Equivalence Classes of Graphons

• Equivalence classes of graphons are collections of symmetric, measurable functions yielding the same graph limit under measure-preserving transformations.
• They provide a canonical framework by using pure graphons, where each equivalence class has a unique representative up to isometric rearrangement.
• The analytic approach employs cut norms, homomorphism densities, and quotient spaces to enable robust approximation and analysis of both random and deterministic network models.

An equivalence class of graphons is the collection of all symmetric, measurable functions (graphons) that yield the same limiting structure for a converging sequence of dense graphs, where “sameness” is formalized via invariance under measure-preserving transformations. The theory of equivalence classes of graphons underpins the analytic framework for graph limits and provides the structural foundation for modern approaches to large network analysis, exchangeable arrays, random graphs, and statistical models for networks.

1. Formal Definition and Cut Metric

Let $(\Omega, \mathcal{F}, \mu)$ be a probability space and $W: \Omega^2 \rightarrow [0,1]$ a symmetric, measurable function. The central metric-theoretic object in this theory is the cut norm: $$\|W\|_{\square,1} = \sup_{S, T \subset \Omega} \left| \int_{S \times T} W(x, y)\, d\mu(x)d\mu(y) \right|$$ and its function-variant: $$\|W\|_{\square,2} = \sup_{f,g: \Omega \to [-1,1]} \left| \int_{\Omega^2} W(x, y)f(x)g(y)\, d\mu(x)d\mu(y) \right|$$ These norms are equivalent and robust under measure-preserving transformations. The cut metric between $W$ and $W'$ is defined as: $$\delta_\square(W, W') = \inf_\phi \|W - W'^\phi\|_\square$$ where the infimum is over all measure-preserving bijections $\phi$ of $\Omega$. The metric is a pseudometric: $\delta_\square(W, W') = 0$ defines an equivalence relation.

2. Equivalence Problem: Characterizations

The equivalence problem asks: under what condition do two graphons represent the same limit object? The main equivalent characterizations are:

• Cut Metric Zero: $W_1$ and $W_2$ are equivalent if $\delta_\square(W_1,W_2) = 0$.
• Measure–Preserving Chains: There exists a finite chain of measure-preserving maps (or "pull-backs") linking $W_1$ to $W_2$.
• Homomorphism Densities: $W_1$ and $W_2$ are equivalent if and only if

$t(F, W_1) = t(F, W_2)$

for every (simple or loopless multigraph) $F$, where

$$t(F, W) = \int_{\Omega^{|V(F)|}} \prod_{\{i, j\} \in E(F)} W(x_i, x_j) d\mu(x_1)... d\mu(x_{|F|})$$

• Random Graph Distributions: $W_1$ and $W_2$ are equivalent if, for every $n$, the distributions of $G(n, W_1)$ and $G(n, W_2)$ coincide.

The interrelation with the Aldous–Hoover representation for exchangeable arrays further reinforces this equivalence (Janson, 2010).

3. Canonical Representatives and Uniqueness

Graphon equivalence classes admit canonical representatives via pure graphons. For $W$, define $w(x) = W_x \in L^1(\Omega)$, where $W_x(y) = W(x, y)$. The pushforward of $\mu$ under $x \mapsto W_x$ yields a canonical support $Q_W \subset L^1$. A graphon $W$ is pure if $x \mapsto W_x$ is injective a.e.

Theorems 9.7 and 9.9 (Janson, 2010) state:

• Every graphon is equivalent to a pure graphon.
• Two pure graphons are equivalent if and only if they are a.e. rearrangements of one another by a measure-preserving bijection.

The new proof of the uniqueness theorem via purification gives, for each equivalence class, a unique canonical representative up to isometric rearrangements.

