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Symmetry and moments in the random-matrix discussion

PDF p. 31: Section 6 paragraph before (6.1)

Statement

In its discussion of the centered adjacency matrix $Δ_{A}$ of $G (n, p)$ , the article describes centered Bernoulli entries as nonsymmetrically distributed and says that, if the distribution were symmetric, all even moments would vanish and much of the machinery in [7,47] would give precise bounds. The parity assertion is incorrect: symmetry about zero forces integrable odd moments to vanish, not even moments in general.

For distinct vertices, an entry of $Δ_{A}$ is $B - p$ , where $B$ is Bernoulli $(p)$ : $B = 1$ with probability $p$ and $B = 0$ with probability $1 - p$ . For $0 < p < 1$ , this centered law is symmetric about zero exactly when $p = 1 / 2$ . The diagonal entry is the deterministic value $- p$ under the loop-free convention.

A concrete symmetric counterexample to the printed parity assertion is $W = B - B^{'}$ , where $B, B^{'}$ are independent Bernoulli $(1 / 2)$ variables on the four-outcome space ${0, 1}^{2}$ with each outcome having probability $1 / 4$ . Its law is symmetric about zero, but $E [W^{2}] = 1 / 2$ . More precisely, let $n, ℓ \geq 1$ be integers and let $M = (M_{a b})_{1 \leq a, b \leq n}$ be a real symmetric random matrix on one probability space. Suppose its upper-triangular entries, including the diagonal, are mutually independent, each has a law symmetric about zero, and $E ∣ M_{a b} ∣^{2 ℓ} < \infty$ for every $a \leq b$ . In the closed-walk expansion of $E [Tr M^{2 ℓ}]$ , every term containing an unordered edge, including a loop, with odd multiplicity vanishes.

Statement references

Erdős–Rényi random graph

Centered graph matrices

Bernoulli variables

Bernoulli difference moments

Trace expansion and parity cancellation

Integrable expectation

Expectation as integration against the law

Linearity of the integrable expectation

Finite probability spaces

Independence of finite-valued variables

Facts & Assumptions

[F1]

In the loop-free graph $G (n, p)$ , distinct-edge indicators are Bernoulli $(p)$ , and $Δ_{A} = A - p J$ because $d^{*} = p n$ (Erdős–Rényi random graph and high probability, Centered graph matrices and reference degree, Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials).

[F2]

A finite probability space has nonnegative outcome weights summing to one, and event probabilities are sums of their outcome weights (Finite probability spaces, outcome weights, events, and event probabilities). Two finite-valued variables are independent when their joint assignment probabilities factor into their marginal probabilities (Pairwise and mutual independence of finite-valued random variables). For $p \in [0, 1]$ and independent Bernoulli $(p)$ variables $B, B^{'}$ on one finite probability space, the difference $W = B - B^{'}$ has symmetric law and $E [W^{2}] = 2 p (1 - p)$ (Finite Bernoulli difference laws and parity of moments). Finite expectation is the probability-weighted sum of values (Expectation of a real random variable on a finite probability space).

[F3]

For integers $n, ℓ \geq 1$ , let $M$ be a real symmetric $n \times n$ random matrix on one probability space, with mutually independent upper-triangular entries including the diagonal and $E ∣ M_{a b} ∣^{2 ℓ} < \infty$ for all $a \leq b$ . The closed-walk expansion of $E [Tr M^{2 ℓ}]$ factors each walk expectation into moments of its distinct unordered edges, including loops. If every entry law is symmetric about zero, each factor whose edge multiplicity is odd vanishes (Trace powers, closed walks and independent edge moments).

[F4]

For an integrable real random variable $Y$ , expectation is the finite integral $E [Y] = \int_{Ω} Y d P$ (Expectation of a nonnegative or integrable random variable). For a real random variable $X$ with law $P_{X}$ and a Borel function $g$ such that $g (X)$ is integrable, $E [g (X)] = \int_{R} g d P_{X}$ (Change of variables for expectation). The Lebesgue integral is linear on $L^{1}$ , so scalar multiples of an integrable variable are integrable and $E [c Y] = c E [Y]$ for real $c$ (The Lebesgue integral is linear on L¹(μ)).

[F5]

For every nonnegative integer $k$ , the real polynomial map $x \mapsto x^{k}$ is continuous on $R$ (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, clause 5). A continuous map between topological spaces has Borel preimages of all Borel sets (A continuous map has Borel preimages of Borel sets). Thus these polynomial maps are measurable from $(R, B (R))$ to itself. Neither result requires a choice assumption.

Verification

Given: The graph convention and variables stated above, and the hypotheses in [F3] when applying the trace expansion.

Goal: Verify the centered entry law, the counterexample, and the corrected parity statement.

1.1F1givenalgebra

By [F1], an off-diagonal entry is $B - p$ , taking $1 - p$ with probability $p$ and $- p$ with probability $1 - p$ . For $0 < p < 1$ both values occur. Equality of this two-point law with its reflection requires $1 - p = p$ , and at $p = 1 / 2$ the two reflected values have equal probability. The loop-free adjacency diagonal is zero, so the diagonal of $A - p J$ is $- p$ .

1.2F2givenalgebra

On $Ω = {0, 1}^{2}$ , assign each of the four outcomes weight $1 / 4$ and set $B (b, b^{'}) = b$ , $B^{'} (b, b^{'}) = b^{'}$ . These nonnegative weights sum to one; each coordinate has probabilities $1 / 2$ at $0$ and $1$ , and every joint coordinate assignment has probability $1 / 4 = (1 / 2) (1 / 2)$ . Thus [F2] verifies a finite probability space and independent Bernoulli $(1 / 2)$ variables. Applying its difference-moment formula at $p = 1 / 2$ gives a symmetric $W = B - B^{'}$ with $E [W^{2}] = 2 (1 / 2) (1 / 2) = 1 / 2$ , disproving the printed even-moment assertion.

1.3F4F5givenalgebra

Let $X$ be a real random variable whose law equals that of $- X$ , and let $k$ be a positive odd integer such that $X^{k}$ is integrable. By [F5], $g (x) = x^{k}$ is continuous on $R$ and has Borel preimages of every Borel set; it is therefore Borel measurable, as required by the law-integration formula in [F4]. Since $(- X)^{k} = - X^{k}$ , [F4] shows that $(- X)^{k}$ is also integrable. Applying the law-integration formula in [F4] to $X$ and to $- X$ therefore gives $E [X^{k}] = \int_{R} x^{k} d P_{X} = E [(- X)^{k}]$ . Linearity from [F4] makes the last expectation $- E [X^{k}]$ . This is a finite real number by integrability, so $E [X^{k}] = 0$ .

2.1F3given∎

Apply [F3] to $M$ and the integers $n, ℓ$ in the Statement. All its hypotheses are imposed there: real symmetry on one probability space, mutual independence of the entire upper triangle including the diagonal, and finite absolute moments of order $2 ℓ$ . In the expansion of $E [Tr M^{2 ℓ}]$ , each walk expectation factors into entry moments; symmetry of each entry law makes an odd edge multiplicity supply a zero factor. Thus the stated even trace power has the claimed odd-multiplicity cancellation, including for loop edges.

Source notes

The printed parity assertion and the conditional claim that much of the machinery in [7,47] would give precise bounds for a symmetric entry distribution are attributed to the article’s Section 6, PDF page 31. The correction and example follow from the stated suppliers; no claim is made here about the precise results or applicability of [7,47].

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Sources

Expander graphs are globally synchronizing · PDF p. 31, Section 6, paragraph before (6.1)