Globally synchronizing graph — Definition 1.1

PDF p. 3: Definition 1.1

Assume countable choice. Let $G=(V,E)$ be a finite undirected graph with nonempty vertex set, at most one edge between any two vertices, and possible self-loops. Its adjacency matrix has $A_{x,y}=1$ if $\{x,y\}\in E$ and $A_{x,y}=0$ otherwise. The phase space is $T_V={\mathbb{R}}^V\mathrm{/}(2\pi \mathbb{Z}{)}^V$, identified with $(S^1{)}^V$, with its flat distance $d$. A phase state $q\in T_V$ is fully synchronized if $q_x=q_y$ in $S^1$ for every $x,y\in V$.

For each initial state $u\in T_V$, let ${\theta }^u$ be the unique trajectory through $u$ of the homogeneous Kuramoto system with common intrinsic frequency $\omega \in \mathbb{R}$: in local real angular lifts, ${\dot{\theta }}_x=\omega -{\sum }_{y\in V}{A}_{x,y}\sin ({\theta }_x-{\theta }_y)$. Its rotating-frame trajectory is ${\varphi }_x^u(t)={\theta }_x^u(t)-\omega t$ modulo $2\pi$. Both trajectories are defined for every real $t$.

Let $\mu$ be normalized phase volume: for $n=\mathrm{\mid }V\mathrm{\mid }$ and the quotient map $q:{\mathbb{R}}^V\to T_V$, it assigns a phase-measurable set $B$ the value $(2\pi {)}^{-n}{\lambda }_n(q^{-1}(B)\cap (-\pi ,\pi {]}^V)$. A uniformly chosen initial state has law $\mu$. The graph $G$ is globally synchronizing if, outside a phase-measurable set $N$ with $\mu (N)=0$, every initial state $u$ has a fully synchronized state $s=s(u)$ such that $d({\varphi }^u(t),s)\to 0$ as $t\to \mathrm{\infty }$. Equivalently, the rotating-frame trajectory from a uniformly chosen initial state converges to a fully synchronized state with probability one. The limiting state may depend on $u$.

The Axiom of Countable Choice (AC_ω)

Adjacency, degrees, graph Laplacian, and spectral notation

Finite phase tori, local lifts and the flat metric

Graph Kuramoto differential equations

Diagonal phase rotations and relative-phase coordinates

Equilibria, stable states, synchronized states, and spurious minima

Normalized phase volume and almost-everywhere assertions

The graph convention gives the stated symmetric adjacency matrix. The quotient construction makes phase equality, the flat distance, and the coordinatewise subtraction used in the rotating frame well defined. Under countable choice, the Kuramoto system supplies a unique trajectory for every initial state, defined for all real time; subtracting the common rotation gives the stated rotating-frame trajectory. Fully synchronized means equality of all phase coordinates, as specified above (Adjacency, degrees, graph Laplacian, and spectral notation, Finite phase tori, local lifts and the flat metric, The Axiom of Countable Choice (AC_ω), Graph Kuramoto differential equations, Diagonal phase rotations and relative-phase coordinates, Equilibria, stable states, synchronized states, and spurious minima).

Normalized phase volume is a complete probability measure, and the uniform initial-state law is that measure. The displayed null-exception formulation therefore says exactly that the event of convergence to some fully synchronized point has probability one: its complement is a subset of the null set $N$ and is measurable by completeness. Conversely, if that event has probability one, its measurable complement can be taken as $N$. This definition asserts point convergence, not merely that the distance to the set of synchronized states tends to zero (Normalized phase volume and almost-everywhere assertions).

The definition and its rotating-frame context are attributed to Expander graphs are globally synchronizing, PDF pages 2–3, Definition 1.1. The justification uses the mathematical suppliers above; the article is cited for attribution.