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Real and complex equilibrium identities

PDF p. 10: Equations (2.3), (2.4)

Statement

Assume countable choice. Let $G = (V, E)$ be a finite undirected graph with nonempty vertex set, possibly with self-loops, and let $A = (A_{x, y})$ be its symmetric adjacency matrix. Let $θ \in (S^{1})^{V}$ be a phase state, with real representatives used in the formulas. The homogeneous Kuramoto energy is $E_{G} (θ) = \frac{1}{2} \sum_{x, y \in V} A_{x, y} (1 - \cos (θ_{x} - θ_{y}))$ . An equilibrium is a critical point of this energy, equivalently a zero of the homogeneous system's rotating-frame vector field.

The following conditions are equivalent:

$θ$ is an equilibrium.
For every $y \in V$ , $\sum_{x \in V} A_{x, y} \sin (θ_{x} - θ_{y}) = 0$ .
For every $y \in V$ there is a real number $r_{y}$ such that $\sum_{x \in V} A_{x, y} e^{i θ_{x}} = r_{y} e^{i θ_{y}}$ .

In condition 3 one may take $r_{y} = \sum_{x \in V} A_{x, y} \cos (θ_{x} - θ_{y})$ . The coefficients $r_{y}$ are real but are not asserted to be nonnegative.

Statement references

Adjacency, degrees, graph Laplacian, and spectral notation

Finite phase tori, local lifts and the flat metric

Homogeneous Kuramoto energy

Graph Kuramoto differential equations

Equilibria, stable states, synchronized states, and spurious minima

Phasors, trigonometric identities and periodic derivatives

The Axiom of Countable Choice (AC_ω)

Facts & Assumptions

[F1]

The adjacency matrix of the finite undirected graph is real and symmetric (Adjacency, degrees, graph Laplacian, and spectral notation).

[F2]

A phase state has real representatives differing coordinatewise by integer multiples of $2 π$ ; the stated energy is well defined on the phase torus (Finite phase tori, local lifts and the flat metric, Homogeneous Kuramoto energy).

[F3]

Under countable choice, an equilibrium is exactly a critical point of the energy, meaning that its gradient vanishes (The Axiom of Countable Choice (AC_ω), Equilibria, stable states, synchronized states, and spurious minima).

[F4]

The $y$ component of the energy gradient is $\sum_{x \in V} A_{y, x} \sin (θ_{y} - θ_{x})$ (Gradient and Hessian formulas).

[F5]

For real $u, v$ , $e^{i u} \overline{e^{i v}} = e^{i (u - v)} = \cos (u - v) + i \sin (u - v)$ and $∣ e^{i v} ∣ = 1$ . These expressions are unchanged by changing phase representatives by multiples of $2 π$ ; real and imaginary parts distribute over finite sums (Phasors, trigonometric identities and periodic derivatives).

Proof

Given: A graph and phase state as in the Statement, under countable choice.

Goal: Prove the equivalence of the three conditions and the displayed formula for $r_{y}$ .

1.1F1F3F4F5algebra

By [F3] and [F4], equilibrium is equivalent to the vanishing, for every $y$ , of $\sum_{x} A_{y, x} \sin (θ_{y} - θ_{x})$ . By symmetry of $A$ and the phasor identities in [F5], $\sin (- t) = - \sin t$ ; hence this sum is the negative of $\sum_{x} A_{x, y} \sin (θ_{x} - θ_{y})$ . Thus conditions 1 and 2 are equivalent.

2.1F1F2F5step 1.1algebra∎

Fix $y$ and set $S_{y} = \sum_{x} A_{x, y} e^{i θ_{x}}$ . By [F5] and the reality of $A$ , $S_{y} e^{- i θ_{y}} = \sum_{x} A_{x, y} \cos (θ_{x} - θ_{y}) + i \sum_{x} A_{x, y} \sin (θ_{x} - θ_{y})$ . Since $e^{i θ_{y}} e^{- i θ_{y}} = 1$ , the equation $S_{y} = r_{y} e^{i θ_{y}}$ for a real $r_{y}$ holds exactly when the imaginary part of this product vanishes; then $r_{y}$ is its real part. This proves the equivalence of conditions 2 and 3 and the formula for $r_{y}$ . The formulas are independent of representatives by [F2] and [F5]. Together with step 1.1, this completes the proof.

Source notes

Equations (2.3) and (2.4) and their equivalence are attributed to Expander graphs are globally synchronizing, PDF page 10. The argument above derives the identities from the stated mathematical inputs.

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Used by · 3

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Sources

Expander graphs are globally synchronizing · PDF p. 10, equations (2.3) and (2.4)