| 中文题名： | 双曲空间下有向网络结构与动力学研究 |
| 姓名： | |
| 保密级别： | 公开 |
| 论文语种： | 中文 |
| 学科代码： | 071101 |
| 学科专业： | |
| 学生类型： | 博士 |
| 学位： | 理学博士 |
| 学位类型： | |
| 学位年度： | 2022 |
| 校区： | |
| 学院： | |
| 研究方向： | 复杂网络分析 |
| 第一导师姓名： | |
| 第一导师单位： | |
| 提交日期： | 2022-06-18 |
| 答辩日期： | 2022-05-25 |
| 外文题名： | Structures and Dynamics of Directed Networks under the Perspective of Hyperbolic Space Embedding |
| 中文关键词： | |
| 外文关键词： | Directed networks ; Hyperbolic embedding ; Geometric structures ; Robustness ; Competing dynamics |
| 中文摘要： |
复杂网络分析是研究复杂系统的一种角度和方法，它关注系统中个体相互关联作用的结构，是理解复杂系统性质和功能的一种途径。有向网络是一种重要的网络类型，其连边方向提供系统更多信息的同时也带来了异质性，使得许多数理方法难以直接应用于有向网络。网络嵌入为研究网络结构与功能的关系提供了一种新的途径，其目标是通过将网络拓扑结构映射到低维几何空间，从而获得网络的几何结构信息，使其能从几何的角度揭示复杂系统性质和功能。大多数实证网络具有无标度性、小世界特性、高集聚性、模体和社团结构等特征，并呈现近似树状的层次结构。这些结构特征与双曲空间性质有着密切联系，使得复杂网络的双曲空间嵌入受到广泛的关注。目前在此方向上虽然取得了一系列重要的成果，但鲜有研究关注有向网络双曲空间嵌入及其相关的议题。为此，本文基于双曲空间提出了有向网络嵌入方法与度量几何结构特征的统计指标，并从几何相关性这一新的视角研究了系统功能的相关问题，从而揭示几何结构影响网络鲁棒性与耦合动力学的新规律。
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本文的具体研究内容如下: （1）基于双曲空间研究有向网络的嵌入问题：本文提出了有向网络双曲嵌入的方法，并研究了几何结构特征的统计指标及其实际含义。该方法基于有向网络的二分性将节点在不同方向上的属性转化为二分结构中的两类节点，并结合两类节点的内在关系建立了嵌入方法的数学框架。通过理论与模拟结果的对比分析、嵌入后拓扑结构性质的保留情况及AUC 指标三方面的验证，发现该方法能够有效地将有向网络嵌入双曲空间，并在嵌入过程中保留了原始网络的度分布以及共同邻居分布等拓扑结构特征。然后，本文基于节点在双曲空间的坐标提出了刻画有向网络几何结构的统计指标，包括几何中心性，几何距离，几何相关性等。通过实证分析，本文给出了几何结构与原有拓扑结构性质之间的关系，并发现几何相关性与高阶结构存在强的正相关。最后，本文将该方法用于两个实证系统的研究，并发现网络的双曲几何结构特征能够揭示实际系统的形成机制与演化规律。 （2）基于几何相关性研究多层有向网络的鲁棒性问题：本文将有向网络嵌入方法用于一类特殊的多层有向网络（即多路复用有向网络），通过机制模型与实证分析，研究了几何相关性这一新的宏观结构特征对网络鲁棒性的影响。首先，本文提出了多层有向网络几何相关性的概念（包括层内角度/径向相关性和层间角度/径向相关性），设计了具有几何相关性的多层有向人工网络的生成方法，并基于人工网络讨论了几何相关性与高阶结构之间的关系。研究表明网络的高阶平均度随着层内几何相关性强度的增加而增加，同时层间几何相关性对高阶结构的影响弱于层内几何相关性。然后，通过在多层有向人工网络上模拟级联失效过程，本文系统地分析了各类几何相关性对多层有向网络鲁棒性的影响。研究表明在各类几何相关性中，层内角度相关性对鲁棒性的影响起到主导作用，并且它是通过改变高阶平均度的方式削弱网络的鲁棒性，同时层间几何相关性对网络鲁棒性影响弱于层内几何相关性。这些结果说明了层内角度相关性是能够解码真实系统鲁棒性的预警指标。最后，本文基于实证网络数据验证了在人工网络上得出的模拟结果。 （3）基于几何相关性研究多层有向网络上的耦合动力学问题：本文从几何相关性这一新视角研究了其对耦合动力学特征与临界行为的影响（包括耦合一阶动力学，耦合高阶动力学，“信息-疾病”耦合高阶动力学），通过机制模型与理论分析，揭示了几何相关性与高阶机制协同作用下耦合动力学临界行为涌现的新现象。首先，本文讨论了多层有向网络几何相关性对耦合一阶动力学过程的影响。研究表明层内几何相关性能够降低竞争性耦合一阶动力学的临界阈值，其中层内径向相关性对耦合一阶动力学的促进作用更强。然后，本文将高阶机制引入耦合动力学的研究中，发现在层内几何相关性与高阶机制协同作用下，竞争性耦合动力学也能呈现非连续相变特性。其主要原因是层内角度相关性与网络集聚性存在关联，从而扩大了高阶机制的作用范围。最后，本文将高阶机制引入信息传递与疾病传播的相互作用过程，建立了“信息-疾病”耦合高阶动力学模型，并应用微观马尔可夫链方法分析了疾病传播的临界特征。与以往研究不同，本文研究结果表明在疾病层几何相关性与高阶机制的作用下，疾病传播的相变类型会经历从二级相变到一级相变的转换；并且在疾病层高阶作用极弱的情况下，信息层才能够起到抑制疾病传播的作用。 |
| 外文摘要： |
Complex network analysis focuses on the structure of the interrelated interactions of individuals in complex systems, which is a perspective and approach to studying the nature and function of complex systems. The directed network is an important type of complex network. The direction of links brings heterogeneity, making many existing mathematical methods challenging to apply. Network embedding provides the geometric perspective to unveiling properties and functions of complex systems and a new way of addressing the above challenges. Most empirical networks are characterized by the scale-free property, the small-world property, high clustering, motifs, and community structures, and present the tree-like hierarchical structure. To this end, this paper proposes a novel approach to mapping directed networks into hyperbolic space and a metric system for measuring the characteristics of geometric structures. It is a new view to investigate issues related to complex systems functions, thus revealing new laws about the influences of geometric structures on the network robustness and coupling dynamics.
