SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Ron Levie
Faculty of Mathematics
Technion - Israel Institute of Technology
levieron@technion.ac.il
We present an approach for analyzing message passing graph neural networks (MPNNs) based on an extension of graphon analysis to a so called graphon-signal analysis. A MPNN is a function that takes a graph and a signal on the graph (a graph-signal) and returns some value. Since the input space of MPNNs is non-Euclidean, i.e., graphs can be of any size and topology, properties such as generalization are less well understood for MPNNs than for Euclidean neural networks. We claim that one important missing ingredient in past work is a meaningful notion of graph-signal similarity measure, that endows the space of inputs to MPNNs with a regular structure. We present such a similarity measure, called the graphon-signal cut distance, which makes the space of all graph-signals a dense subset of a compact metric space – the graphon-signal space. Informally, two deterministic graph-signals are close in cut distance if they “look like” they were sampled from the same random graph-signal model. Hence, our cut distance is a natural notion of graph-signal similarity, which allows comparing any pair of graph-signals of any size and topology. We prove that MPNNs are Lipschitz continuous functions over the graphon-signal metric space. We then give two applications of this result: 1) a generalization bound for MPNNs, and, 2) the stability of MPNNs to subsampling of graph-signals. Our results apply to any regular enough MPNN on any distribution of graph-signals, making the analysis rather universal.
In recent years, the need to accommodate non-regular structures in data science has brought a boom in machine learning methods on graphs. Graph deep learning (GDL) has already made a significant impact on the applied sciences and industry, with ground-breaking achievements in computational biology [2, 10, 17, 28], and a wide adoption as a general-purpose tool in social media, e-commerce, and online marketing platforms, among others. These achievements pose exciting theoretical challenges: can the success of GDL models be grounded in solid mathematical frameworks? Since the input space of a GDL model is non-Euclidean, i.e., graphs can be of any size and any topology, less is known about GDL than standard neural networks. We claim that contemporary theories of GDL are missing an important ingredient: meaningful notions of metric on the input space, namely, graph similarity measures that are defined for all graphs of any size, which respect and describe in some sense the behavior of GDL models. In this paper, we aim at providing an analysis of GDL by introducing such appropriate metrics, using graphon theory.
A graphon is an extension of the notion of a graph, where the node set is parameterized by a probability space instead of a finite set. Graphons can be seen as limit objects of graphs, as the number of nodes increases to infinity, under an appropriate metric. One result from graphon theory (that reformulates Szemerédi's regularity lemma from discrete mathematics) states that any sufficiently large graph behaves as if it was randomly sampled from a stochastic block model with a fixed number of classes.
Figure 1: Illustration of the graph-signal cut distance. Left: a stochastic block model (SBM) with a signal. The color of the block represents the value of the signal at this block. The thickness of the edges between the blocks (including self-loops) represents the probability/density of edges between the blocks. Middle: a small graph-signal which looks like was sampled from the SMB. The color of the nodes represents the signal values. Right: a large graph-signal which looks like was sampled from the SMB. In graphon-signal cut distance, these two graph-signals are close to each other.
This result poses an "upper bound" on the complexity of graphs: while deterministic large graphs may appear to be complex and intricate, they are actually approximately regular and behave random-like.
In this paper we extend this regularity result to an appropriate setting for message passing neural networks (MPNNs), a popular GDL model. Since MPNNs take as input a graph with a signal defined over the nodes (a graph-signal), we extend graphon theory from a theory of graphs to a theory of graph-signals. We define a metric, called the graph-signal cut distance (see Figure 1 for illustration), and formalize regularity statements for MPNNs of the following sort.
(1) Any deterministic graph-signal behaves as if it was randomly sampled from a stochastic block model, where the number of blocks only depends on how much we want the graph-signal to look random-like, and not on the graph-signal itself.
(2) If two graph-signals behave as if they were sampled from the same stochastic block model, then any (regular enough) MPNN attains approximately the same value on both.
Formally, (1) is proven by extending Szemerédi's weak regularity lemma to graphon-signals. As a result of this new version of the regularity lemma, we show that the space of graph-signals is a dense subset of the space of graphon-signals, which is shown to be compact. Point (2) is formalized by proving that MPNNs with Lipschitz continuous message functions are Lipschitz continuous mappings from the space of graph-signals to an output space, in the graphon-signal cut distance.
