Grapheurs: Dual Limits for Directed Networks

• Grapheurs are analytic objects representing quotient limits of finite weighted directed graphs via random equipartitions.
• They offer a dual approach to graphons by capturing global connectivity features and hub structures that local subgraph methods miss.
• Edge-based sampling of grapheurs facilitates efficient property testing with convergence rates around O(1/√n) in large network models.

A grapheur is a mathematical object representing the quotient-limit of a sequence of finite weighted directed graphs via random equipartitions of the vertex set. As developed in the theory of "Graph Limits via Quotients," grapheurs are specified as extremal exchangeable probability measures on the unit square, and serve as a dual to graphons for modeling large-scale global connectivity features—particularly hub-and-spoke structures—that are invisible to classical subgraph-density-based graph limit theories (Levin et al., 29 Dec 2025).

1. Mathematical Definition of a Grapheur

The grapheur construction begins by representing a finite directed graph $G$ with $n$ vertices as a weighted adjacency matrix $G = (G_{i,j})_{1 \le i, j \le n}$, where $G_{i,j} \geq 0$ and $\sum_{i,j} G_{i,j} = 1$. For any $k \le n$, a $k$-quotient is obtained via a surjective map $f: [n] \to [k]$: $$(\rho(f)\,G)_{i,j} = \sum_{u \in f^{-1}(i),\; v \in f^{-1}(j)} G_{u,v}.$$ A random quotient is formed by assigning each vertex independently uniformly at random to one of $k$ blocks, i.e., $F_{k,n}(i) \sim \mathrm{Unif}([k])$ i.i.d., and considering $\rho(F_{k,n})\,G$.

Passing to the limit along sequences of graphs with $n \to \infty$ yields the analytic object: a random exchangeable probability measure $\mathcal M$ on $[0,1]^2$ (i.e., $\mathcal M$ such that for every measure-preserving bijection $\sigma: [0,1] \to [0,1]$, $$\mathcal M \overset{d}{=} \mathcal M \circ (\sigma, \sigma)$$). By Kallenberg's representation, every extremal exchangeable measure can be written as: $$\mathcal M = \sum_{i,j \ge 1} E_{i,j} \,\delta_{(T_i,T_j)} + \sum_{i \ge 1} \Big[\sigma_i\,\delta_{T_i} \otimes \lambda + \varsigma_i\,\lambda \otimes \delta_{T_i}\Big] + \theta\,\lambda^2 + \vartheta\,\lambda_D,$$ with nonnegative coefficients summing to $1$. The term "grapheur" is reserved for extremal (i.e., ergodic) exchangeable probability measures of this type (Levin et al., 29 Dec 2025).

2. Duality with Graphons

Graphons are symmetric measurable functions $W(x,y): [0,1]^2 \to [0,1]$ serving as classical graph limits, equipped with the cut-metric $\delta_\square$, and encode the limits of subgraph homomorphism densities. Grapheurs, being dual, govern the limits of quotient statistics rather than subgraph counts.

The duality is formalized via the evaluation pairing: $$\langle\mathcal M, W\rangle = \sum_{i,j \ge 1} E_{i,j}\,\mathrm{Bern}(W(T_i,T_j)) + 2\sum_{i \ge 1} \sigma_i \int_0^1 W(T_i,y)\,dy + \theta \iint_{[0,1]^2} W(x,y)\,dx\,dy,$$ where $\{T_i\}$ are i.i.d. uniform and Bernoulli draws are coupled accordingly. The theorem states that

$\mathcal M_n \to \mathcal M \iff \langle\mathcal M_n, W_G \rangle \overset{d}{\to} \langle \mathcal M, W_G \rangle \quad\text{for all step-graphons %%%%21%%%%},$

and similarly for graphons versus grapheur-labeled step graphs (Levin et al., 29 Dec 2025). This shows a precise dual relationship: graphons model local density (e.g., triangles), grapheurs model global quotient patterns (e.g., hubs).

3. Detection of Hubs and Global Structures

A fundamental failure of subgraph-density limits arises in the presence of hubs. For example, star graphs on $n$ vertices (with all edges from vertex 1) yield vanishing subgraph densities for any fixed finite tree as $n\to\infty$. However, the mass of the hub is nontrivial and becomes visible only in the atomic coefficients of the limit grapheur (the nonzero $\sigma$-parameters), not in the graphon limit. The random quotient of a large star graph is a randomly labeled star on $k$ vertices, directly reflecting the hub (Levin et al., 29 Dec 2025).

This suggests that grapheurs are essential for modeling and property-testing in graphs where hub and global connectivity features are of primary interest, as encountered in large communication, neural, or transportation networks.

4. Edge-Based Sampling and Regularity Theory

For property testing and statistical analysis, grapheurs admit a practical edge-based sampling procedure. Sampling $n$ edges (with replacement, weighted by $G_{i,j}$) gives an empirical measure approaching the limiting grapheur in the dual cut-metric at a universal $O(1/\sqrt n)$ rate: $\E\left[ W_\square(\hat{\mathcal M}_n, \mathcal M) \right] \leq \frac{174}{\sqrt n}.$ The proof employs a VC-inequality for axis-aligned rectangles, analogous to a Szemerédi-type regularity lemma for edges rather than vertices. This enables efficient testing of hub-centric properties in massive graphs by observing a modest random sample of edges (Levin et al., 29 Dec 2025).

5. Equipartition-Consistent Random Graph Models

The grapheur formalism provides a characterization for random-graph models invariant under equipartitions. A sequence of $k\times k$ law $(\mu_k)_{k \ge 1}$ is called equipartition-consistent if, under all balanced equipartitions $d_{k,nk}: [nk] \to [k]$, the law of the quotient on $k$ blocks matches the prescribed law for all $n$. The characterization theorem states: $$\forall\, (\mu_k)_k \text{ equipartition-consistent},~ \exists!\, \mathcal M \text{ grapheur}~\text{such that}~\mu_k = \mathrm{Law}(G_k[\mathcal M]),~\forall k.$$ Consequently, statistical models for large networks that respect equipartition consistency must arise from drawing i.i.d. quotient samples from some grapheur (Levin et al., 29 Dec 2025). This presents a conceptual foundation for random-graph models focused on global structural constraints, notably the reproduction of hubness and heavy-edge configurations under aggregation.

6. Compactness, Inequality Theory, and Open Questions

The space of grapheurs $\mathfrak M$, under the dual cut-metric, is compact. Every sequence of finite graphs admits a subsequence converging in the quotient sense. Quasi-flag-algebra methods can be developed: every continuous inequality on the vector of quotient densities for finite graphs extends to all grapheurs. Open questions include the optimality of the $O(1/\sqrt n)$ edge-sampling rate, rates of convergence for uncoupled random quotients, and the development of a comprehensive inequality/geometric theory for nonlocal properties accessible only to quotient—i.e., grapheur—analysis (Levin et al., 29 Dec 2025).

---

Grapheurs thus extend the analytic repertoire for studying large random graphs, providing necessary tools for quantifying, sampling, and understanding global (hub- and edge-centric) patterns in complex networks, in contexts where classical local-subgraph-based graph limit theory proves inadequate.