动态超网络的同步：通过矩阵的同时块对角化降维
Synchronization of dynamical hypernetworks: dimensionality reduction through simultaneous block-diagonalization of matrices
摘要：
我们提出了一个一般框架来研究耦合动力系统超网络同步解的稳定性。我们能够通过使用矩阵的同时块对角化来降低问题的维数。我们获得了一组低维问题的同步解的稳定性的充分必要条件，并通过数值模拟测试我们的低维分析的预测。在某些条件下，这种技术可能会产生一个实质性的减少问题的维数。例如，对于本文分析的一类动态超网络，我们发现任意大的网络都可以归结为维数不超过2的子系统的集合。我们应用我们的减少techique到一些不同的例子，包括一类无向无权hypermotifs的三个节点。
英文摘要：
We present a general framework to study stability of the synchronous solution for a hypernetwork of coupled dynamical systems. We are able to reduce the dimensionality of the problem by using simultaneous block-diagonalization of matrices. We obtain necessary and sufficient conditions for stability of the synchronous solution in terms of a set of lower-dimensional problems and test the predictions of our low-dimensional analysis through numerical simulations. Under certain conditions, this technique may yield a substantial reduction of the dimensionality of the problem. For example, for a class of dynamical hypernetworks analyzed in the paper, we discover that arbitrarily large networks can be reduced to a collection of subsystems of dimensionality no more than 2. We apply our reduction techique to a number of different examples, including a class of undirected unweighted hypermotifs of three nodes.
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