We argue that the above regularity result is a powerful property of MPNNs. To illustrate this, we use the new regularity result to prove two corollaries. First, a generalization bound of MPNNs, showing that if the learned MPNN performs well on the training graph-signals, it is guaranteed to also perform well on test graph-signals. This is shown by first bounding the covering number of the graphon-signal space, and then using the Lipschitzness of MPNNs. Second, we prove that MPNNs are stable to graph-signal subsampling. This is done by first showing that randomly subsampling a graphon-signal produces a graph-signal which is close in cut distance to the graphon-signal, and then using the Lipschitzness of MPNNs.
As opposed to past works that analyze MPNNs using graphon analysis, we do not assume any generative model on the data. Our results apply to any regular enough MPNN on any distribution of graph-signals, making the analysis rather universal. We note that past works about generalization in GNNs [14, 23, 26, 30] consider special assumptions on the data distribution, and often on the MPNN model. Our work provides upper bounds under no assumptions on the data distribution, and only mild Lipschitz continuity assumptions on the message passing functions. Hence, our theory bounds the generalization error when all special assumptions (that are often simplistic) from other papers are not met. We show that when all assumptions fail, MPNNs still have generalization and sampling guarantees, albeit much slower ones. See Table 1. This is also true for past sampling theorems, e.g., [18, 22, 27, 31, 32].
The problem with graph-signal domains. Since the input space of MPNNs is non-Euclidean, results like universal approximation theorems and generalization bounds are less well developed for MPNNs than Euclidean deep learning models. For example, analysis like in [6] is limited to graphs of fixed sizes, seen as adjacency matrices. The graph metric induced by the Euclidean metric on
adjacency matrices is called edit-distance. This reduction of the graph problem to the Euclidean case does not describe the full complexity of the problem. Indeed, the edit-distance is defined for weighted graphs, and non-isomorphic simple graphs are always far apart in this metric. This is an unnatural description of the reality of machine learning on graphs, where different large non-isomorphic simple graphs can describe the same large-scale phenomenon and have similar outputs for the same MPNN.
Other papers that consider graphs of arbitrary but bounded size are based on taking the union of the Euclidean edit-distance spaces up to a certain graph size [3]. If one omits the assumption that all graphs are limited by a predefined size, the edit-metric becomes non-compact - a topology too fine to explain the behavior of real MPNNs. For example, two graphs with different number of nodes are always far apart in edit-distance, while most MPNN architectures in practice are not sensitive to the addition of one node to a large graph. In [19], the expressivity of GNNs is analyzed on spaces of graphons. It is assumed that graphons are Lipschitz continuous kernels. The metric on the graphon space is taken as the  $L_{\infty}$  distance between graphons as functions. We claim that the Lipschitz continuity of the graphons in [19], the choice of the  $L_{\infty}$  metric, and the choice of an arbitrary compact subset therein, are not justified as natural models for graphs, and are not grounded in theory. Note that graphon analysis is measure theoretic, and results like the regularity lemma are no longer true when requiring Lipschitz continuity for the graphons. Lastly, in papers like [18, 26, 27, 31], the data is assumed to be generated by one, or a few graphons, which limits the data distribution significantly. We claim that this discrepancy between theory and practice is an artifact of the inappropriate choices of the metric on the space of graphs, and the choice of a limiting generative model for graphs.
For  $n \in \mathbb{N}$ , we denote  $[n] = \{1, \dots, n\}$ . We denote the Lebesgue  $p$  space over the measure space  $\mathcal{X}$  by  $\mathcal{L}^p(\mathcal{X})$ , or, in short,  $\mathcal{L}^p$ . We denote by  $\mu$  the standard Lebesgue measure on  $[0, 1]$ . A partition is a sequence  $\mathcal{P}_k = \{P_1, \dots, P_k\}$  of disjoint measurable subsets of  $[0, 1]$  such that  $\bigcup_{j=1}^k P_j = [0, 1]$ . The partition is called equipartition if  $\mu(P_i) = \mu(P_j)$  for every  $i, j \in [k]$ . We denote the indicator function of a set  $S$  by  $\mathbb{1}_S$ . See Appendix A for more details. We summarize our notations in Appendix I.
