SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\mathbb {E} (i k / n - k a _ {i}) ^ {2} = \mathbb {V} (k a _ {i}) = k (i / n) (1 - i / n).
$$
As a result
$$
\begin{array}{l} \mathbb {E} \| W _ {n} - W _ {n} (\Lambda) ^ {\phi} \| _ {\square} \leq 5 \sum_ {i = 1} ^ {n} \sqrt {\frac {(i / n) (1 - i / n)}{k}} \\ \leq 2 \int_ {1} ^ {n} \sqrt {\frac {(i / n) (1 - i / n)}{k}} d i, \\ \end{array}
$$
and for  $n > 10$
$$
\leq 5 \frac {n}{\sqrt {k}} \int_ {0} ^ {1. 1} \sqrt {z - z ^ {2}} d z \leq 5 \frac {n}{\sqrt {k}} \int_ {0} ^ {1. 1} \sqrt {z} d z \leq 1 0 / 3 (1. 1) ^ {3 / 2} \frac {n}{\sqrt {k}} <   4 \frac {n}{\sqrt {k}}.
$$
Now, by  $n = \left\lceil \frac{\sqrt{k}}{r\log(k)}\right\rceil \leq \frac{\sqrt{k}}{r\log(k)} +1$  , for large enough  $k$
$$
\mathbb {E} \| W _ {n} - W _ {n} (\Lambda) ^ {\phi} \| _ {\square} \leq 4 \frac {1}{r \log (k)} + 4 \frac {1}{\sqrt {k}} \leq \frac {5}{r \log (k)}.
$$
Similarly,
$$
\mathbb {E} \| f _ {n} - f _ {n} (\Lambda) ^ {\phi} \| _ {1} \leq \frac {5}{\log (k)}.
$$
Note that in the proof of Corollary B.7, in (16),  $\alpha$  is chosen close to 1, and especially, for small enough  $\epsilon$ ,  $\alpha > 1/2$ . Hence, for large enough  $k$ ,
$$
\begin{array}{l} \mathbb {E} \left(d _ {\square} \left(W, W (\Lambda) ^ {\phi}\right)\right) \leq d _ {\square} \left(W, W _ {n}\right) + \mathbb {E} \left(d _ {\square} \left(W _ {n}, W _ {n} (\Lambda) ^ {\phi}\right)\right) + \mathbb {E} \left(d _ {\square} \left(W _ {n} (\Lambda), W (\Lambda)\right)\right) \\ \leq \alpha \frac {2 \sqrt {2}}{\sqrt {\log (k) - 2 \log (\log (k)) - 2 \log (r)}} + \frac {5}{r \log (k)} + \frac {1 4}{k ^ {1 / 4}} \\ + \alpha \frac {2 \sqrt {2}}{\sqrt {\log (k) - 2 \log (\log (k)) - 2 \log (r)}} \\ \leq \alpha \frac {6}{\sqrt {\log (k)}}, \\ \end{array}
$$
Similarly, for each  $k$ , if  $1 - \alpha < \frac{1}{\sqrt{\log(k)}}$ , then
$$
\begin{array}{l} \mathbb {E} (d _ {\square} (f, f (\Lambda) ^ {\phi})) \leq (1 - \alpha) \frac {2 \sqrt {2}}{\sqrt {\log (k) - 2 \log (\log (k)) - 2 \log (r)}} + \frac {5}{\log (k)} \\ + \frac {2 r}{k ^ {1 / 2}} + (1 - \alpha) \frac {4 \sqrt {2}}{\sqrt {\log (k) - 2 \log (\log (k)) - 2 \log (r)}} \leq \frac {1 4}{\log (k)}. \\ \end{array}
$$
Moreover, for each  $k$  such that  $1 - \alpha > \frac{1}{\sqrt{\log(k)}}$ , if  $k$  is large enough (where the lower bound of  $k$  depends on  $r$ ), we have
$$
\frac {5}{\log (k)} + \frac {2 r}{k ^ {1 / 2}} <   \frac {5 . 5}{\log (k)} <   \frac {1}{\sqrt {\log (k)}} \frac {6}{\sqrt {\log (k)}} <   (1 - \alpha) \frac {6}{\sqrt {\log (k)}}
$$
so, by  $6\sqrt{2} < 9$
$$
