Mean-field analysis of a scaling MAC radio protocol

Horváth, Illés; Horváth, Kristóf; Kovács, Péter; Telek, Miklós

doi:10.3934/jimo.2019111

Cited by 2 publications

(1 citation statement)

References 17 publications

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“…Markov processes are valuable tools for modeling populations from the individual level. Applications range from physics [23] , chemistry [1], engineering [16], biology [5] and social phenomena [29].…”

Section: Introductionmentioning

confidence: 99%

Markov processes on quasi-random graphs

Keliger¹

2021

Preprint

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We study Markov population processes on large graphs, with the local state transition rates of a single vertex being linear function of its neighborhood. A simple way to approximate such processes is by a system of ODEs called the homogeneous mean-field approximation (HMFA). Our main result is showing that HMFA is guaranteed to be the large graph limit of the stochastic dynamics on a finite time horizon if and only if the graph-sequence is quasi-random. Explicit error bound is given and being of order 1 √ N plus the largest discrepancy of the graph. For Erdős Rényi and random regular graphs we show an error bound of order the inverse square root of the average degree. In general, diverging average degrees is shown to be a necessary condition for the HMFA to be accurate. Under special conditions, some of these results also apply to more detailed type of approximations like the inhomogenous mean field approximation (IHMFA). We pay special attention to epidemic applications such as the SIS process.

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Section: Introductionmentioning

confidence: 99%

Markov processes on quasi-random graphs

Keliger¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

Markov processes on quasi-random graphs

Keliger

2024

Acta Math. Hungar.

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We study Markov population processes on large graphs, with the local state transition rates of a single vertex being a linear function of its neighborhood. A simple way to approximate such processes is by a system of ODEs called the homogeneous mean-field approximation (HMFA). Our main result is showing that HMFA is guaranteed to be the large graph limit of the stochastic dynamics on a finite time horizon if and only if the graph-sequence is quasi-random. An explicit error bound is given and it is $$\frac{1}{\sqrt{N}}$$ 1 N plus the largest discrepancy of the graph. For Erdős–Rényi and random regular graphs we show an error bound of order the inverse square root of the average degree. In general, diverging average degrees is shown to be a necessary condition for the HMFA to be accurate. Under special conditions, some of these results also apply to more detailed type of approximations like the inhomogenous mean field approximation (IHMFA). We pay special attention to epidemic applications such as the SIS process.

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Mean-field analysis of a scaling MAC radio protocol

Cited by 2 publications

References 17 publications

Markov processes on quasi-random graphs

Markov processes on quasi-random graphs

Markov processes on quasi-random graphs

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