Graphon-Signals and Continuum Graph Processing

• Graphon-signals are signals defined on graphons, which are bounded, measurable, symmetric kernels representing the limits of dense graph sequences.
• They extend traditional graph filtering by replacing matrix shifts with integral operators, enabling well-defined graphon Fourier transforms and convergence between discrete and continuum spectra.
• The framework supports advanced applications like sampling theory, dimensionality reduction, and graph neural network analysis while addressing challenges in sparse regimes.

Graphon-signals are signals defined on graphons, the infinite-dimensional limit objects of convergent graph sequences. In the standard formulation, a graphon is a bounded, measurable, symmetric kernel $W:[0,1]^2\to\mathbb{R}$, typically with $0\le W(u,v)\le 1$, and a graphon-signal is a pair $(W,X)$ with $X\in L^2([0,1])$. Graphon signal processing extends finite-graph signal processing by replacing matrix shifts with the integral operator induced by $W$, defining graphon Fourier transforms and linear shift-invariant filters at the continuum level. The resulting framework gives well-defined limits for graph Fourier analysis and graph filtering on convergent dense graph families, and it has since been extended to sampling theory, dimensionality reduction, graph neural network analysis, sparse graph limits, and applications in neuroscience (Ruiz et al., 2020, Ruiz et al., 2019).

1. Measure-theoretic formulation and graph limits

A graphon is a symmetric measurable kernel

$W\in L^\infty([0,1]^2),\qquad W(u,v)=W(v,u),$

and graph convergence is typically expressed through homomorphism densities. For a finite simple motif $F$, the homomorphism density into a graph $G=(V,E)$ with adjacency $S$ is

$t(F,G)=\frac{\mathrm{hom}(F,G)}{|V(G)|^{|V(F)|}},$

while the graphon counterpart is

$0\le W(u,v)\le 1$0

A dense graph sequence $0\le W(u,v)\le 1$1 converges to $0\le W(u,v)\le 1$2 when $0\le W(u,v)\le 1$3 for all finite simple graphs $0\le W(u,v)\le 1$4. Equivalently, after suitable measure-preserving relabelings, the induced step-graphons converge in cut norm (Ruiz et al., 2020).

The induced step-graphon of a finite graph with adjacency $0\le W(u,v)\le 1$5 is obtained from the regular partition $0\le W(u,v)\le 1$6 by

$0\le W(u,v)\le 1$7

A graph signal $0\le W(u,v)\le 1$8 induces the piecewise-constant graphon-signal

$0\le W(u,v)\le 1$9

This construction is the standard bridge from finite graphs to graphon-signals, and it makes spectral convergence statements precise (Ruiz et al., 2019).

The graphon acts on signals through the integral operator

$(W,X)$0

Because $(W,X)$1 for bounded graphons, $(W,X)$2 is a compact, self-adjoint Hilbert–Schmidt operator. The graphon is only defined up to measure-preserving relabelings, so graphon-signals are compared modulo this equivalence; later work on graphon-signal spaces makes this explicit through joint cut distances on pairs $(W,X)$3 (Herbst et al., 18 Mar 2025).

2. Spectral structure and the graphon Fourier transform

The spectral theory of graphon-signals is inherited from compact self-adjoint operator theory. There exist real eigenvalues $(W,X)$4 and an orthonormal eigenbasis $(W,X)$5 such that

$(W,X)$6

The nonzero spectrum accumulates only at $(W,X)$7, and each nonzero eigenvalue has finite multiplicity (Ruiz et al., 2020).

The graphon Fourier transform is the expansion of a signal in this eigenbasis:

$(W,X)$8

Parseval’s identity holds in $(W,X)$9, so the transform is the direct continuum analogue of the graph Fourier transform on finite graphs (Ruiz et al., 2019).

Bandlimitedness is defined spectrally. A signal is $X\in L^2([0,1])$0-bandlimited when

$X\in L^2([0,1])$1

Because the spectrum of a compact operator accumulates only at $X\in L^2([0,1])$2, this implies finite spectral support. Bandlimited graphon-signals play a central role in the original convergence theorems for graph Fourier coefficients and filter outputs (Ruiz et al., 2020).

