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Second-order necessary conditions for stable states

PDF p. 10: Section 2 stability condition; PDF p. 11: equations (2.5), (2.6)

Statement

Assume countable choice. Let $G = (V, E)$ be a finite undirected graph with nonempty vertex set, with self-loops allowed and symmetric binary adjacency matrix $A$ . A phase state is a point $θ \in (S^{1})^{V}$ , with real representatives used in the formulas. Its homogeneous Kuramoto energy is $E_{G} (θ) = \frac{1}{2} \sum_{x, y \in V} A_{x, y} (1 - \cos (θ_{x} - θ_{y}))$ . A stable state means a local minimum of this energy on the phase torus.

At a stable state $θ$ , its Hessian $H = \nabla^{2} E_{G} (θ)$ in flat angular coordinates satisfies $H ⪰ 0$ : that is, $⟨ f, H f ⟩ \geq 0$ for every $f \in C^{V}$ , where $⟨ f, g ⟩ = \sum_{x} \overline{f (x)} g (x)$ . In particular,

$\sum_{x, y \in V} A_{x, y} \cos (θ_{x} - θ_{y}) (1 - \cos (θ_{x} - θ_{y})) \geq 0.$

For every $y \in V$ , the real coefficient $r_{y} = \sum_{x \in V} A_{x, y} \cos (θ_{x} - θ_{y})$ in the equilibrium identity $\sum_{x} A_{x, y} e^{i θ_{x}} = r_{y} e^{i θ_{y}}$ also satisfies $r_{y} \geq 0$ .

Facts & Assumptions

[F1]

The adjacency matrix is real, symmetric and binary; in particular $A_{y, y} \geq 0$ , including when a self-loop is present (Adjacency, degrees, graph Laplacian, and spectral notation).

[F2]

The phase torus has local real angular charts with Euclidean flat coordinates, and the energy is the stated well-defined finite sum (Finite phase tori, local lifts and the flat metric, Homogeneous Kuramoto energy).

[F3]

Under countable choice, a stable state is a local energy minimum and is an equilibrium (The Axiom of Countable Choice (AC_ω), Equilibria, stable states, synchronized states, and spurious minima).

[F4]

A $C^{2}$ real function at a local minimum has a real symmetric Hessian with nonnegative complex quadratic form (Local minima and nonnegative second variation). The notation $H ⪰ 0$ expresses this nonnegativity (Loewner order, Hermitian inner product, and operator norm).

[F5]

For every complex vertex function $f$ , the energy Hessian obeys $⟨ f, H f ⟩ = \frac{1}{2} \sum_{x, y} A_{x, y} \cos (θ_{x} - θ_{y}) ∣ f (x) - f (y) ∣^{2}$ ; its diagonal entry at $y$ is $H_{y, y} = \sum_{x \neq y} A_{x, y} \cos (θ_{x} - θ_{y})$ (Gradient and Hessian formulas).

[F6]

For real angles, $e^{i u} = \cos u + i \sin u$ , $∣ e^{i u} ∣ = 1$ , and $e^{i (u - v)} = e^{i u} \overline{e^{i v}}$ ; the expressions are independent of phase representatives (Phasors, trigonometric identities and periodic derivatives).

[F7]

At an equilibrium the neighboring phasor sum has the stated form, with $r_{y} = \sum_{x} A_{x, y} \cos (θ_{x} - θ_{y})$ (Real and complex equilibrium identities).

Proof

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

Goal: Prove Hessian positive semidefiniteness and both displayed necessary inequalities.

1.1F2F3F4F6

By [F2], choose a local angular chart around $θ$ . In that chart the energy is a finite sum of cosine functions and is $C^{2}$ by [F6]. The local minimum in [F3] is a local minimum of this coordinate function. Applying [F4] gives $⟨ f, H f ⟩ \geq 0$ for every complex $f$ , hence $H ⪰ 0$ .

2.1F5F6step 1.1algebra

Take $f (x) = e^{i θ_{x}}$ . By [F6], $∣ f (x) - f (y) ∣^{2} = 2 - 2 \cos (θ_{x} - θ_{y})$ . Substitution into [F5] and the nonnegativity from step 1.1 give the first inequality. Periodicity in [F6] makes the calculation independent of the chosen representatives.

3.1F1F3F5F6F7step 1.1algebra∎

Fix $y$ and take $f$ to be the coordinate indicator: $f (y) = 1$ and $f (x) = 0$ for $x \neq y$ . Step 1.1 gives $H_{y, y} = ⟨ f, H f ⟩ \geq 0$ . By [F5] and $\cos 0 = 1$ from [F6], $r_{y} = H_{y, y} + A_{y, y} \geq 0$ , using [F1]. By [F3] and [F7], this is also the coefficient in the stated equilibrium identity. The diagonal term is included even when $y$ has a self-loop; it cancels from the Hessian but is nonnegative in $r_{y}$ . This proves the remaining assertion.

Source notes

The stable-state Hessian condition and the two tests are attributed to Expander graphs are globally synchronizing, Section 2, equations (2.5)–(2.6), PDF pp. 10–11. The argument above derives them from the stated mathematical inputs.

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Sources

Expander graphs are globally synchronizing · Section 2, stability condition and equations (2.5)–(2.6), PDF pp. 10–11