Invariant Synchrony Subspaces of Sets of Matrices

Abstract

A synchrony subspace of R n is defined by setting certain components of the vectors equal according to an equivalence relation. Synchrony subspaces invariant under a given set of square matrices form a lattice. Applications of these invariant synchrony subspaces include equitable and almost equitable partitions of the vertices of a graph used in many areas of graph theory, balanced and exo-balanced partitions of coupled cell networks, and coset partitions of Cayley graphs. We study the basic properties of invariant synchrony subspaces and provide many examples of the applications. We also present what we call the split and cir algorithm for finding the lattice of invariant synchrony subspaces. Our theory and algorithm is further generalized for non-square matrices. This leads to the notion of tactical decompositions studied for its application in design theory.2000 Mathematics Subject Classification. 15A72, 34C14, 34C15, 37C80, 06B23, 90C35, 1 0 0 5 6 0 1 0 7 8 0 0 1 7 8 0 0 0 0 0 of [P (A) | Q] has leading columns in the first three columns. So Col(Q) ⊆ Col(P (A)) = sys(A). Proposition 6.4. Let M = {M 1 , . . . , M m } ⊆ R m×n , A ∈ Π(m), and B ∈ Π(n). Then M l sys(B) ⊆ sys(A) for all l if and only if A ≤ ψ( M 1 P (B) · · · M m P (B) ). Proof. First assume that M l sys(B) ⊆ sys(A) for all l. Then Col(M l P (B)) = M l Col(P (B)) ⊆ Col(P (A))for all l. Proposition 6.1 (3) and (1) implyNow assume that A ≤ ψ( M 1 P (B) · · · M m P (B) ). Then A ≤ ψ(M l P (B)) by Proposition 6.1(2). Hence Col(M l P (B)) ⊆ sys(A) by Lemma 6.2.The following result is the special case of Proposition 6.4 when m = n and B = A. Coarsest invariant refinementIn this section we develop an algorithm that finds the coarsest invariant partition that is not larger than a given partition.Proposition 7.1. If M ⊆ R n×n and A ∈ Π(n), then (Π M (n)∩ ↓ A) is in Π M (n).Proof. Note that the down-set ↓ A is taken in Π(n). Proposition 3.5 implies thatThe following is a generalization of [49, Theorem 5]. It is a recursive process for finding cir M (A), so we refer to it as the cir algorithm. Proposition 7.3. Let M = {M 1 , . . . , M m } ⊆ R n×n . Given a partition A 0 := A ∈ Π(n), recursively define A k+1 := ψ( P (A k ) Q k ), where Q k := M 1 P (A k ) · · · M m P (A k ) .There is a k 0 such that A k = cir M (A) for all k ≥ k 0 .

One-way dependent clusters and stability of cluster synchronization in directed networks

“…Invariant partitions, also called balanced partitions or equitable partitions, have a large number of applications in many fields such as graph, network, and control theories (see ref. 34 and references therein). An invariant partition is one in which nodes with the same color follow the same time evolutions.…”

“…Each balanced partition corresponds to a cluster-synchronous solution where all the nodes in the same cluster follow the same time evolution, but nodes in different clusters do not. These solutions can be expressed in terms of the so-called invariant synchrony subspaces (ISS), i.e., the polydiagonal subspaces under all admissible vector fields 19 , 34 , 38 – 41 .…”

“…Our method to build up the irreducible transformation matrix T and the corresponding matrix B is based in part on the methods for the stability analysis of CS solutions 18 , 20 – 23 , 33 and in part on the methods for finding and analyzing ISS in networks of dynamical systems 19 , 34 , 38 , 40 , 41 , 57 , 58 . In reference to the latter class of methods, in particular, the correspondence between CS and the concepts of ISS and balanced partition was already established in refs.…”

Some aspects of the Markovian SIRS epidemic on networks and its mean‐field approximation

“…This partition of the node set can be adopted for representing a population divided in communities, a framework that captures some of the most salient structural inhomogeneities in contact patterns in many applied contexts 36,37 . For an overview of the use of equitable partitions, from a theoretical and practical point of view, see, e.g., the literature 27,37‐39 . One can identify the set of all nodes in V i as the i th community of the whole population.…”

“…36,37 For an overview of the use of equitable partitions, from a theoretical and practical point of view, see, e.g., the literature. 27,[37][38][39] One can identify the set of all nodes in V i as the ith community of the whole population. In particular, each V i induces a subgraph, G i , of G that is necessarily regular.…”