2.1 Message passing neural networks
Most graph neural networks used in practice are special cases of MPNN (see [15] and [11] of a list of methods). MPNNs process graphs with node features, by repeatedly updating the feature at each node using the information from its neighbors. The information is sent between the different nodes along the edges of the graph, and hence, this process is called message passing. Each node merges all messages sent from its neighbors using an aggregation scheme, where typical choices is to sum, average or to take the coordinate-wise maximum of the messages. In this paper we focus on normalized sum aggregation (see Section 4.1). For more details on MPNNs we refer the reader to Appendix E.
2.2 Szemerédi weak regularity lemma
The following is taken from [13, 25]. Let  $G = \{V, E\}$  be a simple graph with nodes  $V$  and edges  $E$ . For any two subsets  $U, S \subset V$ , denote the number of edges with one end point at  $U$  and the other at  $S$  by  $e_G(U, S)$ . Let  $\mathcal{P} = \{V_1, \ldots, V_k\}$  be a partition of  $V$ . The partition is called equipartition if  $||V_i| - |V_j|| \leq 1$  for every  $i, j \in [k]$ . Given two node sets  $U, S \subset V$ , if the edges between each pair of classes  $V_i$  and  $V_j$  were random, we would expect the number of edges of  $G$  connecting  $U$  and  $S$  to be close to the expected value  $e_{\mathcal{P}(U, S)} := \sum_{i=1}^{k} \sum_{j=1}^{k} \frac{e_G(V_i, V_j)}{|V_i||V_j|} |V_i \cap U| |V_j \cap S|$ . Hence, the irregularity, that measures how non-random like the edges between  $\{V_j\}_{j=1}^k$  are, is defined to be
$$
\operatorname {i r r e g} _ {G} (\mathcal {P}) = \max  _ {U, S \subset V} \left| e _ {G} (U, S) - e _ {\mathcal {P}} (U, S) \right| / | V | ^ {2}. \tag {1}
$$
Theorem 2.1 (Weak Regularity Lemma [13]). For every  $\epsilon >0$  and every graph  $G = (V,E)$ , there is an equipartition  $\mathcal{P} = \{V_1,\ldots ,V_k\}$  of  $V$  into  $k\leq 2^{c / \epsilon^2}$  classes such that  $\mathrm{irreg}_G(\mathcal{P})\leq \epsilon$ . Here,  $c$  is a universal constant that does not depend on  $G$  and  $\epsilon$ .
Theorem 2.1 asserts that we can represent any large graph  $G$  by a smaller, coarse-grained version of it: the weighted graph  $G^{\epsilon}$  with node set  $V^{\epsilon} = \{V_{1},\ldots ,V_{k}\}$ , where the edge weight between the nodes  $V_{i}$  and  $V_{j}$  is  $\frac{e_G(V_i,V_j)}{|V_i|,|V_j|}$ . The "large-scale" structure of  $G$  is given by  $G^{\epsilon}$ , and the number of edges between any two subsets of nodes  $U_{i}\subset V_{i}$  and  $U_{j}\subset V_{j}$  is close to the "expected value"  $e_{\mathcal{P}}(U_i,U_j)$ . Hence, the deterministic graph  $G$  "behaves" as if it was randomly sampled from  $G^{\epsilon}$ .
2.3 Graphon analysis
A graphon [4, 24] can be seen as a weighted graph with a "continuous" node set, or more accurately, the nodes are parameterized by an atomless standard probability space called the graphon domain. Since all such graphon domains are equivalent to  $[0,1]$  with the standard Lebesgue measure (up to a measure preserving bijection), we take  $[0,1]$  as the node set. The space of graphons  $\mathcal{W}_0$  is defined to be the set of all measurable symmetric function  $W:[0,1]^2\to [0,1], W(x,y) = W(y,x)$ . The edge weight  $W(x,y)$  of a graphon  $W\in \mathcal{W}_0$  can be seen as the probability of having an edge between the nodes  $x$  and  $y$ .
Graphs can be seen as special graphons. Let  $\mathcal{I}_m = \{I_1, \ldots, I_m\}$  be an interval equipartition: a partition of  $[0, 1]$  into intervals of equal length. The graph  $G = \{V, E\}$  with adjacency matrix  $A = \{a_{i,j}\}_{i,j=1}^{m}$  induces the graphon  $W_G$ , defined by  $W_G(x,y) = a_{[xm], [ym]}$ . Note that  $W_G$  is piecewise constant on the partition  $\mathcal{I}_m$ . We hence identify graphs with their induced graphons. A graphon can also be seen as a generative model of graphs. Given a graphon  $W$ , a corresponding random graph is generated by sampling i.i.d. nodes  $\{X_n\}$  from the graphon domain, and connecting each pair  $X_n, X_m$  in probability  $W(X_n, X_m)$  to obtain the edges of the graph.
