Phase oscillator interpretation and homogeneous modeling assumption

PDF p. 1: Abstract; PDF p. 2: Section 1, equation (1.1) and model interpretation; PDF p. 4: Section 1, random graphs as proxies for realistic complex networks

Adopt the graph Kuramoto equations as an idealized model of a finite nonempty population of coupled oscillators. An oscillator is a system exhibiting repeated motion; its phase ${\theta }_x$ records angular position modulo $2\pi$ in its cycle. The finite undirected graph $G=(V,E)$, with nonempty vertex set $V$, has one vertex $x$ for each oscillator, and an edge $\{x,y\}$ represents a pairwise coupling. Its symmetric binary adjacency matrix has $A_{x,y}=1$ when that edge is present and $0$ otherwise. On local real representatives of the phases, the adopted dynamics are ${\dot{\theta }}_x={\omega }_x-{\sum }_{y\in V}{A}_{x,y}\sin ({\theta }_x-{\theta }_y)$. The intrinsic frequency ${\omega }_x$ is the angular rate assigned to oscillator $x$ when coupling is absent. Self-loop terms, if present, contribute zero.

In the homogeneous specialization all intrinsic frequencies are identical: ${\omega }_x=\omega$ for every $x\in V$. For the early complete-graph model, in which every distinct pair is coupled with equal strength, the intrinsic frequencies are instead modeled as random values with a prescribed probability density $g(\omega )$ that is unimodal and symmetric about its mean. For this random-frequency formulation specify a probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ and real random variables ${\omega }_x:\mathrm{\Omega }\to \mathbb{R}$, measurable for the Borel sigma-algebra of $\mathbb{R}$. Their common law has density $g$ with respect to Lebesgue measure $\lambda$ on $\mathbb{R}$: $g$ is nonnegative and measurable, ${\int }_{\mathbb{R}}gd\lambda =1$, and $\mathbb{P}({\omega }_x\in B)={\int }_{B}gd\lambda$ for every Borel set $B$. Require $\mathbb{E}[\mathrm{\mid }{\omega }_x\mathrm{\mid }]<\mathrm{\infty }$ for every $x$ and a common finite mean $a=\mathbb{E}[{\omega }_x]$, where expectation is integration against $\mathbb{P}$. The finite first absolute moment is an assumption of this finite-mean formulation, not a consequence of symmetry or unimodality. Symmetry about $a$ means $g(a+u)=g(a-u)$, and unimodality about $a$ means that $g$ is nondecreasing to the left of $a$ and nonincreasing to its right.

Random graph models may be used as proxies for realistic complex networks: the graph is sampled from a specified probability distribution and represents an idealized pattern of connections. For fixed $V$, let ${\mathcal{G}}_V$ be the finite set of undirected graphs on $V$ with the preceding binary adjacency convention, allowing self-loops. Specify weights $q_H\ge 0$ for $H\in {\mathcal{G}}_V$ with ${\sum }_{H\in {\mathcal{G}}_V}{q}_H=1$. The graph sample space is ${\mathcal{G}}_V$, every subset is an event, and an event $\mathcal{A}\subseteq {\mathcal{G}}_V$ has probability ${\sum }_{H\in \mathcal{A}}{q}_H$. A sampled graph is an outcome $H$ of this finite probability space; the adopted dynamics then use its adjacency matrix. The phase description, the sinusoidal interaction, the frequency assumptions, and the use of a random graph as a proxy are modeling choices. They do not assert that a particular physical system obeys the equations exactly or that the proxy has been empirically calibrated. The mathematical interfaces used here assume countable choice for Lebesgue measure and the global trajectory construction.

Graph Kuramoto differential equations

Graph notation

Finite phase tori, local lifts and the flat metric

Scope of the phase-only sinusoidal oscillator idealization

Empirical phase measures and Fourier order statistics

The Axiom of Countable Choice

Probability measures and probability spaces

Random elements and real random variables

Law or distribution of a random element

Expectation of a nonnegative or integrable random variable

Finite probability spaces, events and outcome weights

Lebesgue measure on the real line

Lebesgue measure under countable choice

Expander graphs are globally synchronizing, PDF pages 1, 2 and 4, supplies attribution for the interpretation and modeling choices. This is an adopted model, not a reported experiment or a proof of physical applicability.