Alphabeta MathRandom 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.
1 · Prerequisites
- Single-vertex degree laws and the shared-edge dependence
- Asymptotic quantifiers and uniform parameter ranges
- Normalized phase volume and almost-everywhere assertions
- Loop coupling preserves the dynamics and centered Laplacian
- Simultaneous degree deviations on the squared logarithmic scale
- The unnormalized Laplacian needs its own degree deviation
- An elementary sufficient budget for the synchronization criterion
- Every fixed positive squared logarithmic density satisfies the budget
- Finite phase tori, local lifts and the flat metric
- Analytic potentials on a finite phase torus
- Differentiation of symmetric periodic pair potentials
- Gradient trajectories and complete compact phase dynamics
- Diagonal phase rotations and relative-phase coordinates
- The Axiom of Countable Choice (AC_ω)
- The finite union bound
- On the Landscape of Synchronization Networks: A Perspective from Nonconvex Optimization · Energy equation (1.1), PDF p. 1; periodic phase domain and synchronized configuration, PDF p. 2; Theorem 3.3, PDF p. 10; Proposition 5.1, PDF pp. 12–13
- A global synchronization theorem for oscillators on a random graph · Published Chaos 32, 093119 (2022): rotating frame and equation (1), p. 093119-2 (PDF p. 3); equation (2), same page; Theorem 11, p. 093119-8 (PDF p. 9); Theorem 13, p. 093119-9 (PDF p. 10)
- Concentration and regularization of random graphs · Section 1.1 equation (1.2), PDF p. 2; Theorem 1.1 and Section 1.4 example 1, PDF p. 3
- Logarithmic tangent bound and a quadratic entropy envelope
- Entropy endpoint limits and the two monotone branches
- Differentiable entropy inverse branches and shifted contracts
- Parameter bounds giving the signs of shifted entropy roots
- On an interval I, for f continuous on I and differentiable at every interior point: f' ≥ 0 throughout gives f nondecreasing, f' > 0 gives f increasing, f' ≤ 0 and f' < 0 give the two decreasing forms; conversely a nondecreasing f has f' ≥ 0 and a nonincreasing f has f' ≤ 0 wherever it is differentiable, and no strict converse is claimed
- Triangular Gaussian coordinates: metric and moment control
- Expected extreme eigenvalues from finite Gaussian comparison
- Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation
- Closure properties of measurable functions used by the integral
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- The Axiom of Choice
- Additivity of the nonnegative Lebesgue integral
- Positive real powers preserve measurability
- Probability measures and probability spaces
- The integral of a nonnegative simple function
- The nonnegative integral agrees with the simple integral on simple functions
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- Independent random elements
- Standard real and normalized complex Gaussian matrix laws
- Expectation of a nonnegative or integrable random variable
- Moments, variance, and covariance on a probability space
- Normalized graph Laplacian and its nontrivial spectrum
- A spectral window giving condition number below two
- Exponential clocks and the uniform edge-order process
- An isolated vertex before three quarters of the connectivity scale
- The natural logarithm as the inverse of the exponential function
- Finite vertex Hilbert space in the article convention
- Gaussian entry moments and triangular independence
- Minkowski's inequality for integrals, including p = ∞
- Even trace moments transfer from the operator norm
- Random elements and real random variables
- Afonso S. Bandeira and Ramon van Handel, Sharp nonasymptotic bounds on the norm of random matrices with independent entries · Lemma 2.2, printed/PDF pp. 6–7; Gaussian concentration display at top of p. 7; triangular matrix convention in Proposition 2.1 and Lemma 2.2 on p. 6
- Uniform edge orderings, graph-process times and independent labels
- Support connectivity and the unnormalized Laplacian kernel
- Phasors, trigonometric identities and periodic derivatives
- Christopher Hoffman, Matthew Kahle and Elliot Paquette, Spectral gaps of random graphs and applications · Section 6, Theorem 6.1 specialized to k=1 and delta=1/4; process and t0 definitions immediately preceding it (PDF pp. 22–23); compare Theorem 1.3 and following paragraph (PDF p. 3)
- Andrew D. McRae, Benign landscapes for synchronization on spheres via normalized Laplacian matrices · Theorem 3.4, positive implication for normalized Laplacian (PDF p. 10); Theorem 2.1 and complete proof Section 6.1 (PDF pp. 19–22); dynamics interface Section 1.2 (PDF p. 3); process application Section 3.4.2 (PDF p. 11)
- Vishesh Jain, Clayton Mizgerd and Mehtaab Sawhney, The random graph process is globally synchronizing · Theorem 1.2 and preceding process definition (PDF p. 2); Section 1.3 (PDF p. 4); Theorem 1.6 (PDF p. 3) and Sections 2–3 (PDF pp. 4–11)
- Elementary binomial probability envelopes
- The Erdős-Rényi finite random graph G(n,p)
- The Binomial count has its usual probability mass function
- Stirling's formula for factorials
- Chebyshev's inequality on a finite probability space
- Variance and covariance identities for random variables
- The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t
- Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
Remark
Earlier thresholds with explicit source quantifiers and a fixed-scale deduction
Lemma
Inward comparison and small-shift limits of entropy roots
Lemma
Extreme-value Lipschitz bounds and centered norm reduction
Lemma
Uniform spectral control transfers past connectivity
Proposition
Entropy rates, inverse branches and shifted root contracts
Theorem
An independent symmetric-Gaussian norm moment input
Theorem
Simultaneous global synchronization from connectivity
Proposition