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Self-loops do not affect Kuramoto dynamics

PDF p. 18: Footnote 7

Statement

Let $G = (V, E)$ be a finite undirected graph with nonempty vertex set, symmetric binary adjacency matrix $A$ , and possible self-loops. Let $G^{\circ}$ be the graph obtained by deleting every self-loop and retaining all other edges. A phase state is $θ \in (S^{1})^{V}$ , with each $S^{1}$ represented by angles modulo $2 π$ . For the homogeneous Kuramoto energy

$E_{G} (θ) = \frac{1}{2} \sum_{x, y \in V} A_{x, y} (1 - \cos (θ_{x} - θ_{y})),$

the gradient is taken in the flat angular coordinates. The homogeneous Kuramoto dynamics in the rotating frame is the differential equation $d θ / d t = - \nabla E_{G} (θ)$ . This equation is unchanged when $G$ is replaced by $G^{\circ}$ : self-loops contribute zero to the gradient, so the two equations have the same solutions on every common time interval. In the degree convention $d (x) = \sum_{y \in V} A_{x, y}$ , a self-loop at $x$ is counted once.

Statement references

Adjacency, degrees, graph Laplacian, and spectral notation

Homogeneous Kuramoto energy

Finite phase tori, local lifts and the flat metric

Gradient and Hessian formulas

Facts & Assumptions

[F1]

For the stated graph convention, $A_{x, x} = 1$ exactly when $x$ has a self-loop, and $d (x) = \sum_{y} A_{x, y}$ (Adjacency, degrees, graph Laplacian, and spectral notation).

[F2]

The energy in the Statement is well defined on phase states; angular lifts provide flat local coordinates (Homogeneous Kuramoto energy, Finite phase tori, local lifts and the flat metric).

[F3]

For any such graph and phase state, $(\nabla E_{G})_{x} = \sum_{y \in V} A_{x, y} \sin (θ_{x} - θ_{y})$ in these coordinates (Gradient and Hessian formulas).

[F4]

The sine function satisfies $\sin 0 = 0$ (Phasors, trigonometric identities and periodic derivatives).

Proof

Given: A graph $G$ , its loop-deleted graph $G^{\circ}$ , and a phase state $θ$ as in the Statement.

Goal: Show equality of the two gradient vector fields and the asserted degree count.

1.1givenF2F3F4algebra

Let $A^{\circ}$ be the adjacency matrix of $G^{\circ}$ . It agrees with $A$ off the diagonal and has zero diagonal. Apply [F3] to both graphs in the same local angular coordinates. Their $x$ -components differ only by $A_{x, x} \sin (θ_{x} - θ_{x}) = A_{x, x} \sin 0 = 0$ , using [F4]. Thus their gradients agree at every phase state; [F2] identifies these coordinate expressions with the vector fields on the phase space.

2.1F1step 1.1algebra∎

The two differential equations therefore have identical right-hand sides at every state, so a differentiable phase path satisfies one exactly when it satisfies the other on the same interval. By [F1], a loop contributes the single summand $A_{x, x} = 1$ to $d (x)$ , establishing the degree convention.

Source notes

The claim and degree convention are attributed to Expander graphs are globally synchronizing, PDF page 18, opening paragraph of Section 3 and footnote 7. The argument above derives the claim from the listed mathematical suppliers.

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Sources

Expander graphs are globally synchronizing · PDF p. 18, opening paragraph of Section 3 and footnote 7; pp. 2–3, equations (1.1)–(1.2)