Regular synchrony lattices for product coupled cell networks.
Regular synchrony lattices for product coupled cell networks.
复制标题
用于乘积耦合单元网络的规则同步晶格。
作者：
M. Aguiar;A. Dias
有几种方法可以从较小的网络构建（更大的）网络。我们在这里考虑笛卡尔和克罗内克（张量）积网络。我们的主要目的是确定产品网络的同步子空间格与产品的组成网络之间的关系。在这个意义上，我们展示了如何从组件网络的同步子空间的格中得到乘积网络的规则同步子空间的格。具体地说，我们证明了子空间张量对于乘积网络是同步的，当且仅当张量所涉及的子空间是乘积的分量网络的同步子空间。我们还证明，一般情况下，乘积网络的（不规则）同步子空间不是由组件网络的同步子空间描述的，结论是，一般情况下，不可能从组件网络的相应格中获得产品网络的所有同步格。关于笛卡尔积和克罗内克积作为图运算是完全不同的这一事实，我们也作了以下评论，这意味着相关的耦合细胞系统具有不同的结构。虽然期望发生的动力学类型难以比较，但我们在笛卡尔积和克罗内克积的同步子空间格之间建立了包含关系。
There are several ways for constructing (bigger) networks from smaller networks. We consider here the cartesian and the Kronecker (tensor) product networks. Our main aim is to determine a relation between the lattices of synchrony subspaces for a product network and the component networks of the product. In this sense, we show how to obtain the lattice of regular synchrony subspaces for a product network from the lattices of synchrony subspaces for the component networks. Specifically, we prove that a tensor of subspaces is of synchrony for the product network if and only if the subspaces involved in the tensor are synchrony subspaces for the component networks of the product. We also show that, in general, there are (irregular) synchrony subspaces for the product network that are not described by the synchrony subspaces for the component networks, concluding that, in general, it is not possible to obtain the all synchrony lattice for the product network from the corresponding lattices for the component networks. We also make the following remark concerning the fact that the cartesian and Kronecker products, as graph operations, are quite different, implying that the associated coupled cell systems have distinct structures. Although, the kinds of dynamics expected to occur are difficult to compare, we establish an inclusion relation between the lattices of synchrony subspaces for the cartesian and Kronecker products.