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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=738110
• 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.

复杂的系统是一类由多个元素组成的系统，这些元素以非微调方式相互作用，使集体行为出现。这些系统可以具有各种本质：生物学（例如生态系统，微生物组，大脑），技术（例如，互联网，电网）或社交（例如，联系/社交网络）。它们的紧急行为同样多样：生态系统在外部扰动下的稳定性，鸟类的植入行为，大脑的认知功能以及人群维持传染病流行的敏感性。至关重要的是，这些系统从根本上远远超过其各个部分的总和，这意味着它们的集体动力学不是在单个元素中编码，而是这些元素之间相互作用的复杂结构的结果。由于这些结构位于顺序和随机性之间，因此它们不能轻易完全以简洁的合成数学可观察物为特征。结果，当前的最新建模方法仍然需要将完整的结构指定为输入。但是，这些广泛的模型未能提供有关许多结构特征在最动态过程的结果中所起的作用的见解。使用图理论，统计物理学和非线性动力学的概念，该研究计划将开发新的工具和技术，以揭示这些结构在其支持的动态上所扮演的作用。第一种方法包括降低降低技术，该技术将在给定的动力学过程中系统地重新组合具有相似作用的元素。对于拓扑的不可知论者，这种方法将使我们能够探究这些元素组并确定其共同特性。第二种理论方法将将这些结构的复杂性编码为双曲几何形状的元素位置。这些地图将使这些相互作用的组织一目了然，并将被利用以开发出更好的复杂系统行为预测模型。尽管该研究计划的长期目标扎根于理论物理学中的经典概念，但将对物理学领域之外的许多学科产生持久影响 - 例如生物学，流行病学，经济学，经济学和遗传学 - - 因此，物理学研究的范围，并将有助于拓宽现代学术研究的新跨学科动力学。