Nonlinear Markov processes in big networks

Quan-Lin Li

Special Matrices (2016)

  • Volume: 4, Issue: 1, page 202-217
  • ISSN: 2300-7451

Abstract

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Big networks express multiple classes of large-scale networks in many practical areas such as computer networks, internet of things, cloud computation, manufacturing systems, transportation networks, and healthcare systems. This paper analyzes such big networks, and applies the mean-field theory and the nonlinear Markov processes to constructing a broad class of nonlinear continuous-time block-structured Markov processes, which can be used to deal with many practical stochastic systems. Firstly, a nonlinear Markov process is derived from a large number of big networks with weak interactions, where each big network is described as a continuous-time block-structured Markov process. Secondly, some effective algorithms are given for computing the fixed points of the nonlinear Markov process by means of the UL-type RG-factorization. Finally, the Birkhoff center, the locally stable fixed points, the Lyapunov functions and the relative entropy are developed to analyze stability or metastability of the system of weakly interacting big networks, and several interesting open problems are proposed with detailed interpretation. We believe that the methodology and results given in this paper can be useful and effective in the study of big networks.

How to cite

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Quan-Lin Li. "Nonlinear Markov processes in big networks." Special Matrices 4.1 (2016): 202-217. <http://eudml.org/doc/277111>.

@article{Quan2016,
abstract = {Big networks express multiple classes of large-scale networks in many practical areas such as computer networks, internet of things, cloud computation, manufacturing systems, transportation networks, and healthcare systems. This paper analyzes such big networks, and applies the mean-field theory and the nonlinear Markov processes to constructing a broad class of nonlinear continuous-time block-structured Markov processes, which can be used to deal with many practical stochastic systems. Firstly, a nonlinear Markov process is derived from a large number of big networks with weak interactions, where each big network is described as a continuous-time block-structured Markov process. Secondly, some effective algorithms are given for computing the fixed points of the nonlinear Markov process by means of the UL-type RG-factorization. Finally, the Birkhoff center, the locally stable fixed points, the Lyapunov functions and the relative entropy are developed to analyze stability or metastability of the system of weakly interacting big networks, and several interesting open problems are proposed with detailed interpretation. We believe that the methodology and results given in this paper can be useful and effective in the study of big networks.},
author = {Quan-Lin Li},
journal = {Special Matrices},
keywords = {Nonlinear Markov process; Big network; Mean-field theory; Fixed point; Metastability; nonlinear Markov processes; big networks; mean-field theory; fixed point; metastability},
language = {eng},
number = {1},
pages = {202-217},
title = {Nonlinear Markov processes in big networks},
url = {http://eudml.org/doc/277111},
volume = {4},
year = {2016},
}

TY - JOUR
AU - Quan-Lin Li
TI - Nonlinear Markov processes in big networks
JO - Special Matrices
PY - 2016
VL - 4
IS - 1
SP - 202
EP - 217
AB - Big networks express multiple classes of large-scale networks in many practical areas such as computer networks, internet of things, cloud computation, manufacturing systems, transportation networks, and healthcare systems. This paper analyzes such big networks, and applies the mean-field theory and the nonlinear Markov processes to constructing a broad class of nonlinear continuous-time block-structured Markov processes, which can be used to deal with many practical stochastic systems. Firstly, a nonlinear Markov process is derived from a large number of big networks with weak interactions, where each big network is described as a continuous-time block-structured Markov process. Secondly, some effective algorithms are given for computing the fixed points of the nonlinear Markov process by means of the UL-type RG-factorization. Finally, the Birkhoff center, the locally stable fixed points, the Lyapunov functions and the relative entropy are developed to analyze stability or metastability of the system of weakly interacting big networks, and several interesting open problems are proposed with detailed interpretation. We believe that the methodology and results given in this paper can be useful and effective in the study of big networks.
LA - eng
KW - Nonlinear Markov process; Big network; Mean-field theory; Fixed point; Metastability; nonlinear Markov processes; big networks; mean-field theory; fixed point; metastability
UR - http://eudml.org/doc/277111
ER -

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