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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=715406
• 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.

复杂系统是一类由多个元素以非简单的方式相互作用而产生集体行为的系统。这些系统可以具有不同的性质：生物（如生态系统、微生物组、大脑）、技术（如互联网、电网）或社会（如联系/社会网络）。它们的突发行为同样多样：生态系统在外部扰动下的稳定性，鸟类的群集行为，大脑的认知功能，以及种群对传染病流行的易感性。