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Refined H mass bound in a regular graph

Statement

Assume countable choice. Let $G = (V, E)$ be a finite connected undirected graph, with $n = ∣ V ∣ > 0$ , symmetric binary adjacency matrix $A$ , and self-loops allowed and counted once in each row sum. Suppose every row sum is $d \geq 0$ , let $α \geq 0$ , and assume $∥ Δ ∥ \leq α d$ , where $Δ = A - (d / n) J$ and $J$ is the all-ones matrix; the norm is the Euclidean operator norm. Let $θ$ be a stable state, meaning a local minimum on the phase torus of $E_{G} (θ) = \frac{1}{2} \sum_{x, y} A_{x y} (1 - \cos (θ_{x} - θ_{y}))$ . Choose representatives in $(- π, π]$ after common rotation so that $ρ_{1} \geq 0$ is real, where $ρ_{k} = n^{- 1} \sum_{x} e^{i k θ_{x}}$ . Put $X = ρ_{1}^{2}$ , $r = ∣ ρ_{2} ∣$ , and $M = n^{- 1} \sum_{x} H (θ_{x})$ , where $H (t) = \sin^{2} ∣ t ∣$ for $∣ t ∣ \leq π / 2$ and $H (t) = 1$ for $π / 2 \leq ∣ t ∣ \leq π$ .

If $d > 0$ , then $X M \leq α^{2} (1 - X)$ . If $X > 0$ , this gives $M \leq α^{2} (1 - X) / X$ and $r \geq 1 - 2 α^{2} (1 - X) / X$ . For $d = 0$ one has $X = r = 1$ and $M = 0$ .

Facts & Assumptions

[F1]

Put $f \in C^{V}$ with $f_{x} = e^{i θ_{x}}$ for every $x \in V$ . The regular phase projection gives $∥ Δ f ∥_{2}^{2} \leq α^{2} d^{2} n (1 - X)$ (Mean-zero projection of the regular phase vector).

[F2]

For $d > 0$ , apply the normalized stable-state pointwise inequality with $d^{*} = d$ and $S_{y} = \sum_{x} Δ_{x, y} e^{i (θ_{x} - θ_{y})}$ to obtain $∣ S_{y} ∣^{2} \geq d^{2} X H (θ_{y})$ for every $y \in V$ (Centered-adjacency pointwise phase bounds).

[F3]

For $z = a + i b$ with real $a, b$ , $Re z = a$ and $∣ z ∣ = \sqrt{a^{2} + b^{2}} \geq 0$ (Real and imaginary parts, complex conjugation, and modulus).

[F4]

The truncated sine-square function satisfies $H (t) \geq \sin^{2} t$ on $(- π, π]$ (Elementary properties of the function H).

[F5]

Apply connectedness to the loop-free support graph obtained by deleting self-loops: its vertices are joined by finite edge paths (Connected graphs and connected components defined by the existence of vertex paths). Deleting a self-loop changes no reachability between distinct vertices, since a step from a vertex to itself may be removed from a walk.

[F10]

Definition or convention used in the Statement: Adjacency, degrees, graph Laplacian, and spectral notation.

[F11]

Definition or convention used in the Statement: Adjacency spectral expander — Definition 1.3.

[F12]

Definition or convention used in the Statement: Equilibria, stable states, synchronized states, and spurious minima.

[F13]

Definition or convention used in the Statement: Daido order parameters.

[F14]

Definition or convention used in the Statement: Bounds, saturation, and rotation of order parameters.

[F15]

Definition or convention used in the Statement: Truncated sine-square function.

[F16]

Definition or convention used in the Statement: Centered graph matrices and reference degree.

[F17]

Definition or convention used in the Statement: Loewner order, Hermitian inner product, and operator norm.

[F18]

Definition or convention used in the Statement: Phasors, trigonometric identities and periodic derivatives.

[F19]

Definition or convention used in the Statement: The Axiom of Countable Choice (AC_ω).

Proof

Given: The graph, stable normalized state, parameters and notation in the Statement, and the phasor vector $f$ defined in [F1].

Goal: Establish all the asserted estimates.

1.1F1F2F10F13F15F16F18givenalgebra

Suppose $d > 0$ . Since $Δ$ is symmetric, the definitions in [F1] and [F2] give $S_{y} = e^{- i θ_{y}} \sum_{x} Δ_{y, x} f_{x} = e^{- i θ_{y}} (Δ f)_{y}$ . The modulus of the unit factor $e^{- i θ_{y}}$ is one. Sum [F2] over $y$ and apply [F1] to obtain $d^{2} n X M \leq ∥ Δ f ∥_{2}^{2} \leq α^{2} d^{2} n (1 - X)$ . Divide by $d^{2} n > 0$ to obtain $X M \leq α^{2} (1 - X)$ ; if $X > 0$ , divide again by $X$ to obtain the stated bound on $M$ .

2.1F3F4F13F18step 1.1algebra

For $z = a + i b$ , write $m = ∣ z ∣$ . By [F3], $m \geq 0$ and $m^{2} = a^{2} + b^{2}$ . If $m < a$ , then $a > 0$ and $0 < (a - m) (a + m) = - b^{2} \leq 0$ , a contradiction. Thus $∣ z ∣ \geq Re z$ . By [F18], squaring $e^{i t} = \cos t + i \sin t$ and using the exponential addition identity and unit modulus gives $\cos (2 t) = 1 - 2 \sin^{2} t$ . Hence $r \geq Re ρ_{2} = 1 - 2 n^{- 1} \sum_{y} \sin^{2} θ_{y} \geq 1 - 2 M$ by [F4]. For $X > 0$ , substitute the bound on $M$ from step 1.1 to obtain $r \geq 1 - 2 α^{2} (1 - X) / X$ .

3.1F5F10F13F14F15F18givenalgebra∎

If $d = 0$ , nonnegative adjacency entries and zero row sums force $A = 0$ . By [F5], connectedness then forces $n = 1$ : if two distinct vertices existed, the loop-free support would have no path joining them. The normalized single unit phasor is one, so its chosen phase representative is zero. Consequently $X = r = 1$ and $M = H (0) = 0$ . This proves the remaining case.

Depends on

Used by · 3

Dependency tree · 2 levels

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