Randomized urn models revisited using stochastic approximation

Laruelle, Sophie; Pagès, Gilles

doi:10.1214/12-aap875

Cited by 55 publications

(96 citation statements)

References 28 publications

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“…We will prove the almost sure convergence of the normalized composition vector by using the ODE method. This proof is the generalization of the similar proof of Laruelle and Pagès in [11].…”

Section: A General Urn Modelsupporting

confidence: 61%

“…In Section 2, we introduce the perturbed Barabási-Albert random graph and then formulate the main result of this paper. In Section 3, we generalize a result on an urn model by Laruelle and Pagès [11]. By using the asymptotic properties of this general urn model, we are able to prove the main theorem.…”

mentioning

confidence: 74%

“…Laruelle and Pagès in [11]. We remark that the generalization of the results of [9] could also be used for our purposes.…”

Section: A General Urn Modelmentioning

confidence: 90%

“…Theorem A (Almost sure convergence with ODE method, Theorem A.1 in [11]). Assume that we have r n a.s.…”

Section: A General Urn Modelmentioning

confidence: 99%

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Barabási–Albert random graph with multiple type edges and perturbation

Backhausz

Rozner

2019

Acta Math. Hungar.

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In this paper we introduce the perturbed version of the Barabási-Albert random graph with multiple type edges and prove the existence of the (generalized) asymptotic degree distribution. Similarly to the non-perturbed case, the asymptotic degree distribution depends on the almost sure limit of the proportion of edges of different types. However, if there is perturbation, then the resulting degree distribution will be deterministic, which is a major difference compared to the non-perturbed case. one has to understand quantities like the number of vertices of a given type with a given number of edges. This kind of analysis is performed in the papers mentioned above. Now we consider models where it is not the vertices, but the edges that have different types. This can be used to model different kind of relationships between the individuals -in a social, biological or financial network, connections are usually not of the same nature, and this can be important from the point of view of contagion or epidemic spread. In our model, the type of an edge is an element of a fixed finite set, and it does not change with time. It is chosen randomly when the edge is born, with a distribution that depends on the current state of the graph and on the types of the edges going out from the neighbours of the new vertex. A general family of preferential attachment random graphs with multi-type edges has been examined in our previous work [3]. Growing networks with two different types of edges can be considered as directed graphs. In this case the type of an edge is its orientation. That is, when a new vertex is born, then it is attached to the graph with an edge directed from the new vertex to the already existing ones or directed from the existing vertices to the new one, and this corresponds to the two different types. Directed preferential attachment models were introduced and examined in [5,14], but with different dynamics than in [3]. In this paper we are interested in a version of robustness in preferential attachment graph models with multi-type edges. The goal is to compare (i) a model in which the probability of choosing a type is exactly the proportion of the current type among the edges going out from the endpoint of the new edge; and (ii) its modified version, when, after this step, types can change with certain probability. In particular, we introduce perturbation in the multi-type Barabási-Albert random graph, and prove that this shows different phenomena than the original version. That is, errors in the dynamics of multi-type random graphs can lead to essential changes in the asymptotic behaviour of the model. The multi-type Barabási-Albert random graph model has been described in [3], which is a generalization of the Barabási-Albert random graph model introduced in [4], specified in [6]. We prove the existence of the asymptotic degree distribution in the perturbed Barabási-Albert random graph, and we also provide recurrence equations for the asymptotic degree distribution. The main difference between the perturbed and...

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Section: A General Urn Modelsupporting

confidence: 61%

mentioning

confidence: 74%

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Barabási–Albert random graph with multiple type edges and perturbation

Backhausz

Rozner

2019

Acta Math. Hungar.

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“…[Ben99,LP13]). Classically, such rescaled algorithms converge to Normal distributions (or linear diffusion processes); see e.g.…”

Section: Introductionmentioning

confidence: 99%

Ergodicity of inhomogeneous Markov chains through asymptotic pseudotrajectories

Benaı̈m¹,

Bouguet²,

Cloez³

2017

Ann. Appl. Probab.

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In this work, we consider an inhomogeneous (discrete time) Markov chain and are interested in its long time behavior. We provide sufficient conditions to ensure that some of its asymptotic properties can be related to the ones of a homogeneous (continuous time) Markov process. Renowned examples such as a bandit algorithms, weighted random walks or decreasing step Euler schemes are included in our framework. Our results are related to functional limit theorems, but the approach differs from the standard "Tightness/Identification" argument; our method is unified and based on the notion of pseudotrajectories on the space of probability measures.

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Urn Model-Based Adaptive Multi-arm Clinical Trials: A Stochastic Approximation Approach

Laruelle¹,

Pagès²

2014

Econophysics of Agent-Based Models

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Randomized urn models revisited using stochastic approximation

Cited by 55 publications

References 28 publications

Barabási–Albert random graph with multiple type edges and perturbation

Barabási–Albert random graph with multiple type edges and perturbation

Ergodicity of inhomogeneous Markov chains through asymptotic pseudotrajectories

Urn Model-Based Adaptive Multi-arm Clinical Trials: A Stochastic Approximation Approach

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