Chan Hoi-Yeung = 網絡與競爭系統的物理 : 個體為本模型中的網絡效應 / 陳凱揚. / Thesis submitted in: September 2005. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 191-197). / Text in English; abstracts in English and Chinese. / Chan Hoi-Yeung = Wang luo yu jing zheng xi tong de wu li : ge ti wei ben mo xing zhong de wang luo xiao ying / Chen Kaiyang. / Abstract --- p.i / Acknowledgments --- p.v / Contents --- p.vii / Chapter 1 --- Overview --- p.1 / Chapter I --- Networks --- p.3 / Chapter 2 --- Networks in nature --- p.4 / Chapter 2.1 --- Introduction --- p.4 / Chapter 2.2 --- Terminology of the networks studies --- p.6 / Chapter 2.2.1 --- Nodes --- p.6 / Chapter 2.2.2 --- Links --- p.6 / Chapter 2.2.3 --- Adjacency matrix --- p.9 / Chapter 2.2.4 --- Connectivity --- p.10 / Chapter 2.2.5 --- Clustering coefficient --- p.11 / Chapter 2.2.6 --- Shortest path --- p.11 / Chapter 2.2.7 --- Connectivity correlation --- p.12 / Chapter 2.3 --- Topology in the real-world networks --- p.13 / Chapter 2.3.1 --- The Internet --- p.13 / Chapter 2.3.2 --- The WWW --- p.15 / Chapter 2.3.3 --- Collaboration networks --- p.15 / Chapter 2.3.4 --- Food webs --- p.16 / Chapter 2.3.5 --- Power grids --- p.17 / Chapter 2.4 --- Discussion --- p.17 / Chapter 3 --- Review on Network Models --- p.19 / Chapter 3.1 --- Introduction --- p.19 / Chapter 3.2 --- Graph Theory --- p.20 / Chapter 3.2.1 --- Classical random graph --- p.20 / Chapter 3.3 --- Evolving networks --- p.23 / Chapter 3.3.1 --- Random growing network --- p.23 / Chapter 3.3.2 --- Fitness growing network --- p.25 / Chapter 3.3.3 --- Barabasi-Albert model --- p.27 / Chapter 3.3.4 --- Fitness model --- p.31 / Chapter 3.4 --- Lattice --- p.33 / Chapter 3.4.1 --- Regular hypercubic lattices (Periodic) --- p.33 / Chapter 3.4.2 --- Regular hypercubic lattices (Free boundary conditions) . --- p.35 / Chapter 3.5 --- Discussion --- p.35 / Chapter 4 --- Network Properties --- p.38 / Chapter 4.1 --- More derivations on existing models --- p.38 / Chapter 4.1.1 --- Classical random graphs --- p.38 / Chapter 4.1.2 --- Barabasi-Albert model --- p.40 / Chapter 4.1.3 --- Fitness Model --- p.42 / Chapter 4.1.4 --- Regular hypercubic lattices (Periodic) --- p.45 / Chapter 4.2 --- New model --- p.48 / Chapter 4.2.1 --- Fitness-BA hybrid model --- p.48 / Chapter 4.3 --- Link removal --- p.55 / Chapter 4.3.1 --- Introduction --- p.55 / Chapter 4.3.2 --- Formalism in connectivity --- p.55 / Chapter 4.3.3 --- Pruned BA Model --- p.56 / Chapter 4.4 --- Link addition --- p.58 / Chapter 4.4.1 --- Introduction --- p.58 / Chapter 4.4.2 --- Regular hypercubic lattices (Periodic) --- p.58 / Chapter 4.5 --- Discussion --- p.60 / Chapter II --- Games --- p.62 / Chapter 5 --- Review on Agent-based models of competing population --- p.63 / Chapter 5.1 --- Introduction --- p.63 / Chapter 5.2 --- The El Farol Bar attendance problem --- p.65 / Chapter 5.2.1 --- Model --- p.65 / Chapter 5.2.2 --- Strategies --- p.66 / Chapter 5.2.3 --- Discussion --- p.66 / Chapter 5.3 --- Minority game --- p.67 / Chapter 5.3.1 --- Model --- p.67 / Chapter 5.3.2 --- Strategies --- p.68 / Chapter 5.3.3 --- Attendance --- p.69 / Chapter 5.3.4 --- History and quasi-Eulerian state --- p.69 / Chapter 5.3.5 --- Success rate and Hamming distance --- p.71 / Chapter 5.3.6 --- Volatility --- p.73 / Chapter 5.3.7 --- Crowd-anticrowd theory --- p.75 / Chapter 5.3.8 --- Discussion --- p.76 / Chapter 6 --- B-A-R model : Dynamics --- p.78 / Chapter 6.1 --- Model --- p.78 / Chapter 6.2 --- Results: Plateaux and periodicity --- p.81 / Chapter 6.3 --- A microscopic view: Agents' decisions and strategy performance --- p.86 / Chapter 6.4 --- A macroscopic view: Bit-string patterns --- p.92 / Chapter 6.4.1 --- The history space --- p.92 / Chapter 6.4.2 --- Bit-string statistics of different states --- p.94 / Chapter 6.5 --- The (max = 1 states --- p.97 / Chapter 6.5.1 --- Values of wm3iX --- p.97 / Chapter 6.5.2 --- "Strategy ranking evolvement: