Differential equation approximations of stochastic network processes: An operator semigroup approach
András Bátkai Istvan Z. Kiss Eszter Sikolya Péter L. Simon
Networks & Heterogeneous Media 2012, 7(1): 43-58 doi: 10.3934/nhm.2012.7.43
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 $\mathcal{O}(1/N)$ 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.
keywords: one-parameter operator semigroup birth-and-death process mean field approximation. Dynamic network
Vertex control of flows in networks
Klaus-Jochen Engel Marjeta Kramar Fijavž Rainer Nagel Eszter Sikolya
Networks & Heterogeneous Media 2008, 3(4): 709-722 doi: 10.3934/nhm.2008.3.709
We study a transport equation in a network and control it in a single vertex. We describe all possible reachable states and prove a criterion of Kalman type for those vertices in which the problem is maximally controllable. The results are then applied to concrete networks to show the complexity of the problem.
keywords: operator semigroups boundary control transport equation networks

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