Normandie Université CORIA, Campus Universitaire du Madrillet, F-76800 Saint-Etienne du Rouvray, France.
Complex Systems Group and GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain.
Phys Rev E. 2018 Aug;98(2-1):020303. doi: 10.1103/PhysRevE.98.020303.
A faithful description of the state of a complex dynamical network would require, in principle, the measurement of all its d variables, an infeasible task for high dimensional systems due to practical limitations. However the network dynamics might be observable from a reduced set of measured variables but how to reliably identify the minimum set of variables providing full observability still remains an unsolved problem. In order to tackle this issue from the Jacobian matrix of the governing equations, we construct a pruned fluence graph in which the nodes are the state variables and the links represent only the linear dynamical interdependences after having ignored the nonlinear ones. From this graph, we identify the largest connected subgraphs with no outgoing links in which every node can be reached from any other node in the subgraph. In each one of them, at least one node must be measured to correctly monitor the state of the system in a d-dimensional reconstructed space. Our procedure is here tested by investigating large-dimensional reaction networks. Our results are validated by comparing them with the determinant of the observability matrix which provides a rigorous assessment of the system's observability.
一个复杂动力网络的准确描述原则上需要测量其所有的 d 个变量，但由于实际限制，对于高维系统来说这是一项不可行的任务。然而，网络动态可能可以从一组经过测量的变量中观察到，但如何可靠地识别提供完全可观测性的最小变量集仍然是一个未解决的问题。为了解决这个问题，我们从控制方程的雅可比矩阵出发，构建了一个修剪后的通量图，其中节点是状态变量，链接仅代表在忽略非线性之后的线性动态相互依赖关系。从这个图中，我们确定了没有传出链接的最大连通子图，其中任何节点都可以从子图中的任何其他节点到达。在每个子图中，至少需要测量一个节点才能在 d 维重构空间中正确监测系统的状态。我们的程序通过研究大维度反应网络进行了测试。我们的结果通过与可观性矩阵的行列式进行比较得到验证，该行列式提供了对系统可观性的严格评估。
