Nonlinear Markov Processes in Big Networks

TL;DR

This paper develops a theoretical framework using nonlinear Markov processes and mean-field theory to analyze large-scale networks across various practical domains, providing algorithms and stability analysis tools.

Contribution

It introduces a broad class of nonlinear continuous-time block-structured Markov processes derived from large interacting networks, with algorithms for fixed point computation and stability analysis.

Findings

Derived nonlinear Markov processes from symmetric large networks

Provided algorithms for fixed point computation using UL-type RG-factorization

Analyzed stability and metastability using Birkhoff center, Lyapunov functions, and entropy

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.