Nonlinear Markov processes in big networks

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.

MICRO-DYNAMIC ASSOCIATED WITH TWO-STATE NONLINEAR MARKOV PROCESSES: WITH AN APPLICATION TO FREE RECALL

We examine the relationship between time-discrete nonlinear Markov processes defined in terms of nonlinear Markov chains and corresponding micro-dynamic models describing many-body systems composed of a finite number of units interacting with each other via a mean field. To this end, we consider a two-state model and examine appropriately defined measures for attractor strength and noise amplitude using variational calculus. We focus on a two-state model and demonstrate an application to free recall data from 8 participants.