A refinement becomes necessary when $X\in L^2([0,1])$3 has repeated eigenvalues. The noncommutative formulation replaces scalar coefficients attached to individual eigenfunctions by orthogonal projections onto eigenspaces. If $X\in L^2([0,1])$4 are the distinct nonzero eigenvalues and $X\in L^2([0,1])$5 the corresponding spectral projections, then the graphon Fourier transform may be written as

$X\in L^2([0,1])$6

with inverse

$X\in L^2([0,1])$7

This projection-valued formulation is designed to remain well-posed when multiplicities prevent canonical eigenfunction selection (Ghandehari et al., 2021).

For Cayley graphons on compact groups, graphon-signal analysis admits an explicit harmonic-analytic form. If $X\in L^2([0,1])$8 for a symmetric class kernel $X\in L^2([0,1])$9, then the spectrum of $W$0 is determined by the representation-theoretic matrices $W$1, and in the class-function case the eigenvalues reduce to

$W$2

This identifies a class of graphon-signals for which the graphon Fourier basis is furnished by matrix coefficients of irreducible representations rather than by abstract spectral approximation (Ghandehari et al., 2021).

3. Filters, shift invariance, and finite-to-limit convergence

Linear shift-invariant graphon filters are defined as operator polynomials,

$W$3

with spectral response

$W$4

More generally, if $W$5 is analytic on the spectrum, then $W$6 via functional calculus. In the vertex domain,

$W$7

This is the graphon counterpart of polynomial graph filters and diagonal spectral filtering on finite graphs (Ruiz et al., 2020).

The finite/discrete and continuum spectra align through the induced step-graphon. If $W$8 is the symmetric adjacency used as graph shift, then

$W$9

and the eigenfunctions of $W\in L^\infty([0,1]^2),\qquad W(u,v)=W(v,u),$0 are block-indicator lifts of the eigenvectors of $W\in L^\infty([0,1]^2),\qquad W(u,v)=W(v,u),$1. Correspondingly,

$W\in L^\infty([0,1]^2),\qquad W(u,v)=W(v,u),$2

Thus the spectrum of the scaled adjacency matrix converges to the spectrum of the graphon operator, and the graph Fourier transform converges to the graphon Fourier transform under appropriate conditions (Ruiz et al., 2019).

The main convergence theorem of graphon signal processing states that if $W\in L^\infty([0,1]^2),\qquad W(u,v)=W(v,u),$3 induces step graphon-signals $W\in L^\infty([0,1]^2),\qquad W(u,v)=W(v,u),$4, and there exist relabelings $W\in L^\infty([0,1]^2),\qquad W(u,v)=W(v,u),$5 such that $W\in L^\infty([0,1]^2),\qquad W(u,v)=W(v,u),$6 in cut norm and $W\in L^\infty([0,1]^2),\qquad W(u,v)=W(v,u),$7 in $W\in L^\infty([0,1]^2),\qquad W(u,v)=W(v,u),$8, with $W\in L^\infty([0,1]^2),\qquad W(u,v)=W(v,u),$9 $F$0-bandlimited and $F$1 non-derogatory, then the graph Fourier transform converges coefficientwise to the graphon Fourier transform, and the inverse transforms converge in $F$2. The proof combines cut-norm to operator-norm control, eigenvalue convergence, and Davis–Kahan perturbation arguments for eigenspaces (Ruiz et al., 2020).

Filter convergence is stronger. For a polynomial filter with taps $F$3, the spectral responses satisfy

$F$4

and for bandlimited signals,

$F$5

If $F$6 is Lipschitz on the spectral interval, then vertex-domain convergence extends to general finite-energy signals without assuming bandlimitedness or a non-derogatory spectrum. This removes the principal obstruction caused by eigenvalue multiplicity and the accumulation of eigenvalues near zero (Ruiz et al., 2020).

A complementary line of work studies filter design directly at the graphon level. In the Fourier–Galerkin approach, one chooses an orthonormal basis of $F$7, forms the Galerkin matrix

$F$8

and then designs a polynomial response either by Chebyshev approximation of a target spectral response or by solving

$F$9

This yields filters that depend only on the graphon, rather than on any particular sampled graph, and it formalizes graph filter design “in the limit” (Morency et al., 2020).