ni, (w)" --- p.101 / Chapter 6.5.3 --- Substates . --- p.105 / Chapter 7 --- B-A-R model : Formalism --- p.108 / Chapter 7.1 --- Resource level at transitions of Cmax = 0 state --- p.108 / Chapter 7.2 --- Resource levels at transitions of Cmax 二 1 states --- p.109 / Chapter 7.2.1 --- Method --- p.109 / Chapter 7.2.2 --- Lmin for upper substate --- p.110 / Chapter 7.2.3 --- Lmin for lower substate --- p.113 / Chapter 7.3 --- Discussion --- p.116 / Chapter 8 --- B-A-R model : Statistics --- p.121 / Chapter 8.1 --- Problem --- p.121 / Chapter 8.2 --- Bit-string statistics --- p.122 / Chapter 8.2.1 --- Allowed transitions --- p.122 / Chapter 8.2.2 --- Grouping the history space --- p.122 / Chapter 8.2.3 --- "Grouping the states, Cmax" --- p.127 / Chapter 8.2.4 --- "Labelling each state, /(C)" --- p.129 / Chapter 8.3 --- Discussion --- p.130 / Chapter III --- Networked games --- p.131 / Chapter 9 --- Networked minority game --- p.132 / Chapter 9.1 --- Model --- p.132 / Chapter 9.2 --- Preliminary results: Agents' success rates --- p.133 / Chapter 9.3 --- Ranking the strategies --- p.135 / Chapter 9.3.1 --- Ranking pattern --- p.136 / Chapter 9.3.2 --- Fraction of strategies in each rank --- p.140 / Chapter 9.4 --- Number of agents using a best strategy belonging to rank r --- p.141 / Chapter 9.4.1 --- Unconnected population --- p.141 / Chapter 9.4.2 --- Networked population . --- p.142 / Chapter 9.5 --- Application: Mean success rate --- p.143 / Chapter 9.6 --- Mean success rate of agents with degree k --- p.147 / Chapter 9.7 --- Application in other networks --- p.149 / Chapter 9.8 --- Discussion --- p.151 / Chapter 10 --- Interacting agents: Networked B-A-R model --- p.154 / Chapter 10.1 --- Model --- p.154 / Chapter 10.2 --- The quasi-Eulerian state (wmax = 1/2 state) --- p.155 / Chapter 10.3 --- The emergent states --- p.159 / Chapter 10.3.1 --- General results --- p.159 / Chapter 10.3.2 --- The Cmax = 0 state --- p.160 / Chapter 10.3.3 --- The Cmax = 1 state --- p.161 / Chapter 10.4 --- Discussion --- p.162 / Chapter IV --- Conclusion --- p.164 / Chapter 11 --- Conclusion --- p.165 / Chapter V --- Appendices --- p.172 / Chapter A --- List of symbols --- p.173 / Chapter A.1 --- Networks --- p.173 / Chapter A.2 --- Games --- p.174 / Chapter A.3 --- Networked games --- p.176 / Chapter B --- Distance distribution in classical random graphs --- p.177 / Chapter B.1 --- Method --- p.177 / Chapter B.2 --- Distance distribution --- p.177 / Chapter B.3 --- Behaviour at small L --- p.178 / Chapter B.4 --- Behaviour at large L --- p.179 / Chapter C --- Co-ordination number in infinite hypercubic lattice --- p.181 / Chapter C.1 --- Method --- p.181 / Chapter C.1.1 --- ID lattice --- p.181 / Chapter C.1.2 --- 2D square lattice --- p.182 / Chapter C.1.3 --- Higher dimension hypercubic lattices --- p.183 / Chapter C.2 --- Coefficients --- p.185 / Chapter D --- Connectivity distribution in fitness-BA hybrid model --- p.187 / Chapter D.1 --- Mean field approach --- p.187 / Chapter D.2 --- Connectivity distribution --- p.188 / Chapter D.3 --- Power-law exponent --- p.190 / Bibliography --- p.191

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Mathematical concepts have aided the progression of many different fields of study. Math is not only helpful in science and engineering, but also in the humanities and social sciences. Therefore, it seemed quite natural to apply my preliminary work with set intersections to voting theory, and that application has helped to focus my thesis. Rather than studying set intersections in general, I am attempting to study set intersections and what they mean in a voting situation. This can lead to better ways to model preferences and to predict which campaign platforms will be most popular. Because I feel that allowing people to only vote for one candidate results in a loss of too much information, I consider approval voting, where people can vote for as many platforms as they like.

