We start by analysing the effect of random perturbations on non-hyperbolic scattering dynamics.  We show that, under small random fluctuations of the field, trajectories starting inside KAM islands, not only escape but also gain hyperbolic-like time decay.  We show a random walk modelling this distribution to the amplitude of noise, what is expected to be a universal quadratic power law.  Due to the effect of the random perturbations, we show that the typical dimension of invariant sets is that which is obtained for hyperbolic dynamics. For dissipative systems, we show that when the dissipation is decreased, the number of periodic attractors increase.  Furthermore, we show that the dynamics is then better described by invariants that are time and length scale dependent. Employing random maps we described the escape of a random orbit from an attractor in terms of a dynamical system with a chosen “hole”. We show that the mean escape time depends on the measure of this hole, and we analytically obtain one universal power law describing such dependence.  Once we have described the escape from one attractor, we tackle dynamics of a system having finitely many coexisting attractors in terms of conditional invariant measures.  We study conditions for the hopping process to happen and the sojourn time distribution, according to hyperbolic properties. In the second part of this thesis, we study random dynamical systems from a mathematical perspective. Using set dynamic, we prove the existence of a finite number of random attractors.

by Chung-ping Kwong. / Bibliography : leaves 141-152 / Thesis (Ph.D.)--Chinese University of Hong Kong, 1982

Li Chun Wah. / Thesis (M.Phil.)--Chinese University of Hong Kong. / Bibliography: leaves 65-66.

A reliable inference of networks from data is of key interest in many scientific fields. Several methods have been suggested in the literature to reliably determine links in a network. These techniques rely on statistical methods, typically controlling the number of false positive links, but not considering false negative links. In this thesis new methodologies to improve network inference are suggested. Initial analyses demonstrate the impact of false positive and false negative conclusions about the presence or absence of links on the resulting inferred network. Consequently, revealing the importance of making well-considered choices leads to suggest new approaches to enhance existing network reconstruction methods. A simulation study, presented in Chapter 3, shows that different values to balance false positive and false negative conclusions about links should be used in order to reliably estimate network characteristics. The existence of type I and type II errors in the reconstructed network, also called biased network, is accepted. Consequently, an analytic method that describes the influence of these two errors on the network structure is explored. As a result of this analysis, an analytic formula of the density of the biased vertex degree distribution is found (Chapter 4). In the inverse problem, the vertex degree distribution of the true underlying network is analytically reconstructed, assuming the probabilities of type I and type II errors. Chapters 4-5 show that the method is robust to incorrect estimates of α and β within reasonable limits. In Chapter 6, an iterative procedure to enhance this method is presented in the case of large errors on the estimates of α and β. The investigations presented so far focus on the influence of false positive and false negative links on the network characteristics. In Chapter 7, the analysis is reversed - the study focuses on the influence of network characteristics on the probability of type I and type II errors, in the case of networks of coupled oscillators. The probabilities of α and β are influenced by the shortest path length and the detour degree, respectively. These results have been used to improve the network reconstruction, when the true underlying network is not known a priori, introducing a novel and advanced concept of threshold.

本論文討論了幾個重要的用於網絡分析的圖結構以及相闋的算法設計問題，並著重討論了其中的組合結構及其在大規模網絡中之計算問題。根據這些結構在網絡所處之層次，我們將它們分成局部結構和全局結構來分別討論。 / 對於在大規模網絡中的緊密子圖探測問題和極大完全子圖枚舉問題，本論文提出了新概念和新算法。其一，對於一種名為k-truss的新型緊密子圖，提出了更快的內存算法和適用於在大規模網絡中尋找此種子圖的方法。其二，針對於傳統完全子圖枚舉算法的輸出過大的問題，提出了用於緊湊表示圖中一切極大完全子圖的新方法。采用此種方法，我們能夠在保證重要信息得到保留的情況下，顯著地減小輸出的體積同時提高計算速度。 / This thesis discusses a number of important structures for network analysis and their algorithmic results. Special attentions are paid to combinatorial structures and related computation problems in large networks. According to the granularity of the concepts, we shall distinguish between local and global objects and present them separately. / New algorithms and concepts are proposed for interesting arising problems in cohesive subgraph detection and maximal clique enumeration (MCE). Specifically, algorithms for k-truss, a new type of cohesive subgraphs, are proposed including a fast in-memory algorithm and techniques that handle large disk-resident networks. Motivated by the sheer size of output by classic MCE algorithms, a novel notion for a low-redundancy representation of the set of all maximal cliques is proposed, which enables effective use of maximal cliques and a faster computation of the reduced output. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Wang, Jia. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 58-62). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Preliminaries --- p.2 / Chapter 3 --- Global Structures --- p.3 / Chapter 3.1 --- Component Decomposition --- p.4 / Chapter 3.2 --- Structures with Hierarchy --- p.6 / Chapter 3.2.1 --- k-cores --- p.6 / Chapter 3.2.2 --- k-trusses --- p.8 / Chapter 3.3 --- Improved Truss Computation [45] --- p.10 / Chapter 3.3.1 --- Notations --- p.11 / Chapter 3.3.2 --- Faster In-Memory Algorithm --- p.11 / Chapter 3.3.3 --- Handling Massive Graphs --- p.16 / Chapter 3.3.4 --- Bottom-Up Approach --- p.17 / Chapter 3.3.5 --- Top-k Trusses --- p.23 / Chapter 3.3.6 --- Empirical Study --- p.28 / Chapter 3.4 --- Summary --- p.33 / Chapter 4 --- Local Structures --- p.33 / Chapter 4.1 --- Small Cycles and Clustering Coefficients --- p.33 / Chapter 4.2 --- Maximal Cliques --- p.36 / Chapter 4.3 --- Redundancy-Aware MCE [46] --- p.36 / Chapter 4.3.1 --- Motivations --- p.38 / Chapter 4.3.2 --- τ-visible MCE --- p.41 / Chapter 4.3.3 --- Applications --- p.50 / Chapter 4.3.4 --- Empirical Study --- p.52 / Chapter 4.4 --- Summary --- p.56 / Chapter 5 --- Structures in Mid-Zone --- p.56 / Chapter 5.1 --- Densest Subgraph --- p.56 / Chapter 5.2 --- Relaxed Cliques --- p.57 / Chapter 6 --- Concluding Remarks --- p.58

Thesis (M. Phil.)--University of Hong Kong, 2007. / Title proper from title frame. Also available in printed format.

by Michael Athans. / Bibliography: .26-28. / Prepared under AFOSR grant 72-2273, NASA Ames Research Center grant NGL-22-009-124, NSF grant ENG 75-14103 and ONR contract N00014-76-C-0346.

Thesis (Ph. D.)--Industrial & Systems Engineering, Georgia Institute of Technology, 2004. / Erera, Alan, Committee Member ; Liang, Peng, Committee Member ; Hayter, Anthony, Committee Member ; Lu, Jye-Chyi, Committee Co-Chair ; Kvam, Paul, Committee Co-Chair. Vita. Includes bibliographical references (leaves 79-83).

Thesis--Marburg. / Vita. Includes bibliographical references (p. 127-144).

Thesis (M.S.)--West Virginia University, 2003. / Title from document title page. Document formatted into pages; contains vi, 97 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 97).