Invariant Polydiagonal Subspaces of Matrices and Constraint Programming

TL;DR

The paper addresses the challenge of finding all invariant polydiagonal subspaces, which include synchrony and anti-synchrony structures, by introducing a coloring-vector representation of tagged partitions Δ_𝒫. It reduces the problem to a constraint-satisfaction formulation and demonstrates that state-of-the-art solvers can compute these subspaces much faster than prior methods, enabling analysis of large matrices and complex graphs. The authors provide multiple solver implementations, extensive benchmarks, and compelling examples (e.g., Buckminsterfullerene and Petersen graphs), establishing a practical, scalable approach for network dynamics and bifurcation studies. This work significantly broadens the tractable range of invariant subspaces, with direct implications for symmetry-reduced dynamics and BLIS-based bifurcation analysis.

Abstract

In a polydiagonal subspace of the Euclidean space, certain components of the vectors are equal (synchrony) or opposite (anti-synchrony). Polydiagonal subspaces invariant under a matrix have many applications in graph theory and dynamical systems, especially coupled cell networks. We describe invariant polydiagonal subspaces in terms of coloring vectors. This approach gives an easy formulation of a constraint satisfaction problem for finding invariant polydiagonal subspaces. Solving the resulting problem with existing state-of-the-art constraint solvers greatly outperforms the currently known algorithms.

Figures (8)

• Figure 1: Weighted digraph with adjacency matrix and lattice of coloring vectors of $M$-invariant polydiagonal subspaces. The shading indicates the synchrony subspaces.
• Figure 2: Constraint satisfaction problem for the coloring vectors $c$ of $M$-invariant polydiagonal subspaces for $M\in\mathbb{Z}^{n\times n}$.
• Figure 3: The Hasse diagram of the poset of $\Gamma$-orbits of $M$-invariant synchrony subspaces for the Buckyball graph. The shaded orbits are described in Figures \ref{['fig:w1012']} through \ref{['fig:1012subs']}. The relationship between orbits 30, 31, 33, and 35, as indicated by the red figure eight, shows that this poset is not a lattice.
• Figure 4: The invariant synchrony subspace $W$ of Example \ref{['exa:Bucky']}. The 2D embedding uses an azimuthal equidistant projection of the 3D embedding. The fixed point subspace $W$ has point stabilizer isomorphic to $D_{3d}\cong D_{3}\times\mathbb{Z}_{2}$. The $D_{3}$ symmetry can be seen with the help of the shading in the 2D picture of $W$, while the $\mathbb{Z}_{2}$ symmetry is the $180^{\circ}$ rotation around the black axis shown in the 3D embedding of $W$.
• Figure 5: The quotient digraph $G_c$ of $W$ from Figure \ref{['fig:w1012']} with its adjacency matrix $M_{c}$ and lattice of coloring vectors of $M_{c}$-invariant polydiagonal subspaces. The four shaded coloring vectors of synchrony subspaces correspond to the shaded orbits of Figure \ref{['fig:BuckyHasse']}.

Theorems & Definitions (24)

• Definition 3.1
• Remark 3.2
• Example 3.3
• Proposition 3.4
• Definition 3.5
• Proposition 3.6
• proof
• Proposition 3.7
• proof
• Example 3.8