A New Upper Bound for the Critical Probability of the Frog Model on Homogeneous Trees

Lebensztayn, Élcio; Machado, Fábio Prates; Popov, Serguei

doi:10.1007/s10955-019-02294-4

Cited by 12 publications

(4 citation statements)

References 18 publications

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“…Jiang et al (2019) studied the contact process on trees with periodic degree sequences from solving the cubic equation by Cardano's formula. Later on, Lebensztayn and Utria (2019) investigated a system of branching random walk and established a new upper bound for the critical parameter of the model, which involves the Cardano's formula.…”

Section: Introductionmentioning

confidence: 99%

Generating roots of cubic polynomials by Cardano's approach on correspondence analysis

Lestari

Pasaribu

Indratno

et al. 2020

Heliyon

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Cardano's formula is among the most popular cubic formula to solve any third-degree polynomial equation. In this paper, we propose the Cardano's approach as the alternative solution to generate the roots of the cubic characteristic polynomial analytically. In the context of correspondence analysis, these roots referred to eigenvalues, which play an important role in assessing the quality of the correspondence plot. Considering the correspondence analysis on the I Â J contingency table for I ¼ 4 and J ¼ 4; 5; ⋯, we obtained a cubic characteristic polynomial (since zero is one of its eigenvalues). Therefore, Cardano's formula allows us to obtain the eigenvalues directly without involving numerical processes, e.g., using singular value decomposition. We note several advantages of using Cardano's approach, such as (1) it produces the roots with the same result as singular value decomposition, as well more precise because without errors involving, (2) the algorithm is simpler and does not depend on initial guess, hence the computation time becomes shorter than numerical process, and (3) the manual calculation is easy because it uses a formula. The results show that the matrix operations on correspondence analysis can be replaced by a formula for determining eigenvalues and eigenvectors of the standard residual matrix directly. Some mathematical results are also presented.

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Section: Introductionmentioning

confidence: 99%

Generating roots of cubic polynomials by Cardano's approach on correspondence analysis

Lestari

Pasaribu

Indratno

et al. 2020

Heliyon

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“…In 2005 there was the first improvement of the upper bound of p c by Lebensztayn, Machado and Popov [18]. Recently, Lebensztayn and Utria improved the result again in [20] and proved an upper bound for p c on biregular trees in [19]. Another modification of the frog model was considered by Deijfen, Hirscher and Lopes in [5] and by Deijfen and Rosengren in [6].…”

Section: Introductionmentioning

confidence: 99%

On transience of frogs on Galton-Watson trees

Müller¹,

Wiegel²

2020

Preprint

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We consider a random interacting particle system, known as the frog model, on infinite Galton-Watson trees allowing offspring zero and one. The system starts with one awake particle (frog) at the root of the tree and a random number of sleeping particles at the other vertices. Awake frogs move according to simple random walk on the tree and as soon as they encounter sleeping frogs, those will wake up and move independently according to simple random walk. The frog model is called transient, if there are almost surely only finitely many particles returning to the root. In this paper we prove a zero-one law for transience of the frog model and show the existence of a transient phase for certain classes of Galton-Watson trees.

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“…In the case when the frogs have geometrically distributed lifetimes, there exists a critical value of the parameter p below which the system dies out almost surely. This critical phenomenon is an important topic of research; see Alves et al [1], Fontes et al [14], Lebensztayn and Utria [30,31], and references therein. The issue of survival is also addressed for similar models on Z in Bertacchi et al [6] and Lebensztayn et al [33].…”

Section: Introductionmentioning

confidence: 99%

Laws of large numbers for the frog model on the complete graph

Lebensztayn

Estrada

2019

Journal of Mathematical Physics

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The frog model is a stochastic model for the spreading of an epidemic on a graph, in which a dormant particle starts to perform a simple random walk on the graph and to awake other particles, once it becomes active. We study two versions of the frog model on the complete graph with N + 1 vertices. In the first version we consider, active particles have geometrically distributed lifetimes. In the second version, the displacement of each awakened particle lasts until it hits a vertex already visited by the process. For each model, we prove that as N → ∞, the trajectory of the process is well approximated by a three-dimensional discrete-time dynamical system. We also study the long-term behavior of the corresponding deterministic systems.2010 Mathematics Subject Classification. 60K35, 60J10, 92B05.

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A New Upper Bound for the Critical Probability of the Frog Model on Homogeneous Trees

Cited by 12 publications

References 18 publications

Generating roots of cubic polynomials by Cardano's approach on correspondence analysis

Generating roots of cubic polynomials by Cardano's approach on correspondence analysis

On transience of frogs on Galton-Watson trees

Laws of large numbers for the frog model on the complete graph

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