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

复杂系统是由多个元素以一种非简单的方式相互作用，从而产生集体行为的一类系统。这些系统可以具有各种性质：生物(例如，生态系统、微生物、大脑)、技术(例如，互联网、电网)或社会(例如，联系/社交网络)。它们的紧急行为也是多种多样的：生态系统在外部扰动下的稳定性，鸟类的集群行为，大脑的认知功能，以及种群维持传染病流行的敏感性。关键的是，这些系统从根本上不仅仅是它们各部分的总和，这意味着它们的集体动力不是编码在单个元素中，而是这些元素之间相互作用的复杂结构的结果。因为这些结构介于有序性和随机性之间，所以很难用一组简明的综合数学观测数据来完全描述它们。因此，当前最先进的建模方法仍然需要将完整的结构指定为输入。然而，这些广泛的模型未能提供关于许多结构特征在大多数动态过程的结果中所起作用的洞察力。利用图论、统计物理和非线性动力学的概念，这项研究计划将开发新的工具和技术，以揭示这些结构对它们所支持的动力学所起的作用。第一种方法是一种降维技术，它将在给定的动态过程中系统地重新组合具有相似角色的元素。由于与拓扑无关，这种方法将允许我们探测这些元素组并确定它们的共同属性。第二种理论方法将这些结构的复杂性编码为其元素在双曲几何中的位置。这些地图将允许一目了然地了解这些交互的组织，并将被用于为复杂系统的行为开发更好的预测模型。尽管这一研究计划牢牢植根于理论物理学的经典概念，但该研究计划的长期目标将对物理学领域以外的许多学科产生持久的影响-生物学、流行病学、经济学和遗传学-从而拓宽物理学研究的范围，并将有助于塑造现代学术研究的新的跨学科动态。

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.