Construction of Lattices of Balanced Equivalence Relations for Regular Homogeneous Networks Using Lattice Generators and Lattice Indices
Construction of Lattices of Balanced Equivalence Relations for Regular Homogeneous Networks Using Lattice Generators and Lattice Indices
复制标题
使用格生成器和格索引构造正则齐次网络的平衡等价关系格
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发表时间：
2009
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通讯作者：
H. Kamei
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作者：
H. Kamei
正则齐次网络是一类耦合胞元网络，它由一类胞元（节点）和一类耦合（箭头）组成，每个胞元具有相同数量的输入箭头（称为网络的价）。在耦合细胞网络中，鲁棒同步（流不变多对角）对应于一种特殊的细胞划分，称为平衡等价关系。平衡的等价关系完全由网络结构决定。众所周知，给定有限网络上的平衡等价关系的集合形成完备格。在本文中，我们考虑了规则的齐次网络，其中每个细胞的内部动力学是一维的，其相关的邻接矩阵具有简单的特征值（真实的或复杂的）。我们构造明确的形式，这种网络的平衡等价关系的格引入关键的积木，称为格生成器，沿着与整数称为格指数。格指数的性质允许构造所有可能的格结构，以平衡任何数量的细胞与任何价的正则齐次网络的等价关系。作为一个例子，我们显示了所有14个可能的晶格结构的平衡等价关系的四细胞定期齐次网络。
Regular homogeneous networks are a class of coupled cell network, which comprises one type of cell (node) with one type of coupling (arrow), and each cell has the same number of input arrows (called the valency of the network). In coupled cell networks, robust synchrony (a flow-invariant polydiagonal) corresponds to a special kind of partition of cells, called a balanced equivalence relation. Balanced equivalence relations are determined solely by the network structure. It is well known that the set of balanced equivalence relations on a given finite network forms a complete lattice. In this paper, we consider regular homogeneous networks in which the internal dynamics of each cell is one-dimensional, and whose associated adjacency matrices have simple eigenvalues (real or complex). We construct explicit forms of lattices of balanced equivalence relations for such networks by introducing key building blocks, called lattice generators, along with integer numbers called lattice indices. The properties of lattice indices allow construction of all possible lattice structures for balanced equivalence relations of regular homogeneous networks of any number of cells with any valency. As an illustration, we show all 14 possible lattice structures of balanced equivalence relations for four-cell regular homogeneous networks.