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Complete-graph positive-semidefinite equilibria

PDF p. 7: Remark 1.7 continuation; PDF p. 8: Remark 1.7 continuation

Statement

Let $K_{n}$ be the unweighted graph on $n \geq 1$ vertices with one edge between each pair of distinct vertices, and let $θ \in (R / 2 π Z)^{n}$ . Its energy is $E_{K_{n}} (θ) = \sum_{1 \leq x < y \leq n} (1 - \cos (θ_{x} - θ_{y}))$ . An equilibrium is a state with $\nabla E_{K_{n}} (θ) = 0$ in angular coordinates. Its Hessian is positive semidefinite, written $\nabla^{2} E_{K_{n}} (θ) ⪰ 0$ , when its real quadratic form is nonnegative on every real vector. Fully synchronized means that all phases agree modulo $2 π$ .

Every equilibrium of $K_{n}$ with positive-semidefinite energy Hessian is fully synchronized. The equilibrium hypothesis is necessary: positive semidefiniteness alone does not imply synchronization.

Statement references

Homogeneous Kuramoto energy

Loewner order, Hermitian inner product, and operator norm

Real symmetric quadratic forms tested on phasors

A complete-graph second-order landscape application

Facts & Assumptions

[F1]

For the complete graph, the ordered-pair energy in the energy definition equals the sum over unordered distinct pairs displayed above, since the adjacency matrix is symmetric and the diagonal contributions vanish (Homogeneous Kuramoto energy).

[F2]

For this unordered-pair energy, every equilibrium whose Hessian is positive semidefinite is fully synchronized (A complete-graph second-order landscape application).

[F3]

The Hessian quadratic form for a cosine-difference pair potential is the sum over unordered edges of $\cos (θ_{x} - θ_{y}) (v_{x} - v_{y})^{2}$ ; the derivatives of sine and cosine have their usual values (Differentiation of symmetric periodic pair potentials, Phasors, trigonometric identities and periodic derivatives).

[F4]

Positive semidefiniteness means nonnegativity of the quadratic form on real vectors; a real symmetric Hessian satisfying this condition also satisfies the stated Hermitian convention (Loewner order, Hermitian inner product, and operator norm, Real symmetric quadratic forms tested on phasors).

Proof

Given: A phase state on $K_{n}$ with vanishing energy gradient and positive-semidefinite Hessian.

Goal: Establish synchronization and show that the stationarity hypothesis cannot be omitted.

1.1givenF1F2

By [F1], the energy and angular derivatives are those in [F2]. The given gradient and Hessian hypotheses therefore meet that result's hypotheses, so all phases agree modulo $2 π$ .

2.1F3F4choosealgebra∎

For the necessity of stationarity, take $K_{2}$ and phases $θ_{1} = 0$ , $θ_{2} = π / 3$ . By [F3], the Hessian quadratic form is $\frac{1}{2} (v_{1} - v_{2})^{2} \geq 0$ for every real vector $v$ , hence the Hessian is positive semidefinite by [F4]. The phases are distinct modulo $2 π$ ; the gradient is nonzero because its first component is $\sin (- π / 3) \neq 0$ . Thus this state is not synchronized and is not an equilibrium.

Source notes

Remark 1.7 of Expander graphs are globally synchronizing, PDF pp. 7–8, states the configuration-level claim without the equilibrium hypothesis. The counterexample above addresses that wording; the equilibrium result follows from the local supplier cited in [F2].

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Sources

Expander graphs are globally synchronizing · Remark 1.7, PDF pp. 7–8