2.4 Regularity lemma for graphons
A simple way to formulate the regularity lemma in the graphon language is via stochastic block models (SBM). A SBM is a piecewise constant graphon, defined on a partition of the graphon domain [0, 1]. The number of classes of the SBM is defined to be the number of sets in the partition. A SBM is seen as a generative model for graphs, where graphs are randomly sampled from the graphon underlying the SBM, as explained above. Szemerédi weak regularity lemma asserts that for any error tolerance  $\epsilon$ , there is a number of classes  $k$ , such that any deterministic graph (of any size and topology) behaves as if it was randomly sampled from a SBM with  $k$  classes, up to error  $\epsilon$ . Hence, in some sense, every graph is approximately quasi-random.
To write the weak regularity lemma in the graphon language, the notion of irregularity (1) is extended to graphons. For any measurable  $W:[0,1]^2\to \mathbb{R}$  the cut norm is defined to be
$$
\| W \| _ {\square} = \sup  _ {U, S \subset [ 0, 1 ]} \left| \int_ {U \times S} W (x, y) d x d y \right|,
$$
where  $U, S \subset [0,1]$  are measurable. It can be verified that the irregularity (1) is equal to the cut norm of a difference between graphons induced by adequate graphs. The cut metric between two graphons  $W, V \in \mathcal{W}_0$  is defined to be  $d_{\square}(W,V) = \| W - V\|_{\square}$ . The cut distance is defined to be
$$
\delta_ {\square} (W, V) = \inf  _ {\phi \in S _ {[ 0, 1 ]}} \| W - V ^ {\phi} \| _ {\square},
$$
where  $S_{[0,1]}$  is the space of measure-preserving bijections  $[0,1] \to [0,1]$ , and  $V^{\phi}(x,y) = V(\phi(x),\phi(y))$  (see Section 3.1 and Appendix A.3 for more details). The cut distance is a pseudo metric on the space of graphons. By considering equivalence classes of graphons with zero cut distance, we can construct a metric space  $\widetilde{\mathcal{W}_0}$  for which  $\delta_\square$  is a metric. The following version of the weak regularity lemma is from [25, Lemma 7].
Theorem 2.2. For every graphon  $W \in \mathcal{W}_0$  and  $\epsilon > 0$  there exists a step graphon  $W' \in \mathcal{W}_0$  with respect to a partition of at most  $\lceil 2^{c / \epsilon^2} \rceil$  sets such that  $\delta_{\square}(W, W') \leq \epsilon$ , for some universal constant  $c$ .
The exact definition of a step graphon is given in Definition 3.3. It is possible to show, using Theorem 2.2, that  $\widetilde{\mathcal{W}_0}$  is a compact metric space [25, Lemma 8]. Instead of recalling this construction here, we refer to Section 3.4 for the extension of this construction to graphon-signals.
In the definition of  $W_{G}$ , the convention is that  $\lceil 0\rceil = 1$ .
3 Graphon-signal analysis
A graph-signal  $(G, \mathbf{f})$  is a graph  $G$ , that may be weighted or simple, with node set  $[n]$ , and a signal  $\mathbf{f} \in \mathbb{R}^{n \times k}$  that assigns the value  $f_j \in \mathbb{R}^k$  for every node  $j \in [n]$ . A graphon-signal will be defined in Section 3.1 similarly to a graph-signal, but over the node set  $[0, 1]$ . In this section, we show how to extend classical results in graphon theory to a so-called graphon-signal theory. All proofs are given in the appendix.
3.1 The graphon signal space
For any  $r > 0$ , define the signal space
$$
\mathcal {L} _ {r} ^ {\infty} [ 0, 1 ] := \left\{f \in \mathcal {L} ^ {\infty} [ 0, 1 ] \mid \forall x \in [ 0, 1 ], | f (x) | \leq r \right\}. \tag {2}
$$
We define the following "norm" on  $\mathcal{L}_r^\infty [0,1]$  (which is not a vector space).
