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Year:: 2019
Contributor:: Sudijono, Timothy (creator); Ramanan, Kavita (thesis advisor); Wang, Hui (reader); Brown University. Applied Mathematics (sponsor)
Genre:: theses
Subject:: Stochastic processes--Computer simulation; Conditional expectations (Mathematics); Graphical modeling (Statistics)
DOI: https://doi.org/10.26300/czs0-t212

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Abstract:: Finite-state interacting Markov chains are a well-studied class of stochastic processes on networks that have been used to model diverse phenomena, such as gas particles and magnetism from physics, voting, reputation systems, and the spread of epidemics. The nature of the interaction of the dynamics of these chains is governed by an underlying graph. As such, these processes are closely related to graphical models from statistics and machine learning and Gibbs measures from statistical mechanics. In studying these models, which for physical applications requires large collections of interacting chains, it is often of interest to autonomously characterize the typical dynamics of a given chain. Current approximations used in practice include Monte Carlo methods, the dynamic cavity method, and the mean field approximation, the latter of which is asymptotically exact on sequences of dense graphs. In a recent work by Lacker, Ramanan, and Wu, a coupled non-Markovian system of equations referred to as the local recursion is introduced, characterizing the dynamics of a typical particle and its neighborhood on infinite tree-like graphs. This thesis investigates the local recursion as an approximation to the dynamics of a particle and its neighborhood on a finite graph. It also investigates conditions needed for ergodicity of the local recursions. To do the latter, we first study a simplification of the local recursion as a chain of infinite order called the frozen local recursion. Infinite order chains are a class of processes whose ergodic properties have been studied; we apply the results of several papers on infinite-order chains to establish results on the frozen local recursion. We then introduce a reformulation of the local recursion as a nonlinear Markov process, a well-known class of processes with existing theory on stationarity and ergodicity. Finally, we discuss a nonlinear approximation of the nonlinear Markov formulation, which we call the k-approximation, in terms of the previous k timesteps of the local recursion’s history. Under suitable conditions we demonstrate existence of stationary distributions for the k-approximation, and we also discuss some of its other properties. We also give numerical evidence of ergodicity for the k-approximations, as well as a numerical comparison of the k-approximations and the nonlinear Markov chain to other approximation methods. Throughout the work, we pose questions to guide future research.
Notes:: Senior thesis (ScB)--Brown University, 2019; Concentration: Applied Mathematics

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Sudijono, Timothy, "Stationarity and Ergodicity of Local Dynamics of Interacting Markov Chains on Large Sparse Graphs" (2019). Applied Mathematics Theses and Dissertations. Brown Digital Repository. Brown University Library. https://doi.org/10.26300/czs0-t212

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Applied Mathematics Theses and Dissertations

Theses and Dissertations for the Applied Mathematics department.
...

Stationarity and Ergodicity of Local Dynamics of Interacting Markov Chains on Large Sparse Graphs