\begin{array}{l} \mathbb {E} \left(d _ {\square} \left(f, f (\Lambda) ^ {\phi}\right)\right) \leq (1 - \alpha) \frac {2 \sqrt {2}}{\sqrt {\log (k) - 2 \log \left(\log (k)\right) - 2 \log (r)}} + \frac {2}{\log (k)} \\ + \frac {2 r}{k ^ {1 / 2}} + (1 - \alpha) \frac {4 \sqrt {2}}{\sqrt {\log (k) - 2 \log (\log (k)) - 2 \log (r)}} \\ \leq (1 - \alpha) \frac {1 5}{\sqrt {\log (k)}}. \\ \end{array}
$$
Lastly, by Corollary D.2,
$$
\begin{array}{l} \mathbb {E} \left(d _ {\square} \left(W, \mathbb {G} (W, \Lambda) ^ {\phi}\right)\right) \leq \mathbb {E} \left(d _ {\square} \left(W, W (\Lambda) ^ {\phi}\right)\right) + \mathbb {E} \left(d _ {\square} \left(W (\Lambda) ^ {\phi}, \mathbb {G} (W, \Lambda) ^ {\phi}\right)\right) \\ \leq \alpha \frac {6}{\sqrt {\log (k)}} + \frac {1 1}{\sqrt {k}} \leq \alpha \frac {7}{\sqrt {\log (k)}}, \\ \end{array}
$$
As a result, for large enough  $k$ ,
$$
\mathbb {E} \left(\delta_ {\square} \big ((W, f), (W (\Lambda), f (\Lambda)) \big)\right) <   \frac {1 5}{\sqrt {\log (k)}},
$$
E Graphon-signal MPNNs
In this appendix we give properties and examples of MPNNs.
E.1 Properties of graphon-signal MPNNs
Consider the construction of MPNN from Section 4.1. We first explain how a MPNN on a grpah is equivalent to a MPNN on the induced graphon.
Let  $G$  be a graph of  $n$  nodes, with adjacency matrix  $A = \{a_{i,j}\}_{i,j\in [n]}$  and signal  $\mathbf{f}\in \mathbb{R}^{n\times d}$ . Consider a MPL  $\theta$ , with receiver and transmitter message functions  $\xi_{\mathrm{r}}^{k},\xi_{\mathrm{t}}^{k}:\mathbb{R}^{d}\to \mathbb{R}^{p}$ , for  $k\in [K]$ , where  $K\in \mathbb{N}$ , and update function  $\mu :\mathbb{R}^{d + p}\rightarrow \mathbb{R}^s$ . The application of the MPL on  $(G,\mathbf{f})$  is defined as follows. We first define the message kernel  $\Phi_{\mathbf{f}}:[n]^2\to \mathbb{R}^p$ , with entries
$$
\Phi_ {\mathbf {f}} (i, j) = \Phi (\mathbf {f} _ {i}, \mathbf {f} _ {j}) = \sum_ {k = 1} ^ {K} \xi_ {\mathrm {r}} ^ {k} (\mathbf {f} _ {i}) \xi_ {\mathrm {t}} ^ {k} (\mathbf {f} _ {j}).
$$
We then aggregate the message kernel with normalized sum aggregation
$$
\left(\operatorname {A g g} (G, \Phi_ {\mathbf {f}})\right) _ {i} = \frac {1}{n} \sum_ {j \in [ n ]} a _ {i, j} \Phi_ {\mathbf {f}} (i, j).
$$
Lastly, we apply the update function, to obtain the output  $\theta (G,\mathbf{f})$  of the MPL with value at each node  $i$
$$
\theta (G, \mathbf {f}) _ {i} = \eta \Big (\mathbf {f} _ {i}, \big (\mathrm {A g g} (G, \Phi_ {\mathbf {f}}) \big) _ {i} \Big) \in \mathbb {R} ^ {s}.
$$
Lemma E.1. Consider a MPL  $\theta$  as in the above setting. Then, for every graph signal  $(G,A,\mathbf{f})$ ,
$$
\theta \Big ((W, f) _ {(G, \mathbf {f})} \Big) = (W, f) _ {\theta (G, \mathbf {f})}.
