Homogeneous Kuramoto energy

PDF p. 2: Section 1, homogeneous model; PDF p. 3: Equation (1.2); PDF p. 10: Section 2, energy formula

Let $G=(V,E)$ be a finite undirected graph with nonempty vertex set, allowing self-loops, and let $A=(A_{x,y}{)}_{x,y\in V}$ be its symmetric adjacency matrix: $A_{x,y}=1$ when $\{x,y\}\in E$ and $A_{x,y}=0$ otherwise. A phase state is an element $\theta$ of $(S^1{)}^V$, identified with ${\mathbb{R}}^V\mathrm{/}(2\pi \mathbb{Z}{)}^V$; a lift of $\theta$ is a real vector representing its equivalence class.

The homogeneous Kuramoto energy is the function ${\mathcal{E}}_G:(S^1{)}^V\to \mathbb{R}$ defined by

${\mathcal{E}}_G(\theta )=\frac{1}{2}{\sum }_{x,y\in V}{A}_{x,y}(1-\cos ({\theta }_x-{\theta }_y))\mathrm{.}$

The sum is over ordered pairs of vertices, and the phase differences in the cosine are evaluated using any real lift of $\theta$.

Adjacency, degrees, graph Laplacian, and spectral notation

Finite phase tori, local lifts and the flat metric

Phasors, trigonometric identities and periodic derivatives

Given: A graph and phase state as in the definition, together with two real lifts of that state.

Goal: Show that the displayed formula assigns one real number to the phase state independently of the lift.

1.1 By [F1], the two lifts differ coordinatewise by integer multiples of $2\pi$. For any ordered pair $(x,y)$, the corresponding differences of their $x$ and $y$ coordinates therefore differ by an integer multiple of $2\pi$. By [F2], the two cosine values are equal. [given, F1, F2]

2.1 By [F3], the ordered-pair sum has finitely many terms and each coefficient is real; by [F2], each cosine value is real. Thus the formula is a real number. Step 1.1 shows that every summand is unchanged by replacing the lift, so the resulting real number is independent of the lift. This defines the stated function on $(S^1{)}^V$. [F2, F3, step 1.1] ∎

The formula and its use for the homogeneous model are attributed to Expander graphs are globally synchronizing, PDF page 3, equation (1.2), and page 10. The justification uses the mathematical suppliers above.

The formula and its use for the homogeneous model are attributed to Expander graphs are globally synchronizing, PDF page 3, equation (1.2), and page 10. The justification uses the mathematical suppliers above.