复杂网络模型间的等价关系及优质网络的构建研究

In the past ten years, a large number of network research literature indicates that people can be summed up and sorted out methods used in network research. There are some phenomenons: an object has two or more descriptions, or even conflicting statements; the nature of an object has two or more different assertions. These phenomena greatly affected the network research and learning, misleading, delay the study. People need to use precise definitions to describe the research objects, need strict and normative methods to describe the research objects in order to cope with the rapid development of networks. We understand dynamic networks are greatly the product of topology and algebra; and network operations exist naturally, waiting for people to find out; and deeper characteristics of networks are waiting people to detect them by mathematics that serves as fundamental knowledge. Our researching tasks are: 1. To explore the transformation of existing methods in network research, the equivalence relation, modify the wrong methods and eliminate contradictory phenomenons, and reveal the hidden methods, and find new methods for researching networks. 2. From the existing network mathematical research conclusions, to establish the superior performance of network models, trying to apply these network models in the real situation. 3. To establish network operation system by mathematical rules in order to construct network models and characterize them; focus on those operations having stronger backgrounds and being simple, practical and closed to network's excellent natures.

大量的网络研究文献标志着人们可以归纳、整理其中的数学方法。网络研究中存在一个对象有着多种描述，甚至是相互抵触的表述，一个对象的性质有着相异的断言等，这些现象极大地影响了网络研究和学习，误导、延缓研究。人们需要用精确定义来表述研究对象，需要严格、规范的方法来刻画研究对象，以应对网络的快速发展。本申请项目认为动态网络是拓扑与代数合成的产物，网络运算自然地存在，等待着人们的发现，网络的更深层次性质有待于人们用数学“探测”出来。研究目标是：1. 探索网络研究中已有方法之间的变换、等价关系，挖掘深层次的内涵，纠正错误、消除矛盾，发现研究网络的新方法。2. 由现有的网络数学研究结论出发，来建立性能优越的网络模型，并将这些网络模型应用到实际当中。3. 建立规范的网络运算系统，依此来建立网络模型并刻画之；注重应用背景强、简单实用、对网络的优良性质封闭的网络运算。

本申请项目认为动态网络是拓扑与代数合成的产物，利用图的运算和代数群、矩阵的运算，定义了图格，图群格，Topsnut-矩阵格，匹配型图格，图格序列，度序列格（等价于非负整数格）等其他各种类型的图格，基本实现本项目的预定目标：网络研究的基本数学方法，并揭示隐含的方法、发现新的方法；由数学理论来创建具有优良性质的网络，力图将这些网络模型应用到实际当中；用网络运算来规范网络模型的表述，注重应用背景强、简单实用、对网络的优良性质封闭的网络运算。.为图论学科和网络学科提供新运算、新对象以及新问题；建立离散数学分支《拓扑编码学》的基础；建立汉字数学模型，创立汉字图编码，使用者仅需要记忆简短的中文，就可以直接用中文产生数百位、千万位的数字密码与口令；深入研究汉字图编码技术，将拓扑编码向实际应用推进；开展网络、软件和图结构的相似研究。开发、创新了基于图的着色、标号、瑕标号、混合着色、集合着色、序列着色、群着色、同态着色等数百种编码技术，为非对称密码体制提供计算速度快、通信开销小、无条件安全性强、抗量子计算的拓扑编码技术。