4. Generalization to Arbitrary Probability Spaces

Graphons need not be defined on $([0,1], \mathcal{L}, \mu_{Leb})$. Theorem 7.1 (Janson, 2010) demonstrates that every graphon on any probability space $(\Omega, \mathcal{F}, \mu)$ is equivalent to one on $([0,1], \mathcal{L}, \mu_{Leb})$ by an appropriate coupling. The equivalence relation and cut metric are thus intrinsic and not dependent on the choice of underlying space; equivalence classes are defined purely by the structural properties of the graphon and the induced null set.

This fact enables the transfer of combinatorial settings and probabilistic limit objects into the analytic graphon framework.

5. Special Subclasses: {0,1}-Valued and Random-Free Graphons

Special attention is given to random-free (i.e., {0,1}-valued) graphons and to pure graphons.

• Random-Free Graphons: These are graphons that are a.e. $0$ or $1$. Every graph limit can be approximated by random-free graphons; in the $L^1$ metric, the set of random-free graphons is closed and dense in the cut metric (Lemma 10.6 (Janson, 2010)).
• Purification and Canonical Forms: For every graphon $W$, a canonical pure graphon can be constructed, and the equivalence class is uniquely represented by this pure version up to isometric measure-preserving rearrangement. This canonical form is especially useful in settings where deterministic structure is preferred, such as threshold and interval graphs.

These constructions provide canonical forms illuminating the minimal sufficient structure required for a graphon to represent a graph limit.

6. Functional Approach: Quotients and Invariance

From the quotient construction perspective, as clarified in (Paulson, 2019), if $G$ is the set of graphons and $r$ is the equivalence relation given by measure-preserving isomorphism, one studies the quotient space $G/r$. Functions of interest—such as subgraph densities $t(F,W)$ or the cut norm—are invariant on equivalence classes. This framework allows one to define "lifted" functions on equivalence classes using the formula: $f([W]) = f(W)$ whenever $f$ respects the equivalence relation. For multi-argument functions (e.g., those depending on two graphons), similar congruence properties hold.

This supports the formal manipulation and computation in quotient spaces without selecting arbitrary representatives.

7. Computational and Topological Aspects

The equivalence of different representations—$L^1$ («$d_1$»-names), cut norm («$\delta_\square$»-names), invariant measures—exhibits further structure (Ackerman et al., 2018):

• $L^1$-representations contain more computable information than cut norm representations; an oracle for the halting problem ($0'$) is generally required to pass from cut norm to $L^1$ representations, except in the random-free case.
• Invariance under the equivalence relation may obscure (or lose) computational specifics present in function-level details.
• There exist random-free graphons with an $L^1$ computable name that are not weakly isomorphic to any a.e. continuous graphon, indicating that the equivalence classes can possess fractal-like or highly non-smooth representatives, and that passage to equivalence class may elide nontrivial topological or computational detail.

Summary Table: Key Notions and Characterizations

Concept	Defining Property	Notes
Equivalence	$\delta_\square(W_1, W_2) = 0$	Implies same homomorphism densities
Pure Graphon	$x \mapsto W_x$ injective a.e.	Canonical representative for each class
Random-Free	$W$ is a.e. $\in \{0,1\}$	Closed in $L^1$, dense in cut-metric
Quotient Space	$G/r$ where $r$ is measure-preserving equivalence	Admits lifted functions respecting invariance
Homomorphism	$t(F,W_1) = t(F,W_2)$ for all $F$	Equivalent to zero cut-metric distance

This synthesis shows that equivalence classes of graphons are defined via zero cut metric distance or, equivalently, agreement of all pattern (homomorphism) densities after rearrangement. Each equivalence class contains a unique (up to isometry) pure graphon, and every graphon is equivalent to one on $([0,1],\mathcal{L},\mu_{Leb})$. The analytic structure of these classes is subtle, preserving structural invariants but potentially differing in computational or topological complexity, especially in $L^1$ representations or under additional restrictions such as random-freeness. The framework enables a comprehensive, canonical, and robust description of discrete structures in the continuum limit (Janson, 2010).