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The specific research content of this paper is as follows: (1) Directed networks embedding under hyperbolic space: The new method is firstly proposed to embed directed networks into hyperbolic space. This method transforms the node's attributes of different directions into two classes of nodes in the bipartite structure of directed networks and develops network embedding of the mathematical framework by combining the intrinsic relationship between two types of nodes. Topological features of the original network are preserved, including the degree distribution and the common neighbor distribution. Analyzing through three aspects (the comparative analysis of theoretical and simulation results, the preservation of topological structure properties after embedding, and the AUC index), results show that the mapping method can effectively map directed networks into hyperbolic space. Then, this paper proposes the indicators system to quantify the geometric structure based on the nodes' coordinations under hyperbolic space, including geometric centrality, geometric distance, and geometric correlation. The relationship between the geometric structure and the nature of the topology structures is analyzed by empirical data. The results show a strong positive correlation between geometric correlation and higher-order structure. Finally, this paper applies the embedding method to analyze empirical systems and finds that the hyperbolic geometric structures can reveal the formation mechanism and evolution laws. (2)The geometric correlation perspective on multi-layer directed networks to study the robustness: This paper applies the embedding method of directed networks to study a special class of multi-layer directed networks (i.e., multiplex directed networks) and focuses on how geometric correlations affect network robustness through mechanism modeling and empirical analysis. Firstly, the concept of geometric correlations is proposed for multiplex directed networks (including intra-layer angular/radial correlations and inter-layer angular/radial correlations) and designs a method for generating multiplex directed artificial networks with geometric correlations. And we discuss the relationship between geometric correlations and high-order structures. The results show that the average of the high-order degree increases as the strength of intra-layer geometric correlations rises, and inter-layer geometric correlations have a weaker effect on the higher-order structures than intra-layer geometric correlations. Moreover, the effect of geometric correlations on the robustness of multiplex directed networks is analyzed by simulating the cascade failure process. The results show that intra-layer angular correlations play a dominant role in the influence of network robustness, and it is weakening the network robustness by changing the higher-order averages. Also, the intra-layer radial correlation has a greater influence than intra-layer angular correlations on the network robustness. These results also indicate that the intra-layer angular correlation is an early warning indicator that can decode the robustness of real systems. Finally, this paper validates the simulation results of the artificial networks based on empirical networks. (3)The geometric correlations perspective on multiplex directed networks to study the coupling dynamics: This paper studies the influence of geometric correlations on coupled dynamics characteristics and critical behavior, including the coupled first-order dynamics, the coupled higher-order dynamics, and the "information-disease" coupled higher-order dynamics. First, this paper discusses the effect of geometric correlations in multiplex-directed networks on coupled first-order dynamics. The results show that intra-layer geometric correlations can reduce the critical threshold of competing first-order dynamics, where intra-layer radial correlation has a greater influence than intra-layer angular correlations on the dynamic processes. Then, this paper introduces the higher-order mechanism into the study of coupled dynamics. The results show that competing dynamics can exhibit the first-order phase transition property under the synergistic effect of intra-layer geometric correlation and higher-order mechanism. The main reason is that intra-layer angular correlation is related to clustering properties, thus extending the range of the higher-order mechanism. Finally, this paper develops the "information-disease" higher-order coupling dynamics model and discusses the critical feature of disease transmission by micro-Markov chain analysis. Different from previous studies, the results show that the type of phase transition for disease spreading changes the second-order phase transition into the first-order phase transition. And it is only when the higher-order effects of the diseased layer are extremely weak that the information layer can inhibit the disease transmission. |
| 参考文献总数： | 211 |
| 馆藏地： | 图书馆学位论文阅览区（主馆南区三层BC区） |
| 馆藏号： | 博071101/22002 |
| 开放日期： | 2023-06-18 |
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