### Abstract

The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its master equation, which is a system of linear ODEs with large state space size (N). We derive a single non-linear ODE (called mean-field approximation) for the expected value that yields a good approximation as N tends to infinity. Using only elementary semigroup theory we can prove the order O(1/N) convergence of the solution of the system to that of the mean-field equation. The proof holds also for cases that are somewhat more general than the usual density dependent one. Moreover, for Markov chains where the transition rates satisfy some sign conditions, a new approach using a countable system of ODEs for proving convergence to the mean-field limit is proposed.

Original language | English |
---|---|

Pages (from-to) | 43-58 |

Number of pages | 16 |

Journal | Networks and Heterogeneous Media |

Volume | 7 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2012 |

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### Keywords

- Birth-and-death process
- Dynamic network
- Mean field approximation
- One-parameter operator semigroup

### ASJC Scopus subject areas

- Applied Mathematics
- Statistics and Probability
- Engineering(all)
- Computer Science Applications

### Cite this

*Networks and Heterogeneous Media*,

*7*(1), 43-58. https://doi.org/10.3934/nhm.2012.7.43

**Differential equation approximations of stochastic network processes : An operator semigroup approach.** / Bátkai, András; Kiss, Istvan Z.; Sikolya, Eszter; Simon, L. P.

Research output: Contribution to journal › Article

*Networks and Heterogeneous Media*, vol. 7, no. 1, pp. 43-58. https://doi.org/10.3934/nhm.2012.7.43

}

TY - JOUR

T1 - Differential equation approximations of stochastic network processes

T2 - An operator semigroup approach

AU - Bátkai, András

AU - Kiss, Istvan Z.

AU - Sikolya, Eszter

AU - Simon, L. P.

PY - 2012

Y1 - 2012

N2 - The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its master equation, which is a system of linear ODEs with large state space size (N). We derive a single non-linear ODE (called mean-field approximation) for the expected value that yields a good approximation as N tends to infinity. Using only elementary semigroup theory we can prove the order O(1/N) convergence of the solution of the system to that of the mean-field equation. The proof holds also for cases that are somewhat more general than the usual density dependent one. Moreover, for Markov chains where the transition rates satisfy some sign conditions, a new approach using a countable system of ODEs for proving convergence to the mean-field limit is proposed.

AB - The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its master equation, which is a system of linear ODEs with large state space size (N). We derive a single non-linear ODE (called mean-field approximation) for the expected value that yields a good approximation as N tends to infinity. Using only elementary semigroup theory we can prove the order O(1/N) convergence of the solution of the system to that of the mean-field equation. The proof holds also for cases that are somewhat more general than the usual density dependent one. Moreover, for Markov chains where the transition rates satisfy some sign conditions, a new approach using a countable system of ODEs for proving convergence to the mean-field limit is proposed.

KW - Birth-and-death process

KW - Dynamic network

KW - Mean field approximation

KW - One-parameter operator semigroup

UR - http://www.scopus.com/inward/record.url?scp=84861733991&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84861733991&partnerID=8YFLogxK

U2 - 10.3934/nhm.2012.7.43

DO - 10.3934/nhm.2012.7.43

M3 - Article

VL - 7

SP - 43

EP - 58

JO - Networks and Heterogeneous Media

JF - Networks and Heterogeneous Media

SN - 1556-1801

IS - 1

ER -