Differential equation approximations of stochastic network processes

An operator semigroup approach

András Bátkai, Istvan Z. Kiss, Eszter Sikolya, L. P. Simon

Research output: Contribution to journalArticle

8 Citations (Scopus)

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 languageEnglish
Pages (from-to)43-58
Number of pages16
JournalNetworks and Heterogeneous Media
Volume7
Issue number1
DOIs
Publication statusPublished - 2012

Fingerprint

Operator Semigroups
Stochastic Networks
Stochastic models
Mathematical operators
Differential equations
Mean-field Approximation
Differential equation
Stochastic Model
Approximation
Markov processes
Mean-field Limit
Nonlinear ODE
Mean Field Equation
Semigroup Theory
Master Equation
Expected Value
Linking
Countable
Markov chain
State Space

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

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

In: Networks and Heterogeneous Media, Vol. 7, No. 1, 2012, p. 43-58.

Research output: Contribution to journalArticle

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