Master Stability Function approach to meta-networks

网络的概念对于研究复杂系统非常有用。通过将网络范式扩展到具有不同交织类型节点的系统，科学分析可以获得更广泛的复杂系统集合。为了更好地理解这种元网络，理论工具的发展是必不可少的。我们将用于耦合振子研究的主稳定性函数形式推广到元网络的线性稳定性。这使我们能够解耦两类子网对系统稳定性的影响，从而获得对这些系统更深入的分析理解。此外，这大大简化了数值计算，允许研究更大的系统。与此同时，我们成功地在几个方向上进一步发展了这种形式主义，以便它可以应用于包含几种类型的空间节点或感知不同空间网络的物种的系统。基于这个新开发的工具箱，我们希望在这个项目中研究不同类型元网络的稳定性和模式形成性质，并进一步扩展形式主义。要研究的系统是空间耦合的食物网，具有源和汇的生态系统，空间栖息地网络上的互惠和竞争网络，或由细胞-细胞通信耦合的基因表达网络。该项目要解决的问题包括生物量通量对源库系统稳定性的影响，互惠和拮抗物种相互作用之间转换的空间结构的相关性，以及显示图灵不稳定性的生成网络的构建，从而导致所需的模式。

The concept of networks has been very useful for investigating complex systems. By extending the network paradigm to systems with different interwoven classes of nodes, a much wider ensemble of complex systems has become accessible to scientific analysis. In order to better understand such meta-networks, the development of theoretical tools is indespensible. We have extended the Master Stability function formalism used in the study of coupled oscillators to the linear stability of meta-networks. This enabled us to decouple the influence of the two classes of subnetworks on the stability of the system, thus gaining a deeper analytical understanding of these systems. Furthermore, this greatly simplifies numerical evaluations, allowing to study much larger systems. In the meantime, we succeded in developing this formalism further in several directions, so that it can be applied to systems that contain several types of spatial nodes or species that perceive different spatial networks.Based on this newly developed toolbox we want to investigate in this project thestability and pattern-forming properties of different classes of metanetworks, and to extend the formalism even further. The systems to be investigated are spatially coupled foodwebs, ecosystems with sources and sinks, mutualistic and competitive networks on a spatial habitat network, or gene expression networks coupled by cell-cell communication. The questions to be addressed by this project include the influence of biomass flux on the stability of source-sink systems, the relevance of spatial structure for the switching between mutualistic and antagonistic species interactions, and the construction of genereulatory networks that display a Turing instability that leads to a desired pattern.