[Paper Review] Integration and measures on the space of countable labelled graphs

Apoorva Khare, Bala Rajaratnam|arXiv (Cornell University)|Jun 4, 2015

Graph Labeling and Dimension Problems10 references4 citations

TL;DR

This paper establishes a rigorous foundation for integration and measures on the space of countable labelled graphs, $ \mathscr{G}(V)$, by showing that the dyadic Hamming norm $\|\cdot\|_{\psi,2}$ induces a Haar measure-preserving bijection between the space of infinite graphs and the unit circle. This enables a change of variables that transfers Riemann-Lebesgue theory from $\mathbb{R}$ to graph space, and identifies the Pontryagin dual of $\mathscr{G}(V)$ as the group of Walsh functions, enabling Fourier analysis on graph space.

ABSTRACT

In this paper we develop a rigorous foundation for the study of integration and measures on the space $\mathscr{G}(V)$ of all graphs defined on a countable labelled vertex set $V$. We first study several interrelated $σ$-algebras and a large family of probability measures on graph space. We then focus on a "dyadic" Hamming distance function $\left\| \cdot ight\|_{ψ,2}$, which was very useful in the study of differentiation on $\mathscr{G}(V)$. The function $\left\| \cdot ight\|_{ψ,2}$ is shown to be a Haar measure-preserving bijection from the subset of infinite graphs to the circle (with the Haar/Lebesgue measure), thereby naturally identifying the two spaces. As a consequence, we establish a "change of variables" formula that enables the transfer of the Riemann-Lebesgue theory on $\mathbb{R}$ to graph space $\mathscr{G}(V)$. This also complements previous work in which a theory of Newton-Leibnitz differentiation was transferred from the real line to $\mathscr{G}(V)$ for countable $V$. Finally, we identify the Pontryagin dual of $\mathscr{G}(V)$, and characterize the positive definite functions on $\mathscr{G}(V)$.

Motivation & Objective

• To develop a rigorous theory of integration and measures on the space $\mathscr{G}(V)$ of all graphs with a fixed countable, labelled vertex set $V$.
• To identify and study the Haar measure on $\mathscr{G}(V)$, which is a compact abelian group under symmetric difference.
• To establish a connection between integration on $\mathscr{G}(V)$ and the Riemann-Lebesgue theory on $\mathbb{R}$ via a Haar measure-preserving bijection.
• To characterize the Pontryagin dual of $\mathscr{G}(V)$ and identify positive definite functions on the graph space.
• To enable Fourier analysis on $\mathscr{G}(V)$ by showing that Walsh functions form a complete orthonormal basis in $L^2(\mathscr{G}(V))$.

Proposed method

• Define $\mathscr{G}(V) = \{0,1\}^{K_V}$ as the space of all graphs on a countable vertex set $V$, equipped with the product topology and symmetric difference as group operation.
• Introduce the dyadic Hamming norm $\|G\|_{\psi,2} = \sum_{e \in G} 2^{-\psi(e)}$ for a fixed bijection $\psi: K_V \to \mathbb{N}$, which induces a translation-invariant metric.
• Prove that $\|\cdot\|_{\psi,2}$ is a Haar measure-preserving bijection from the subset of infinite graphs to the unit circle $[0,1)$, thus enabling a change of variables for integration.
• Use this bijection to transfer the Riemann-Lebesgue theory from $\mathbb{R}$ to $\mathscr{G}(V)$, allowing integration on graph space to be reduced to integration on $[0,1]$.
• Characterize the Pontryagin dual $\mathscr{G}^\wedge$ of $\mathscr{G}(V)$ as the group of Walsh functions $\chi_E$ indexed by finite edge sets $E \in \mathscr{G}_0(V)$.
• Apply Bochner’s Theorem to show that every normalized positive definite function on $\mathscr{G}(V)$ arises as a Fourier transform of a probability measure on $\mathscr{G}^\wedge$, which is isomorphic to $\mathscr{G}_0(V)$.

Experimental results

Research questions

• RQ1 How can a consistent theory of integration and measures be developed on the space of countable labelled graphs $\mathscr{G}(V)$?
• RQ2 What is the role of the dyadic Hamming norm $\|\cdot\|_{\psi,2}$ in relating graph space to the unit circle and enabling change of variables?
• RQ3 How does the Haar measure on $\mathscr{G}(V)$ relate to Lebesgue measure on the circle, and what is the significance of this isomorphism?
• RQ4 What is the Pontryagin dual of $\mathscr{G}(V)$, and how does it relate to the structure of positive definite functions on graph space?
• RQ5 Can Fourier analysis on $\mathscr{G}(V)$ be fully characterized using Walsh functions, and how does this relate to classical $L^2$-theory?

Key findings

• The dyadic Hamming norm $\|\cdot\|_{\psi,2}$ induces a Haar measure-preserving bijection from the space of infinite graphs to the unit circle $[0,1)$, enabling a change of variables that transfers integration from $\mathbb{R}$ to $\mathscr{G}(V)$.
• Integration on $\mathscr{G}(V)$ can be reduced to integration on $[0,1]$ via this bijection, effectively transporting the Riemann-Lebesgue theory to graph space.
• The Pontryagin dual $\mathscr{G}^\wedge$ of $\mathscr{G}(V)$ is isomorphic to the group $\mathscr{G}_0(V)$ of finite graphs under symmetric difference.
• The Walsh functions $\chi_E$ for $E \in \mathscr{G}_0(V)$ form a complete orthonormal basis in $L^2(\mathscr{G}(V))$, transforming into the standard Walsh functions on $[0,1]$ under the $\|\cdot\|_{\psi,2}$ map.
• Every normalized positive definite function on $\mathscr{G}(V)$ is of the form $f(G) = \sum_{H \in \mathscr{G}_0(V)} (-1)^{|G \cap H|} \mu(H)$ for some probability measure $\mu$ on $\mathscr{G}_0(V)$, as per Bochner’s Theorem.