Kolmogorov complexity is a theory based on the premise that the complexity of a binary string can be measured by its compressibility; that is, a string’s complexity is the length of the shortest program that produces that string. We explore applications of this measure to graph theory.

Graph Theory is the mathematical study of the structure of abstract relationships between objects. Although these constructions (graphs) are themselves purely theoretical, their ability to model pair-wise relationships in systems of arbitrary complexity yields abundant direct correspondence with numerous important physical and societal systems in the real world. Additionally, the simple discrete nature of fundamental graph structures allows for easy pseudo-geometric visualization of graphs in a wide variety of ways. Taken together, these two properties suggest that graph theory teaching, research, and applications would benefit greatly from the use of a unified software environment for graph construction, interaction, and visualization. Based on this need, a comprehensive survey was undertaken of existing graph theory software packages, programs, and libraries to determine the suitability of each for use as a graph theory teaching and research tool. Some of the desired components (especially in the realm of graph visualization) were found to be implemented in several current tools and systems, but no single system was located with the ability to perform all such functions together in a coordinated way. Graph Shop (the Graph Theory Workshop) is a new software package for graph theory research and applications. It was designed to be usable by students and graph theory beginners yet powerful enough to assist with advanced graph theory research. It runs on a variety of platforms and is available for free under the GNU GPL open source license.

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Relational information visualisation concerns viewing relational data, where the underlying data model is a graph. Hierarchical visualisation is one of hot topics in graph visualisation in which the data is organised in a hierarchical structure. As the amount of information, that we want to visualise, becomes larger and the relations become more complex, classical visualisation techniques and hierarchical drawing methods tend to be inadequate.Traditional hierarchical visualisation algorithms are more concerned with the readability of the layouts. They usually do not consider the efficient utilisation of the geometrical plane for the drawings. Therefore, for most hierarchical layouts, a large portion of display space is wasted as background. The aim of this research is to investigate a space-efficient approach to handle the visualisation of large hierarchies in two-dimensional spaces. This thesis introduces a new graph visualisation approach called enclosure+ connection for visualizing large hierarchies. This approach maximises the space utilisation by taking advantages of the traditional enclosure partitioning approach, while it retains the display of a traditional node-link diagram to hopefully provide users a direct perception of relational structures. The main contribution of this thesis is layout and navigation algorithms for visualising large hierarchies. Two layout algorithms, the space-optimised tree and the EncCon tree, have been developed to achieve the space-efficient visualisation. Both algorithms use the enclosure concept to define layout of hierarchies, which ensure the efficient utilisation of display space. Two focus+context navigation and interaction methods have been proposed to cooperate with the visualization of large hierarchies. Several advanced computer graphics approaches, such as graphic distortion and transparency, are used for the development of these navigation methods. Two case studies have been implemented to evaluate the layout algorithms and the associated navigation methods. The first case study is an application of a shared collaborative workspace which aims to provide users with a better assistance for visual manipulation and navigation of knowledge-based information. The second case study is a visual browser for navigating large-scale online product catalogues. Although the case studies have provided some useful evaluation, formal usability studies would be required to justify fully the effectiveness of these layout and navigation methods. Although this task has not carried out in this research, the author has presented his usability study's plan as a future work.