$$
Proof. Let  $\{I_i, \ldots, I_n\}$  be the equipartition to intervals. For each  $j \in [n]$ , let  $y_j \in I_j$  be an arbitrary point. Let  $i \in [n]$  and  $x \in I_i$ . We have
$$
\begin{array}{l} \operatorname {A g g} (G, \Phi_ {\mathbf {f}}) _ {i} = \frac {1}{n} \sum_ {j \in [ n ]} a _ {i, j} \Phi_ {\mathbf {f}} (i, j) = \frac {1}{n} \sum_ {j \in [ n ]} W _ {G} (x, y _ {j}) \Phi_ {f _ {\mathbf {f}}} (x, y _ {j}) \\ = \int_ {0} ^ {1} W _ {G} (x, y) \Phi_ {f _ {\mathbf {f}}} (x, y) d y = \operatorname {A g g} (W _ {G}, \Phi_ {f _ {\mathbf {f}}}) (x). \\ \end{array}
$$
Therefore, for every  $i \in [n]$  and every  $x \in I_i$ ,
$$
\begin{array}{l} f _ {\theta (G, \mathbf {f})} (x) = f _ {\eta \left(\mathbf {f}, \operatorname {A g g} (G, \Phi_ {\mathbf {f}})\right)} (x) = \eta \left(\mathbf {f} _ {i}, \operatorname {A g g} (G, \Phi_ {\mathbf {f}}) _ {i}\right) \\ = \eta \left(f _ {\mathbf {f}} (x), \operatorname {A g g} \left(W _ {G}, \Phi_ {f _ {\mathbf {f}}} (x)\right)\right) = \theta \left(W _ {G}, f _ {\mathbf {f}}\right) (x). \\ \end{array}
$$
E.2 Examples of MPNNs
The GIN convolutional layer [36] is defined as follows. First, the message function is
$$
\Phi (a, b) = b
$$
and the update function is
$$
\eta (x, y) = M \big ((1 + \epsilon) x + y \big).
$$
where  $M$  is a multi-layer perceptron (MLP) and  $\epsilon$  a constant. Each layer may have a different MLP and different constant  $\epsilon$ . The standard GIN is defined with sum aggregation, but we use normalized sum aggregation.
Given a graph-signal  $(G, \mathbf{f})$ , with  $\mathbf{f} \in \mathbb{R}^{n \times d}$  with adjacency matrix  $A \in \mathbb{R}^{n \times n}$ , a spectral convolutional layer based on a polynomial filter  $p(\lambda) = \sum_{j=0}^{J} \lambda^j C_j$ , where  $C_j \in \mathbb{R}^{d \times p}$ , is defined to be
$$
p (A) \mathbf {f} = \sum_ {j = 0} ^ {J} \frac {1}{n ^ {j}} A ^ {j} \mathbf {f} C _ {j},
$$
followed by a pointwise non-linearity like ReLU. Such a convolutional layer can be seen as  $J + 1$  MPLs. We first apply  $J$  MPLs, where each MPL is of the form
$$
\theta (\mathbf {f}) = \left(\mathbf {f}, \frac {1}{n} A \mathbf {f}\right).
$$
We then apply an update layer
$$
U (\mathbf {f}) = \mathbf {f} C
$$
for some  $C \in \mathbb{R}^{(J + 1)d \times p}$ , followed by the pointwise non-linearity. The message part of  $\theta$  can be written in our formulation with  $\Phi(a, b) = b$ , and the update part of  $\theta$  with  $\eta(c, d) = (c, d)$ . The last update layer  $U$  is linear followed by the pointwise non-linearity.
F Lipschitz continuity of MPNNs
In this appendix we prove Theorem 4.1. For  $v \in \mathbb{R}^d$ , we often denote by  $|v| = \| v\|_{\infty}$ . We define the  $L_{1}$  norm of a measurable function  $h:[0,1] \to \mathbb{R}^d$  by
$$
\| h \| _ {1} := \int_ {0} ^ {1} | h (x) | d x = \int_ {0} ^ {1} \| h (x) \| _ {\infty} d x.
$$
$$
\| h \| _ {\infty} := \sup  _ {x \in \mathbb {R} ^ {d}} | h (x) | = \sup  _ {x \in \mathbb {R} ^ {d}} \| h (x) \| _ {\infty}.
$$
We define Lipschitz continuity with respect to the infinity norm. Namely,  $Z: \mathbb{R}^d \to \mathbb{R}^c$  is called Lipschitz continuous with Lipschitz constant  $L$  if
$$
| Z (x) - Z (y) | = \| Z (x) - Z (y) \| _ {\infty} \leq L \| x - z \| _ {\infty} = L | x - z |.
$$
We denote the minimal Lipschitz bound of the function  $Z$  by  $L_Z$ .