Definition 3.1 (Cut norm of a signal). For a signal  $f:[0,1]\to \mathbb{R}$ , the cut norm  $\| f\|_{\square}$  is defined as
$$
\| f \| _ {\square} := \sup  _ {S \subseteq [ 0, 1 ]} \left| \int_ {S} f (x) d \mu (x) \right|, \tag {3}
$$
where the supremum is taken over the measurable subsets  $S\subset [0,1]$
In Appendix A.2 we prove basic properties of signal cut norm. One important property is the equivalence of the signal cut norm to the  $L_{1}$  norm
$$
\forall f \in \mathcal {L} _ {r} ^ {\infty} [ 0, 1 ], \| f \| _ {\square} \leq \| f \| _ {1} \leq 2 \| f \| _ {\square}.
$$
Given a bound  $r$  on the signals, we define the space of graphon-signals to be the set of pairs  $\mathcal{WL}_r \coloneqq \mathcal{W}_0 \times \mathcal{L}_r^\infty[0,1]$ . We define the graphon-signal cut norm, for measurable  $W, V : [0,1]^2 \to \mathbb{R}$  and  $f, g : [0,1] \to \mathbb{R}$ , by
$$
\| (W, f) \| _ {\square} = \| W \| _ {\square} + \| f \| _ {\square}.
$$
We define the graphon-signal cut metric by  $d_{\square}\big((W,f),(V,g)\big) = \| (W,f) - (V,g)\|_{\square}$ .
We next define a pseudo metric that makes the space of graphon-signals a compact space. Let  $S_{[0,1]}'$  be the set of measurable measure preserving bijections between co-null sets of [0, 1], namely,
$$
S _ {[ 0, 1 ]} ^ {\prime} = \left\{\phi : A \to B \mid A, B \text {c o - n u l l i n} [ 0, 1 ], \text {a n d} \forall S \in A, \mu (S) = \mu (\phi (S)) \right\},
$$
where  $\phi$  is a measurable bijection and  $A, B, S$  are measurable. For  $\phi \in S_{[0,1]}'$ , we define  $W^{\phi}(x,y) \coloneqq W(\phi(x),\phi(y))$ , and  $f^{\phi}(z) = f(\phi(z))$ . Note that  $W^{\phi}$  and  $f^{\phi}$  are only defined up to a null-set, and we arbitrarily set  $W, W^{\phi}, f$  and  $f^{\phi}$  to 0 in their respective null-sets, which does not affect our analysis. Define the cut distance between two graphon-signals  $(W,f)$ ,  $(V,g) \in \mathcal{WL}_r$  by
$$
\delta_ {\square} \big ((W, f), (V, g) \big) = \inf  _ {\phi \in S _ {[ 0, 1 ]} ^ {\prime}} d _ {\square} \big ((W, f), (V, g) ^ {\phi} \big). \tag {4}
$$
Here,  $(V,g)^{\phi} \coloneqq (V^{\phi},g^{\phi})$ . More details on this construction are given in Appendix A.3.
The graphon-signal cut distance  $\delta_{\square}$  is a pseudo-metric, and can be made into a metric by introducing the equivalence relation:  $(W,f)\sim (V,g)$  if  $\delta_{\square}((W,f),(V,g)) = 0$ . The quotient space  $\widetilde{\mathcal{WL}_r} := \mathcal{WL}_r / \sim$  of equivalence classes  $[(W,f)]$  of graphon-signals  $(W,f)$  is a metric space with the metric  $\delta_{\square}([W,f]),[(V,g)]) = \delta_{\square}((W,f),(V,g))$ . By abuse of terminology, we call elements of  $\widetilde{\mathcal{WL}_r}$  also graphon-signals. A graphon-signal in  $\widetilde{\mathcal{WL}_r}$  is defined irrespective of a specific "indexing" of the nodes in [0, 1].
3.2 Induced graphon-signals
Any graph-signal can be identified with a corresponding graphon-signal as follows.
