Logistic Operator Equation and the Induced Stochastic Process for Complex System Modelling
Guo, Qi 1993-
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The logistic operator equation (LOE) is a type of a multidimensional system of nonlinear ordinary differential equations developed from the classical logistic equation (LE), which has been devised as a theoretical model for dynamically changing complex networks. According to the choice of its constituent parameters, the LOE can display a number of essentially distinct dynamical characteristics. The connection between a specific LOE and its corresponding complex network is established by interpreting the dependent variable as an adjacency matrix of the network graph. Preexisting studies of the LOE were based upon the Dirichlet series playing the role of an a priori Ansatz for the form of solutions. In this thesis we extend those results by replacing the Dirichlet series with the power series as well as the Fourier series. This leads to new types of solutions for the LOE and, consequently, new examples of complex network dynamics. The solutions are studied via rigorous theoretical calculations as well as via MATLAB and Cytoscape simulations. In addition, the LOE model admits a natural randomization that transforms a deterministic dynamical solution into a stochastic process. A large part of this work is devoted to the study of stochastic processes of this type. In particular, we have been able to demonstrate that in some special cases the given stochastic model is equivalent to a multi-dimensional stochastic differential equation (SDE). However, the general case is extremely hard to tackle and open questions still abound. To illustrate what is involved we calculate certain expectations related to the general LOE-based stochastic process. This approach may be compared to the study of weak solutions of classical SDE via the Fokker-Plank equation. The study of the LOE and LOE-based stochastic processes is a new direction in Complex Network Theory.
DegreeMaster of Science (M.Sc.)
DepartmentMathematics and Statistics
CommitteeSpiteri, Raymond; Khan, Shahedul; Wu, Fangxiang
Copyright DateAugust 2018