Families with vanishing total spurious-basin measure

PDF p. 39: Section 7, final paragraph

Assume countable choice. Is there a family of finite undirected graphs $G_n=(V_n,E_n)$, indexed by an unbounded set $S$ of positive integers with $\mathrm{\mid }{V}_n\mathrm{\mid }=n$, such that every $G_n$ fails to be globally synchronizing while the aggregate normalized phase volume of the basins of its spurious local minima tends to zero as $n$ tends to infinity through $S$? Identify and analyze such families, if they exist.

Here the graphs have at most one edge between any two vertices and may have self-loops. On each graph use the homogeneous Kuramoto system, in the frame rotating at the common intrinsic frequency. A graph is globally synchronizing if, for all initial phases outside a normalized phase-volume null set, the rotating-frame trajectory converges to a point whose phases are all equal. A spurious local minimum is a local minimum of the Kuramoto energy whose phases are not all equal.

More explicitly, write $Q_n$ for the set of spurious local minima, $B_n(q)$ for the initial phases whose rotating-frame trajectory converges to the point $q$, and $U_n$ for the union of $B_n(q)$ over $q$ in $Q_n$. Require $U_n$ to be phase-measurable, and let ${\mu }_n$ be normalized phase volume on the phase torus of $G_n$. The requested limit is ${\mu }_n(U_n)\to 0$: for every $\epsilon >0$ there is an $N$ such that ${\mu }_n(U_n)<\epsilon$ for every $n$ in $S$ with $n\ge N$. If $Q_n$ is empty, its union of basins is empty. The aggregate is the volume of a union, not a sum over possibly uncountably many minima.

The Axiom of Countable Choice (AC_ω)

Graph Kuramoto differential equations

Globally synchronizing graph — Definition 1.1

Equilibria, stable states, synchronized states, and spurious minima

Point basins, equilibrium-set basins and aggregate volume

Normalized phase volume and almost-everywhere assertions

Asymptotic quantifiers and uniform parameter ranges

Asymptotically global synchrony via basin mass

The article proposes this possibility as a research direction; its cited paragraph supplies neither a family nor an existence proof. The formulation above makes aggregate size precise using measurable unions of point basins, so it is one explicit version of the question. As of 2026-11-13, no solution is claimed here; the limited source inspection does not establish that the question remains unresolved throughout the literature.

Wiley, Strogatz, and Girvan propose the broader study of the synchronous state's basin size and its dependence on network connectivity. Their opening paragraph does not establish the existence of a family satisfying the question above.

The graph-family possibility is attributed to Expander graphs are globally synchronizing, PDF page 39, Section 7, final paragraph. The broader basin-size direction is attributed to The size of the sync basin, page 1, opening paragraph. Neither citation is used as a proof of existence.