Definition 3.2. Let  $(G,\mathbf{f})$  be a graph-signal with node set  $[n]$  and adjacency matrix  $A = \{a_{i,j}\}_{i,j\in [n]}$ . Let  $\{I_k\}_{k = 1}^n$  with  $I_{k} = [(k - 1) / n,k / n)$  be the equipartition of  $[0,1]$  into  $n$  intervals. The graphonsignal  $(W,f)_{(G,\mathbf{f})} = (W_G,f_{\mathbf{f}})$  induced by  $(G,\mathbf{f})$  is defined by
$$
W _ {G} (x, y) = \sum_ {i, j = 1} ^ {n} a _ {i j} \mathbb {1} _ {I _ {i}} (x) \mathbb {1} _ {I _ {j}} (y), \quad \text {a n d} \quad f _ {\mathbf {f}} (z) = \sum_ {i} ^ {n} f _ {i} \mathbb {1} _ {I _ {i}} (z).
$$
We denote  $(W, f)_{(G, \mathbf{f})} = (W_G, f_{\mathbf{f}})$ . We identify any graph-signal with its induced graphon-signal. This way, we define the cut distance between a graph-signal and a graphon-signal. As before, the cut distance between a graph-signal  $(G, \mathbf{f})$  and a graphon-signal  $(W, g)$  can be interpreted as how much the deterministic graph-signal  $(G, \mathbf{f})$  "looks like" it was randomly sampled from  $(W, g)$ .
3.3 Graphon-signal regularity lemma
(a)
Figure 2: Illustration of the graphon-signal regularity lemma. The values of the graphon are in gray scale over  $[0,1]^2$ , and the signal is plotted in color on the diagonal of  $[0,1]^2$ . (a) A graphon-signal. (b) Representation of the same graphon-signal under the "good" permutation/measure preserving bijection guaranteed by the regularity lemma. (c) The approximating step graphon-signal guaranteed by the regularity lemma.
To formulate our regularity lemma, we first define spaces of step functions.
Definition 3.3. Given a partition  $\mathcal{P}_k$ , and  $d \in \mathbb{N}$ , we define the space  $\mathcal{S}_{\mathcal{P}_k}^d$  of step functions of dimension  $d$  over the partition  $\mathcal{P}_k$  to be the space of functions  $F: [0,1]^d \to \mathbb{R}$  of the form
$$
F \left(x _ {1}, \dots , x _ {d}\right) = \sum_ {j = \left(j _ {1}, \dots , j _ {d}\right) \in [ k ] ^ {d}} c _ {j} \prod_ {l = 1} ^ {d} \mathbb {1} _ {P _ {j _ {l}}} \left(x _ {l}\right), \tag {5}
$$
for any choice of  $\{c_j\in \mathbb{R}\}_{j\in [k]^d}$
We call any element of  $\mathcal{W}_0\cap \mathcal{S}_{\mathcal{P}_k}^2$  a step graphon with respect to  $\mathcal{P}_k$ . A step graphon is also called a stochastic block model (SBM). We call any element of  $\mathcal{L}_r^\infty [0,1]\cap \mathcal{S}_{\mathcal{P}_k}^1$  a step signal. We also call  $[\mathcal{WL}_r]_{\mathcal{P}_k}\coloneqq (\mathcal{W}_0\cap \mathcal{S}_{\mathcal{P}_k}^2)\times (\mathcal{L}_r^\infty [0,1]\cap \mathcal{S}_{\mathcal{P}_k}^1)$  the space of SBMs with respect to  $\mathcal{P}_k$ .
In Appendix B.2 we give a number of versions of the graphon-signal regularity lemma. Here, we show one version in which the partition is fixed regardless of the graphon-signal.
Theorem 3.4 (Regularity lemma for graphon-signals - equipartition). For any  $c > 1$ , and any sufficiently small  $\epsilon > 0$ , for every  $n \geq 2^{\lceil \frac{9c}{4\epsilon^2} \rceil}$  and every  $(W, f) \in \mathcal{WL}_r$ , there exists a step graphon-signal  $(W_n, f_n) \in [\mathcal{WL}_r]_{\mathcal{I}_n}$  such that
$$
\delta_ {\square} \left(\left(W, f\right), \left(W _ {n}, f _ {n}\right)\right) \leq \epsilon , \tag {6}
$$
where  $\mathcal{I}_n$  is the equipartition of [0, 1] into  $n$  intervals.
Figure 2 illustrates the graphon-signal regularity lemma. By identifying graph-signals with their induced graphon-signals, (6) shows that the space of graph-signals is dense in the space of graphon-signals with cut distance.