Construction of Lattices of Balanced Equivalence Relations for Regular Homogeneous Networks Using Lattice Generators and Lattice Indices

Abstract

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.

Special Jordan Subspaces and Synchrony Subspaces in Coupled Cell Networks

“…Example 6.1. Taking this last statement into account, we can understand that the lattice structure L 14 in Figure 7, presented by Kamei in [8], for 4-cell regular networks whose adjacency matrices have only simple eigenvalues, can be eliminated. This statement answers the query in Remark 6.1 of [8].…”

“…Taking this last statement into account, we can understand that the lattice structure L 14 in Figure 7, presented by Kamei in [8], for 4-cell regular networks whose adjacency matrices have only simple eigenvalues, can be eliminated. This statement answers the query in Remark 6.1 of [8]. In fact, in that work, the lth level of a lattice includes all codimension-(l − 1) synchrony subspaces (and so, the descriptions of these lattices are upside-down compared with those of all the other lattices in this paper).…”

“…Algebraic methods for finding synchrony subspaces were also motivated by Theorem 4.3 of [6] and were based on the aforementioned relation between synchrony subspaces of regular networks and invariant subspaces of the corresponding adjacency matrix. This relation was studied in detail by Kamei for the special case of simple eigenvalues in [8], and this work was the main motivation of our paper. More recently, Aguiar and Dias [1] extended this work to general networks and provided an algorithm for constructing the lattice of all synchrony subspaces through a small set of synchrony subspaces.…”

“…Recall that in Aguiar and Dias [1] it is proved that all synchrony subspaces of a general network can be obtained using the sum operation over a small set of synchrony subspaces-the sum-irreducible synchrony subspaces. We emphasize that although our initial aim was to generalize the work of Kamei [8] to regular networks, our principal aim has become to demonstrate the close relationship between synchrony subspaces of regular networks and special Jordan subspaces of the corresponding adjacency matrices.…”

“…In lattice theory, the pentagon lattice is very popular and has many important properties, such as that it is one of the simplest examples of nondistributive lattices and is the smallest nonmodular lattice [7]. Kamei [8] proved that when n = 4, there is only one possible pentagon structure, which has two vertices as 2-dimensional synchrony subspaces and one vertex as a 3-dimensional synchrony subspace. When n increases, the number of possible pentagon structures also increases; for example, for n = 5 there are four possible pentagon structures, which are shown in Figure 2.…”