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Pointwise comparisons used in kernel stability

PDF p. 17: Proof of Lemma 2.8, Cases 1–3

Statement

Assume countable choice. Let $G = (V, E)$ be a finite undirected graph with nonempty vertex set, self-loops allowed, and symmetric binary adjacency matrix $A$ . Let $θ$ be a stable state, meaning a local minimum on the phase torus of the energy $E_{G} (θ) = \frac{1}{2} \sum_{x, y \in V} A_{x, y} (1 - \cos (θ_{x} - θ_{y}))$ , and choose representatives $θ_{x} \in (- π, π]$ . Define $K (α, β) = \sin (∣ α ∣ - \min {∣ β ∣, π / 2})$ . For each $y$ , let $r_{y} = \sum_{x} A_{x, y} \cos (θ_{x} - θ_{y})$ .

For every $x, y \in V$ : if $0 \leq θ_{y} \leq π / 2$ , then $K (θ_{x}, θ_{y}) \geq \sin (θ_{x} - θ_{y})$ ; if $- π / 2 \leq θ_{y} \leq 0$ , then $K (θ_{x}, θ_{y}) \geq - \sin (θ_{x} - θ_{y})$ . If $∣ θ_{y} ∣ \geq π / 2$ , then $K (θ_{x}, θ_{y}) = - \cos θ_{x}$ and $\sum_{x} A_{x, y} K (θ_{x}, θ_{y}) = - r_{y} Re (e^{- i θ_{y}}) \geq 0$ .

Facts & Assumptions

[F1]

The kernel has branches $K (α, β) = \sin (∣ α ∣ - ∣ β ∣)$ for $∣ β ∣ \leq π / 2$ and $K (α, β) = - \cos α$ for $∣ β ∣ \geq π / 2$ (Kernel for stability amplification).

[F2]

Sine is nonnegative on $[0, π]$ . Sine is odd, cosine is even, and $\cos t = \sin (π / 2 - t)$ (Angular order and elementary inverse-trigonometric bounds, Parity and the Pythagorean identity for sine and cosine, Quarter-turn values and shifts by pi/2 and pi).

[F3]

For real $u, v$ , $e^{i (u - v)} = e^{i u} e^{- i v} = \cos (u - v) + i \sin (u - v)$ , and real parts distribute over finite sums (Phasors, trigonometric identities and periodic derivatives).

[F4]

Under countable choice, for the graph and stable state in the Statement, $r_{y} \geq 0$ and $\sum_{x} A_{x, y} e^{i θ_{x}} = r_{y} e^{i θ_{y}}$ (The Axiom of Countable Choice (AC_ω), Second-order necessary conditions for stable states). The graph, phase, energy and stable-state conventions are those of Adjacency, degrees, graph Laplacian, and spectral notation, Finite phase tori, local lifts and the flat metric, Homogeneous Kuramoto energy and Equilibria, stable states, synchronized states, and spurious minima.

[F5]

For all real $u, v$ , $\sin (u + v) = \sin u \cos v + \cos u \sin v$ (The addition formulas for sine and cosine).

Proof

Given: The graph, stable state, representatives and kernel in the Statement, under countable choice.

Goal: Prove the two pointwise inequalities and the branch identity and nonnegative sum in the third case.

1.1F1F2F5algebra

Write $a = θ_{x}$ and $b = θ_{y}$ . If $0 \leq b \leq π / 2$ and $a \geq 0$ , [F1] gives $K (a, b) = \sin (a - b)$ . If instead $a < 0$ , put $t = ∣ a ∣ \in [0, π]$ . By [F1], parity and the sine addition formula in [F5], $K (a, b) - \sin (a - b) = \sin (t - b) + \sin (t + b) = 2 \sin t \cos b \geq 0$ . Here $\sin t \geq 0$ by [F2], and $\cos b = \sin (π / 2 - b) \geq 0$ because $π / 2 - b \in [0, π]$ . This proves the first comparison.

2.1F1F2F5step 1.1algebra

If $- π / 2 \leq b \leq 0$ and $a \leq 0$ , [F1] and oddness give $K (a, b) = - \sin (a - b)$ . If $a > 0$ , put $s = ∣ b ∣ \in [0, π / 2]$ . By [F5], $K (a, b) + \sin (a - b) = \sin (a - s) + \sin (a + s) = 2 \sin a \cos s \geq 0$ , by the same sign argument as in step 1.1. This proves the second comparison, including the endpoints.

3.1F1F2F3F4algebra∎

If $∣ b ∣ \geq π / 2$ , [F1] gives $K (a, b) = - \cos a$ . Taking the real part of the phasor identity in [F4] and using [F3] yields $\sum_{x} A_{x, y} \cos θ_{x} = r_{y} \cos θ_{y}$ . Thus the kernel sum is $- r_{y} \cos θ_{y} = - r_{y} Re (e^{- i θ_{y}})$ . Since $∣ θ_{y} ∣ \in [π / 2, π]$ , cosine is nonpositive there: by [F2], $\cos ∣ θ_{y} ∣ = - \sin (∣ θ_{y} ∣ - π / 2) \leq 0$ , and cosine is even. Together with $r_{y} \geq 0$ , this proves the nonnegativity and completes the proof.

Source notes

The three comparisons are attributed to Expander graphs are globally synchronizing, PDF page 17, proof of Lemma 2.8. The proof above uses the stated mathematical suppliers.

Depends on

Used by · 1

Kernel stability condition — Lemma 2.8

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

Sources

Expander graphs are globally synchronizing · PDF p. 17, proof of Lemma 2.8, Cases 1–3