in Groningen.I would like to thank Noga Alon, Arnaud Le Ny and Wioletta Ruzsel to show me that to do research with collaborations is very interesting and delightful.Let me thank Irene, who rented a room for me when I was in Groningen. The room is very confortable, and it was very important for me, since I could say "I'm home" in a country where I have never been before. Thanks to show me many dutch cultures, to help me to solve many bureaucracies, to teach dutch language, and for the dinners. Dank u wel, het was gezellig! Why not also thanks Groningen city? Het is een mooie stad. Ik ben elke dag gefietst en ik altijd vond culturele evenementen. Ik vind Groningen leuk! Dank u wel, Groningen.There are many people to thanks when I lived in the Netherlands. First, I would like to thank all of the group of the Groningen university. Desiree, Ineke and Lineke to help me for the university bureaucracy. Pedro, it was very fun to talk a lot with you, let's go again to eat krokketjes and drink dutch beer. Réka, it was so nice to show me nice places in Groningen and invite me to watch nice movies. Vladimír, thanks a lot to invite me to the game party and pancakes evenings. Further I would like to thanks Mónika, Nikolay, Pan and all other former and present members of our group.I would like to thanks Sylvia and Sonia for many nice talks that we had.Bruno, you are brazilian, we met in Brazil, but it is so nice that we both went to the Netherlands at the same time to do our Ph. D. Thanks to show me Delft and the research group. Let's spread the brazilian's culture and memes in the world! Ric...

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The age-dependent random connection model

et al. 2019

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We investigate a class of growing graphs embedded into the d-dimensional torus where new vertices arrive according to a Poisson process in time, are randomly placed in space and connect to existing vertices with a probability depending on time, their spatial distance and their relative ages. This simple model for a scale-free network is called the age-based spatial preferential attachment network and is based on the idea of preferential attachment with spatially induced clustering. We show that the graphs converge weakly locally to a variant of the random connection model, which we call the age-dependent random connection model. This is a natural infinite graph on a Poisson point process where points are marked by a uniformly distributed age and connected with a probability depending on their spatial distance and both ages. We use the limiting structure to investigate asymptotic degree distribution, clustering coefficients and typical edge lengths in the age-based spatial preferential attachment network.MSc classification (2010): Primary 05C80 Secondary 60K35.

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Spatial Gibbs random graphs

Cited by 6 publications

References 47 publications

Local limits of spatial Gibbs random graphs

Local limits of spatial Gibbs random graphs

Gibbs Measures for Models on Lines and Trees

The age-dependent random connection model

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