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Gradient and Hessian formulas

PDF p. 10: Equations (2.1), (2.2); PDF p. 11: Hessian quadratic form

Statement

Let $G = (V, E)$ be a finite undirected graph with nonempty vertex set, possibly with self-loops, and symmetric adjacency matrix $A$ . Let $θ \in (S^{1})^{V}$ and let $E_{G} (θ) = \frac{1}{2} \sum_{x, y \in V} A_{x, y} (1 - \cos (θ_{x} - θ_{y}))$ be the homogeneous Kuramoto energy. Use local real angular lifts to compute its gradient and Hessian in the flat coordinates. Then, for $x \in V$ ,

$(\nabla E_{G} (θ))_{x} = \sum_{z \in V} A_{x, z} \sin (θ_{x} - θ_{z}) .$

For $x \neq y$ , $(\nabla^{2} E_{G} (θ))_{x, y} = - A_{x, y} \cos (θ_{x} - θ_{y})$ , while $(\nabla^{2} E_{G} (θ))_{x, x} = \sum_{z \neq x} A_{x, z} \cos (θ_{x} - θ_{z})$ . For every $f : V \to C$ , with $⟨ f, g ⟩ = \sum_{x} \overline{f (x)} g (x)$ ,

$⟨ f, \nabla^{2} E_{G} (θ) f ⟩ = \frac{1}{2} \sum_{x, y \in V} A_{x, y} \cos (θ_{x} - θ_{y}) ∣ f (x) - f (y) ∣^{2} .$

Facts & Assumptions

[F1]

The graph has finitely many vertices and real symmetric adjacency matrix $A$ , with diagonal entries allowed for self-loops (Adjacency, degrees, graph Laplacian, and spectral notation).

[F2]

The energy is the displayed ordered-pair sum and is independent of the chosen real lift of the phase state (Homogeneous Kuramoto energy). Angular charts have locally constant translation transitions and the flat metric has Euclidean coefficients in those charts (Finite phase tori, local lifts and the flat metric).

[F3]

Sine and cosine are $2 π$ -periodic, $\cos^{'} (t) = - \sin t$ and $\sin^{'} (t) = \cos t$ ; the phasor identities imply that cosine is even and $\sin 0 = 0$ (Phasors, trigonometric identities and periodic derivatives).

[F4]

For a finite symmetric real array $w$ and an even, twice continuously differentiable $2 π$ -periodic function $v$ , the gradient of $\frac{1}{2} \sum_{x, y} w_{x y} v (θ_{x} - θ_{y})$ has components $\sum_{y} w_{x y} v^{'} (θ_{x} - θ_{y})$ (Differentiation of symmetric periodic pair potentials).

[F5]

The chain rule differentiates a composition by composing its derivatives (The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)). Finite sums and scalar multiples of differentiable functions may be differentiated termwise (Phasors, trigonometric identities and periodic derivatives).

[F6]

If $b$ is a real symmetric array and $L = diag (\sum_{y} b_{x y}) - b$ , then $f^{*} L f = \frac{1}{2} \sum_{x, y} b_{x y} ∣ f (x) - f (y) ∣^{2}$ for every complex vector $f$ (Real symmetric quadratic forms tested on phasors). Matrix action and the Hermitian inner product use the finite-sum convention in the Statement (Loewner order, Hermitian inner product, and operator norm).

Proof

Given: A graph, phase state and complex vertex function as in the Statement.

Goal: Establish the gradient, Hessian entries and complex quadratic-form identity.

1.1F1F2F3F4

Choose a local real angular lift. Set $v (t) = 1 - \cos t$ and $w_{x y} = A_{x, y}$ . By [F1]–[F3], $w$ is symmetric and $v$ is even and $2 π$ -periodic; the derivative formulas give $v^{'} (t) = \sin t$ and $v^{''} (t) = \cos t$ , with continuous derivatives. Thus [F4] applies and yields the asserted gradient. The self-loop term vanishes because $\sin 0 = 0$ .

2.1F2F3F5step 1.1algebra

Differentiate the gradient component with respect to the lift coordinate $θ_{y}$ . By [F5], the chain rule gives $\partial_{θ_{y}} \sin (θ_{x} - θ_{z}) = \cos (θ_{x} - θ_{z}) (δ_{x y} - δ_{z y})$ , where $δ_{a b}$ is $1$ if $a = b$ and $0$ otherwise. For $y \neq x$ , only $z = y$ contributes, giving the stated off-diagonal entry. For $y = x$ , the factor is $1 - δ_{z x}$ , giving the sum over $z \neq x$ . Translation transitions and the Euclidean metric in [F2] identify these coordinate derivatives with the stated gradient and Hessian on the phase torus, independently of the lift.

3.1F1F3F6step 2.1algebra∎

Put $b_{x y} = A_{x, y} \cos (θ_{x} - θ_{y})$ . By symmetry of $A$ and evenness of cosine, $b$ is real symmetric. The entries in step 2.1 are exactly those of $L = diag (\sum_{z} b_{x z}) - b$ : its diagonal is $\sum_{z \neq x} b_{x z}$ , so any loop coefficient cancels. Applying [F6] to this $L$ proves the displayed identity for every complex $f$ . This completes the proof.

Source notes

The formulas are attributed to Expander graphs are globally synchronizing, PDF pages 10–11. The proof above derives them from the stated mathematical inputs.

Depends on

Used by · 5

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Sources

Expander graphs are globally synchronizing · PDF p. 10, equations (2.1)–(2.2); p. 11, opening display