Nonlinear Markov Processes in Big Networks

Abstract

Big networks express various 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 set up a broad class of nonlinear continuous-time block-structured Markov processes, which can be applied to deal with many practical stochastic systems. Firstly, a nonlinear Markov process is derived from a large number of interacting big networks with symmetric interactions, each of which 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 Lyapunov functions and the relative entropy are used to analyze stability or metastability of the big network, and several interesting open problems are proposed with detailed interpretation. We believe that the results given in this paper can be useful and effective in the study of big networks.

28 pages in Special Matrices; 2016

References in corpus (7)

• Differential equation approximations for Markov chains
• Tunneling and Metastability of continuous time Markov chains
• On Mean Field Convergence and Stationary Regime
• A class of nonlinear random walks related to the Ornstein-Uhlenbeck process
• Metastability in communication networks
• Queuing Networks with Varying Topology -- A Mean-Field Approach
• Long Time Results for a Weakly Interacting Particle System in Discrete Time