### 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 |
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Pages (from-to) | 43-58 |

Number of pages | 16 |

Journal | Networks and Heterogeneous Media |

Volume | 7 |

Issue number | 1 |

DOIs | |

Publication status | Published - Dec 18 2012 |

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### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

*Networks and Heterogeneous Media*,

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