4. Sampling, uniqueness, and dimensionality reduction

Sampling theory for graphon-signals extends finite-graph notions of removable and uniqueness sets. For a measurable open set $G=(V,E)$0, removability is defined by the existence of $G=(V,E)$1 such that

$G=(V,E)$2

If $G=(V,E)$3 lies in the $G=(V,E)$4-bandlimited subspace

$G=(V,E)$5

then

$G=(V,E)$6

Hence a measurable set $G=(V,E)$7 is a uniqueness set for $G=(V,E)$8 whenever its complement $G=(V,E)$9 is removable with $S$0; in that case samples on $S$1 determine the bandlimited graphon-signal uniquely (Parada-Mayorga et al., 2024).

This framework also gives a common language for comparing sampling patterns across graphs of different sizes and labelings. If a graph sequence converges to a graphon and the graphon representations of its sampling sets coincide as measurable subsets of $S$2, then the associated removability constants converge as well. On that basis, an algorithm can transfer an approximately near-optimal sampling set computed on a smaller graph to a larger graph from the same graphon family by matching the induced measurable subset in $S$3 (Parada-Mayorga et al., 2024).

Dimensionality reduction can likewise be formulated at the graphon-signal level. Graphon pooling replaces a graph and its signal by block-constant approximations on a partition $S$4 of $S$5, with pooled operator

$S$6

and pooled signal

$S$7

Three constructions were proposed: regular integration (M1), irregular integration (M2), and irregular sampling (M3). The resulting low-dimensional graphons and signals form convergent sequences, and the reduced objects inherit spectral-structural properties of the originals (Parada-Mayorga et al., 2022).

The convergence theory is operator-theoretic. If $S$8 is the pooled step graphon, then

$S$9

and Lipschitz graph filters satisfy perturbation bounds linear in the cut norm. Reported numerical experiments showed that graphon pooling performed significantly better than other approaches proposed in the literature when dimensionality reduction ratios between layers were large, while also exhibiting less overfitting and lower computational cost (Parada-Mayorga et al., 2022).

5. Graphon-signal spaces in graph neural network theory

A separate development embeds attributed graphs into graphon-signal spaces in order to analyze message passing graph neural networks on a size- and topology-varying domain. For a bounded signal $t(F,G)=\frac{\mathrm{hom}(F,G)}{|V(G)|^{|V(F)|}},$0 and graphon $t(F,G)=\frac{\mathrm{hom}(F,G)}{|V(G)|^{|V(F)|}},$1, the graphon-signal cut distance is defined by

$t(F,G)=\frac{\mathrm{hom}(F,G)}{|V(G)|^{|V(F)|}},$2

where the same measure-preserving relabeling $t(F,G)=\frac{\mathrm{hom}(F,G)}{|V(G)|^{|V(F)|}},$3 is applied to both the graphon and the signal. Quotienting by zero cut distance yields a compact metric space, and the space of finite graph-signals is dense in it (Levie, 2023).

On this space, an MPNN layer can be written as an integral operator:

$t(F,G)=\frac{\mathrm{hom}(F,G)}{|V(G)|^{|V(F)|}},$4

with readout

$t(F,G)=\frac{\mathrm{hom}(F,G)}{|V(G)|^{|V(F)|}},$5

Under Lipschitz assumptions on the message, update, and readout maps, MPNNs are Lipschitz continuous with respect to graphon-signal cut distance, which leads to generalization bounds and subsampling stability results (Levie, 2023).

Later refinements extend this theory from scalar to multidimensional signals, from symmetric graphons to non-symmetric kernels, and from networks without readout to networks with readout. The same work replaces the original covering-number argument by a robustness-type generalization bound and preserves the sampling estimate

$t(F,G)=\frac{\mathrm{hom}(F,G)}{|V(G)|^{|V(F)|}},$6

together with the analogous simple-graph sampling bound (Rauchwerger et al., 25 Aug 2025).

