Research collection / 01

Expander graphs are globally synchronizing

Pedro Abdalla, Afonso S. Bandeira, Martin Kassabov, Victor Souza, Steven H. Strogatz, Alex Townsend

This collection studies Expander graphs are globally synchronizing.

Present how expansion controls the Kuramoto landscape and yields deterministic and random-graph synchrony, retaining every original claim and its immediate suppliers for the engine's article pagination.

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Research & notes

22 entries

Prerequisites

Phase dynamics and oscillator models
Build phase geometry, analytic gradient dynamics and basin measure; place oscillator assumptions, historical observations and continuum context in their separate stated scopes.
Random graphs and matrix concentration
Develop degree tails, trace and Gaussian estimates, and concentration; connect them to regular spectra, connectivity and synchronization thresholds with each input's model and asymptotic range.
Spectral estimates and phase landscapes
Supply graph and matrix conventions, centered cut estimates, angular and numerical controls, and graph-specific and vector-spin landscape applications.
Supporting prerequisites · 01 — Spectral edge and vertex expansion
Presents 18 supporting prerequisites, covering published library prerequisites and dependency levels 0–3.
Supporting prerequisites · 02 — A centered adjacency column retains a high-degree lower bound
Presents 19 supporting prerequisites, covering dependency levels 3–10.

Article items and immediate prerequisites

View the article page dependency graph

Level 0

4 pages

Article items · 01 — Additivity of the nonnegative Lebesgue integral28 items
Presents 28 items from the article and its immediate prerequisites, covering published library prerequisites.
Article items · 02 — Expectation of a nonnegative or integrable random variable28 items
Presents 28 items from the article and its immediate prerequisites, covering published library prerequisites.
Article items · 03 — Partial order and partially ordered set28 items
Presents 28 items from the article and its immediate prerequisites, covering published library prerequisites.
Article items · 04 — Every nonempty finite set of reals has a maximum and a minimum28 items
Presents 28 items from the article and its immediate prerequisites, covering published library prerequisites.

Level 1

1 page

Article items · 05 — The principle of mathematical induction29 items
Presents 29 items from the article and its immediate prerequisites, covering published library prerequisites and dependency level 0.

Level 2

1 page

Article items · 06 — Finite phase tori, local lifts and the flat metric28 items
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 0–1.

Level 3

1 page

Article items · 07 — Diagonal phase rotations and relative-phase coordinates28 items
Presents 28 items from the article and its immediate prerequisites, covering dependency level 1.

Level 4

1 page

Article items · 08 — A scoped interacting-particle transformer connection28 items
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 1–2.

Level 5

1 page

Article items · 09 — Bipartite Ramanujan existence and the centered spectrum28 items
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 2–3.

Level 6

1 page

Article items · 10 — Łojasiewicz inequality and analytic point stability29 items
Presents 29 items from the article and its immediate prerequisites, covering dependency levels 3–5.

Level 7

1 page

Article items · 11 — Entropy rates, inverse branches and shifted root contracts28 items
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 5–6.

Level 8

1 page

Article items · 12 — Asymptotically global synchrony via basin mass28 items
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 7–8.

Level 9

1 page

Article items · 13 — Families with vanishing total spurious-basin measure28 items
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 8–11.

Level 10

1 page

Article items · 14 — Angle and mass sequences in the amplification argument29 items
Presents 29 items from the article and its immediate prerequisites, covering dependency levels 11–15.

Assessed 13 Nov 2026, 18:09 UTC
Prerequisite omissions and potential defects
Original-article findings for Expander graphs are globally synchronizing.

Assessed 13 Nov 2026, 18:09 UTC
Open questions and potential solutions
Insights, fruitful questions and potential solutions arising from Expander graphs are globally synchronizing.

All research collections

Research & notes

Prerequisites

Phase dynamics and oscillator models

Random graphs and matrix concentration

Spectral estimates and phase landscapes

Supporting prerequisites · 01 — Spectral edge and vertex expansion

Supporting prerequisites · 02 — A centered adjacency column retains a high-degree lower bound

Article items and immediate prerequisites

Level 0

Level 1

Level 2

Level 3

Level 4

Level 5

Level 6

Level 7

Level 8

Level 9

Level 10

Prerequisite omissions and potential defects

Open questions and potential solutions