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A simple nonsynchronizing family approaching eleven sixteenths

Statement

Assume countable choice. For every integer $k \geq 3$ , take the explicit eight-vertex weighted graph of the eight-phase certificate with $a = b = c = k$ and $d = ⌈ k / 2 ⌉ - 1$ , and form its spinning graph $F_{k}$ . It is a simple unweighted graph with $8 k$ vertices and common degree $5 k + ⌈ k / 2 ⌉ - 2$ . It has a nonsynchronous equilibrium whose Hessian is positive transverse to simultaneous rotation, and is not globally synchronizing, where global synchronization means convergence to all-equal phases for almost every initial phase. Its loop-free minimum-degree density is

$μ_{k} = \frac{5 k + ⌈ k / 2 ⌉ - 2}{8 k - 1} < \frac{11}{16}, \lim_{k \to \infty} μ_{k} = \frac{11}{16} .$

Facts & Assumptions

[F1]

Use the network and spinning definitions of Finite symmetric networks and loop removal and Simple spinning graphs and cluster-constant equilibria.

[F2]

The explicit weighted equilibrium is given by An explicit eight-phase weighted stability certificate, and its lifted strict transverse stability by Transverse positivity for the special spinning family.

[F3]

Under The Axiom of Countable Choice (AC_ω), positive transverse stability of a nonsynchronous equilibrium obstructs almost-everywhere synchronous convergence, by A positive transverse Hessian gives an open nonsynchronous basin, with the phase measure of Normalized phase volume and almost-everywhere assertions.

Proof

Given: An integer $k \geq 3$ and the construction in the Statement.

Goal: Establish size, exact density and the nonsynchronization certificate.

1.1F1givenalgebra

Each base vertex has weighted degree $a + 2 b + c + d = 4 k + d$ . The spinning construction has eight clusters of $k$ vertices, and [F1] gives degree $k - 1 + 4 k + d = 5 k + ⌈ k / 2 ⌉ - 2$ . Its simplicity and unit edge weights are part of the justified construction. Therefore division by the number $8 k - 1$ of other vertices gives the displayed loop-free density.

2.1F1F2F3step 1.1

By [F2] the cluster-constant equilibrium is nonsynchronous and its Hessian is positive on the complement of global constants. The potential is a finite sum of smooth functions of phase differences and is therefore smooth and rotation invariant. Apply [F3] to obtain an open set of positive phase volume converging to nonsynchronous rotations. Thus $F_{k}$ is not globally synchronizing.

3.1step 1.1algebra∎

For even $k$ , $11 (8 k - 1) - 16 (5 k + k / 2 - 2) = 21$ ; for odd $k$ it equals $13$ . Both are positive, proving the strict upper inequality. Since $⌈ k / 2 ⌉ / k \to 1 / 2$ , division of numerator and denominator by $k$ proves the limit $11 / 16$ . Thus the approach is from below under the loop-free density convention as well as under the source's degree-over-order convention.

Source notes

The parameters and special stability argument follow Canale Theorem 2, p. 15. The explicit base edge convention and local Fourier certificate are stated in the earlier helpers. We use degree divided by $8 k - 1$ , required by the loop-free strong-density definition on Canale p. 4; the displayed calculations on pp. 14–15 instead divide by $8 k$ . Both tend to $11 / 16$ , so this correction preserves the reported limiting bound.

Depends on

Used by · 3

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External dependencies

Eduardo A. Canale, From weighted to unweighted graphs in Synchronizing Graph Theory, arXiv:2209.06362v3 · Section II.A, printed p. 4 (strong density); Section V, printed pp. 14–15 (degree, order, parameters and Theorem 2)

Strong density is minimum degree divided by order minus one. Section V gives order 8k and degree a+2b+c+d+k-1, takes (a,b,c,d)=(k,k,k,ceil(k/2)-1), and proves a nonsynchronizing family with limiting density 11/16; its displayed density formulas divide by 8k.

Hypotheses: Finite simple graphs with positive integer edge weights and an integer spinning parameter at least the maximum edge weight; homogeneous Kuramoto dynamics. In the local application k is at least 3, so the chosen weights are positive and the special stability certificate applies.

Reading limits: Read the PDF text extraction of Section II.A, printed p. 4, Proposition 1 on printed p. 7, and the degree and density calculation and Theorem 2 on printed pp. 14–15. This citation documents attribution and the denominator comparison only; the mathematical argument uses the declared local suppliers. No general stability-transfer theorem is imported.

Sources

Eduardo A. Canale, From weighted to unweighted graphs in Synchronizing Graph Theory, arXiv:2209.06362v3 · Section II.A, printed p. 4 (strong density); Section V, printed pp. 14–15 (degree, order, parameters and Theorem 2)