Higher-order graph learning admits a parallel graphon-signal theory. Signal-weighted homomorphism densities

$t(F,G)=\frac{\mathrm{hom}(F,G)}{|V(G)|^{|V(F)|}},$7

extend classical motif densities to graphon-signals, characterize weak isomorphism, and connect graphon-signals to $t(F,G)=\frac{\mathrm{hom}(F,G)}{|V(G)|^{|V(F)|}},$8-WL indistinguishability. Invariant Graphon Networks (IWNs), built from bounded equivariant operators on $t(F,G)=\frac{\mathrm{hom}(F,G)}{|V(G)|^{|V(F)|}},$9, are at least as powerful as the $0\le W(u,v)\le 1$00-WL test, and they satisfy universal approximation results on compact subsets of $0\le W(u,v)\le 1$01 for $0\le W(u,v)\le 1$02 (Herbst et al., 18 Mar 2025).

This higher-order theory also exposes a tension in graphon-signal analysis: typical higher-order GNNs are discontinuous with respect to cut distance, and this discontinuity is tied to multigraph homomorphism densities and the definition of $0\le W(u,v)\le 1$03-WL. Nevertheless, transferability remains achievable, because the relevant functionals can still be controlled through sampling lemmas and approximated by continuous quantities in suitable regimes (Herbst et al., 18 Mar 2025).

6. Sparse regimes, applications, and limitations

Classical graphon-signal processing is intrinsically a dense-graph theory. If a graph sequence is sparse, with edge density tending to zero, then its canonical graphons converge to the zero graphon in classical cut distance, and the limiting operators, spectra, and filters become trivial. To avoid this collapse, generalized graphons are defined on $0\le W(u,v)\le 1$04 as bounded, symmetric $0\le W(u,v)\le 1$05 kernels, together with the stretch normalization

$0\le W(u,v)\le 1$06

which fixes the $0\le W(u,v)\le 1$07 mass. Convergence is then measured by the stretched cut distance $0\le W(u,v)\le 1$08, and the associated operators, spectra, and polynomial filters remain nontrivial for sparse sequences (Carvalho et al., 2023).

In this sparse setting, the generalized graphon operator

$0\le W(u,v)\le 1$09

is still compact and self-adjoint, and if $0\le W(u,v)\le 1$10 in stretched cut distance, then

$0\le W(u,v)\le 1$11

Finite-graph eigenvalues must be rescaled by $0\le W(u,v)\le 1$12 rather than by $0\le W(u,v)\le 1$13:

$0\le W(u,v)\le 1$14

A plausible implication is that “graphon-signal” should be understood as two related theories: the original dense theory on $0\le W(u,v)\le 1$15, and a stretched generalized theory for sparse limits (Carvalho et al., 2023).

Applications have begun to validate the formalism. In graphon signal processing for spiking and biological neural networks, graphon-based spectral projections were used to construct trial-invariant, low-dimensional embeddings for the stimulus identification problem. In spiking simulations, graphon projections on modular graphons yielded stable embeddings across stochastic trials and varying inter-block connectivity. On calcium imaging recordings from cultured modular neuronal networks, ridge regression on graphon embeddings achieved accuracy $0\le W(u,v)\le 1$16, compared with $0\le W(u,v)\le 1$17 for PCA and $0\le W(u,v)\le 1$18 for an RC baseline (Sumi et al., 24 Aug 2025).

The principal limitations remain those repeatedly identified across the literature. Graphons are unique only up to measure-preserving relabelings; estimation must therefore work on equivalence classes rather than on canonical coordinate systems. Classical graphon-signal processing targets dense graph sequences, and sparse networks require generalized graphons or different limit theories. Near-zero spectral components are delicate because eigenvalues accumulate at $0\le W(u,v)\le 1$19, which is why original convergence theorems invoked bandlimitedness or Lipschitz spectral responses. Finally, estimating $0\le W(u,v)\le 1$20 and computing accurate eigenpairs of $0\le W(u,v)\le 1$21 or its discretizations can be computationally expensive, particularly when the graphon is not block-structured or when high spectral resolution is required (Ruiz et al., 2020, Carvalho et al., 2023).

Taken together, these developments establish graphon-signals as a continuum formalism for graph-structured data in which transforms, filters, sampling sets, pooling operators, and neural architectures can be studied independently of any single finite realization. The central idea is stable across the literature: finite graph signal processing becomes transferable when it is interpreted as approximation to signal processing on an operator induced by a graph limit.