Towards a mathematical description of complex networks: Effective structure and latent hyperbolic geometry

走向复杂网络的数学描述：有效结构和潜在双曲几何

• 批准号: RGPIN-2019-05183
• 负责人: Allard, Antoine
• 金额: $ 2.11万
• 依托单位: Université Laval
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=763489
• 关键词: Towards; mathematical; description; complex; networks

Complex systems are a class of systems composed of multiple elements interacting in a nonsimple way such that collective behaviors emerge. These systems can be of various natures: biological (e.g., ecosystems, microbiomes, brains), technological (e.g., the Internet, power grids), or social (e.g., contact/social networks). Their emergent behaviors are equally diverse: the stability of ecosystems under external perturbations, the flocking behavior of birds, the cognitive functions of the brain, and the susceptibility of a population to sustain an epidemic of an infectious disease. Critically, these systems are fundamentally more than the sum of their parts, meaning that their collective dynamics are not encoded in the individual elements, but are the results of the complex structure of the interactions between these elements. Because these structures lie somewhere between order and randomness, they cannot easily be fully characterized by a concise set of synthesizing mathematical observables. As a result, the current state-of-the-art modeling approaches still require the full structure to be specified as an input. These extensive models, however, fail to provide insights on the role played by many structural features in the outcome of most dynamical process. Using concepts from Graph Theory, Statistical Physics and Non-linear Dynamics, this research program will develop new tools and techniques to unveil the role these structures play on the dynamics they support. A first approach consists in a dimensionality reduction technique that will systematically regroup elements with similar roles in a given dynamical process. Being agnostic to the topology, this method will allow us to probe these groups of elements and identify their common properties. The second theoretical approach will encode the complexity of these structures into the positions of their elements in a hyperbolic geometry. These maps will allow to understand the organization of these interactions at a glance, and will be leveraged to develop better forecasting models for the behavior of complex systems. Although firmly rooted in classical concepts from Theoretical Physics, the long-term objectives of this research program will have a lasting impact in many disciplines outside the realm of Physics---such as Biology, Epidemiology, Economics and Genetics---, hence broadening the scope of Physics research, and will help to shape the new transdisciplinary dynamics of modern academic research.

复杂系统是由多个元素组成的一类系统，这些元素以非简单的方式相互作用，从而出现集体行为。这些系统可以具有多种性质：生物系统（例如生态系统、微生物组、大脑）、技术系统（例如互联网、电网）或社交系统（例如联系人/社交网络）。它们的新兴行为同样多种多样：外部扰动下生态系统的稳定性、鸟类的集群行为、大脑的认知功能以及人口对传染病流行的易感性。至关重要的是，这些系统从根本上来说不仅仅是其各个部分的总和，这意味着它们的集体动力不是编码在各个元素中，而是这些元素之间相互作用的复杂结构的结果。由于这些结构介于有序性和随机性之间，因此它们无法轻易地通过一组简洁的综合数学观测值来完全表征。因此，当前最先进的建模方法仍然需要将完整结构指定为输入。然而，这些广泛的模型未能深入了解许多结构特征在大多数动态过程的结果中所扮演的角色。该研究项目将利用图论、统计物理学和非线性动力学的概念，开发新的工具和技术，以揭示这些结构对其支持的动力学所发挥的作用。第一种方法包括降维技术，该技术将系统地重新组合在给定动态过程中具有相似作用的元素。由于与拓扑无关，这种方法将允许我们探测这些元素组并识别它们的共同属性。第二种理论方法将这些结构的复杂性编码到双曲几何中其元素的位置中。这些地图将使人们一目了然地了解这些交互的组织，并将被用来为复杂系统的行为开发更好的预测模型。尽管该研究项目牢固植根于理论物理学的经典概念，但其长期目标将对物理学领域以外的许多学科（例如生物学、流行病学、经济学和遗传学）产生持久影响，从而拓宽研究范围物理学研究的范围，并将有助于塑造现代学术研究的新的跨学科动力。