论文标题
用不对称输入的网络分类
Classification of Networks with Asymmetric Inputs
论文作者
论文摘要
与耦合单元网络相关的耦合单元系统由与网络结构一致的（平滑）向量场确定。在这里，我们遵循Stewart，Golubitsky和Pivato（耦合细胞网络中同步的对称性组和模式）的形式主义，Siam J.Appl。2（4）（2003）609--646），Golubistky，Stewart和Torok（Stewart and Torok和Synchrony in syms in syms in sym sypers sypled artoty sys sypled artory artoty artory artoty artory sys sypled artoty artoty sypled artoty sypled artoty。 （1）（2005）78--100）和场（组合动力学，动力学系统19（2004）（3）（3）217--243）。众所周知，两个非异态N细胞耦合网络可以确定相同的向量场集合 - 据说这些网络是相当的。所有N细胞耦合网络的集合都被分区为ODE等效网络类。在没有进一步限制的情况下，ode类的数量不是有限的，并且每个类都有无限数量的网络。在每个ODE级中，我们可以找到一个有限的网络子类，以最大程度地减少类中的边缘数量，称为最小网络。 在本文中，我们考虑使用不对称输入的耦合细胞网络。也就是说，如果k是不同边缘类型的数量，则这些网络具有每个单元格输入的属性，每种类型之一。固定单元格的数量n，我们证明：ode级的数量是有限的；限制在最大n（n-1）输入中，我们可以覆盖所有ode级；所有具有N（N-1）不对称输入的最小N细胞网络都是相当的。我们还提供了一个简单的标准，以测试网络是否最小，我们猜测较低的n细胞网络的不同ode级网络的数量具有任何数字k的非对称输入。此外，我们介绍了具有三个单元格和两个不对称输入的网络的代表的完整列表。
Coupled cell systems associated with a coupled cell network are determined by (smooth) vector fields that are consistent with the network structure. Here, we follow the formalisms of Stewart, Golubitsky and Pivato (Symmetry groupoids and patterns of synchrony in coupled cell networks, SIAM J. Appl. Dyn. Syst. 2 (4) (2003) 609--646), Golubistky, Stewart and Torok (Patterns of synchrony in coupled cell networks with multiple arrows, SIAM J. Appl. Dynam. Sys. 4 (1) (2005) 78--100) and Field (Combinatorial dynamics, Dynamical Systems 19 (2004) (3) 217--243). It is known that two non-isomorphic n-cell coupled networks can determine the same sets of vector fields -- these networks are said to be ODE-equivalent. The set of all n-cell coupled networks is so partitioned into classes of ODE-equivalent networks. With no further restrictions, the number of ODE-classes is not finite and each class has an infinite number of networks. Inside each ODE-class we can find a finite subclass of networks that minimize the number of edges in the class, called minimal networks. In this paper, we consider coupled cell networks with asymmetric inputs. That is, if k is the number of distinct edges types, these networks have the property that every cell receives k inputs, one of each type. Fixing the number n of cells, we prove that: the number of ODE-classes is finite; restricting to a maximum of n(n-1) inputs, we can cover all the ODE-classes; all minimal n-cell networks with n(n-1) asymmetric inputs are ODE-equivalent. We also give a simple criterion to test if a network is minimal and we conjecture lower estimates for the number of distinct ODE-classes of n-cell networks with any number k of asymmetric inputs. Moreover, we present a full list of representatives of the ODE-classes of networks with three cells and two asymmetric inputs.
