Computation of Balanced Equivalence Relations and Their Lattice for a Coupled Cell Network

Kamei, Hiroko; Cock, Peter

doi:10.1137/100819795

Cited by 42 publications

(71 citation statements)

References 37 publications

(65 reference statements)

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Section: Analyzing Cluster Synchronization Patternsmentioning

confidence: 99%

“…An approach to the construction of all allowed CS patterns in a network has been proposed in [30]. While the method presented in [30] is general, as it applies to any network topology, it is computationally expensive. Here we argue that for the case of symmetric networks, a faster approach can often be followed that takes advantage of computational group theory, which is quite efficient.…”

Section: Analyzing Cluster Synchronization Patternsmentioning

confidence: 99%

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Complete characterization of the stability of cluster synchronization in complex dynamical networks

et al. 2016

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Section: Analyzing Cluster Synchronization Patternsmentioning

confidence: 99%

Section: Analyzing Cluster Synchronization Patternsmentioning

confidence: 99%

Complete characterization of the stability of cluster synchronization in complex dynamical networks

et al. 2016

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“…Note that the orbit partitions for Z 4 and D 4 are the same for P 2 P 2 and P 3 P 3 . An algorithm using the eigenvectors of the adjacency matrix [1,27] can possibly be applied symbolically for all m and n to prove this conjecture. A direct approach might also work.…”

Section: 1mentioning

confidence: 99%

“…The polydiagonal subspace of a cell partition A of a cell network system is ∆ P [1,25,27,39,46]. Note that…”

Section: 2mentioning

confidence: 99%

Invariant Synchrony Subspaces of Sets of Matrices

Neuberger¹,

Sieben²,

Swift³

2020

SIAM J. Appl. Dyn. Syst.

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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 .

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“…They are careful to point out that while the checking stage (computing geometric decompositions and subgroups) is computationally efficient, the number of possible cluster mergings which must be checked grows combinatorially with the number µ of cluster states from symmetry with an upper bound of B µ the µ-th Bell number. Thus the algorithm of Sorrentino et al [10] is much faster than that of Kamei and Cock [40] when the number of cluster states from symmetry is small compared to the size of the network, that is µ N . Both Sorrentino et al [10] and Kamei and Cock [40] note that finding all dynamically valid cluster states for networks with Laplacian coupling may be substantially more difficult than for symmetric networks without self-coupling where connectivity is described by an adjacency matrix and all cluster states arise due to network symmetries.…”

Section: Example: Laplacian Clustersmentioning

confidence: 99%

Clusters in nonsmooth oscillator networks

2018

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For coupled oscillator networks with Laplacian coupling, the master stability function (MSF) has proven a particularly powerful tool for assessing the stability of the synchronous state. Using tools from group theory, this approach has recently been extended to treat more general cluster states. However, the MSF and its generalizations require the determination of a set of Floquet multipliers from variational equations obtained by linearization around a periodic orbit. Since closed form solutions for periodic orbits are invariably hard to come by, the framework is often explored using numerical techniques. Here, we show that further insight into network dynamics can be obtained by focusing on piecewise linear (PWL) oscillator models. Not only do these allow for the explicit construction of periodic orbits, their variational analysis can also be explicitly performed. The price for adopting such nonsmooth systems is that many of the notions from smooth dynamical systems, and in particular linear stability, need to be modified to take into account possible jumps in the components of Jacobians. This is naturally accommodated with the use of saltation matrices. By augmenting the variational approach for studying smooth dynamical systems with such matrices we show that, for a wide variety of networks that have been used as models of biological systems, cluster states can be explicitly investigated. By way of illustration, we analyze an integrate-and-fire network model with event-driven synaptic coupling as well as a diffusively coupled network built from planar PWL nodes, including a reduction of the popular Morris-Lecar neuron model. We use these examples to emphasize that the stability of network cluster states can depend as much on the choice of single node dynamics as it does on the form of network structural connectivity. Importantly, the procedure that we present here, for understanding cluster synchronization in networks, is valid for a wide variety of systems in biology, physics, and engineering that can be described by PWL oscillators.

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Computation of Balanced Equivalence Relations and Their Lattice for a Coupled Cell Network

Cited by 42 publications

References 37 publications

Complete characterization of the stability of cluster synchronization in complex dynamical networks

Complete characterization of the stability of cluster synchronization in complex dynamical networks

Invariant Synchrony Subspaces of Sets of Matrices

Clusters in nonsmooth oscillator networks

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