We extend  $\mathcal{L}_r^\infty [0,1]$  to the space of functions  $f:[0,1]\to \mathbb{R}^d$  with the above  $L_{1}$  norm.
Define the space  $\mathcal{K}_q$  of kernels bounded by  $q > 0$  to be the space of measurable functions
$$
K: [ 0, 1 ] ^ {2} \to [ - q, q ].
$$
The cut norm, cut metric, and cut distance are defined as usual for kernels in  $\mathcal{K}_q$ .
F.1 Lipschitz continuity of message passing and update layers
In this subsection we prove that message passing layers and update layers are Lipschitz continuous with respect to the graphon-signal cut metric.
Lemma F.1 (Product rule for message kernels). Let  $\Phi_f, \Phi_g$  be the message kernels corresponding to the signals  $f, g$ . Then
$$
\| \Phi_ {f} - \Phi_ {g} \| _ {L ^ {1} [ 0, 1 ] ^ {2}} \leq \sum_ {k = 1} ^ {K} \left(L _ {\xi_ {r} ^ {k}} \| \xi_ {t} ^ {k} \| _ {\infty} + \| \xi_ {r} ^ {k} \| _ {\infty} L _ {\xi_ {t} ^ {k}}\right) \| f - g \| _ {1}.
$$
Proof. Suppose  $p = 1$  For every  $x, y \in [0,1]^2$
$$
\begin{array}{l} | \Phi_ {f} (x, y) - \Phi_ {g} (x, y) | = \left| \sum_ {k = 1} ^ {K} \xi_ {r} ^ {k} (f (x)) \xi_ {t} ^ {k} (f (y)) - \sum_ {k = 1} ^ {K} \xi_ {r} ^ {k} (g (x)) \xi_ {t} ^ {k} (g (y)) \right| \\ \leq \sum_ {k = 1} ^ {K} \left| \xi_ {r} ^ {k} (f (x)) \xi_ {t} ^ {k} (f (y)) - \xi_ {r} ^ {k} (g (x)) \xi_ {t} ^ {k} (g (y)) \right| \\ \leq \sum_ {k = 1} ^ {K} \left(\left| \xi_ {r} ^ {k} (f (x)) \xi_ {t} ^ {k} (f (y)) - \xi_ {r} ^ {k} (g (x)) \xi_ {t} ^ {k} (f (y)) \right| + \left| \xi_ {r} ^ {k} (g (x)) \xi_ {t} ^ {k} (f (y)) - \xi_ {r} ^ {k} (g (x)) \xi_ {t} ^ {k} (g (y)) \right|\right) \\ \leq \sum_ {k = 1} ^ {K} \left(L _ {\xi_ {r} ^ {k}} | f (x) - g (x) | \left| \xi_ {t} ^ {k} (f (y)) \right| + \left| \xi_ {r} ^ {k} (g (x)) \right| L _ {\xi_ {t} ^ {k}} | f (y) - g (y) |\right). \\ \end{array}
$$
$$
\begin{array}{l} \left\| \Phi_ {f} - \Phi_ {g} \right\| _ {L ^ {1} [ 0, 1 ] ^ {2}} \\ \leq \sum_ {k = 1} ^ {K} \int_ {0} ^ {1} \int_ {0} ^ {1} \left(L _ {\xi_ {r} ^ {k}} | f (x) - g (x) | \left| \xi_ {t} ^ {k} (f (y)) \right| + \left| \xi_ {r} ^ {k} (g (x)) \right| L _ {\xi_ {t} ^ {k}} | f (y) - g (y) |\right) d x d y \\ \leq \sum_ {k = 1} ^ {K} \left(L _ {\xi_ {r} ^ {k}} \| f - g \| _ {1} \| \xi_ {t} ^ {k} \| _ {\infty} + \| \xi_ {r} ^ {k} \| _ {\infty} L _ {\xi_ {t} ^ {k}} \| f - g \| _ {1}\right) \\ = \sum_ {k = 1} ^ {K} \left(L _ {\xi_ {r} ^ {k}} \| \xi_ {t} ^ {k} \| _ {\infty} + \| \xi_ {r} ^ {k} \| _ {\infty} L _ {\xi_ {t} ^ {k}}\right) \| f - g \| _ {1}. \\ \end{array}
$$
Lemma F.2. Let  $Q, V$  be two message kernels, and  $W \in \mathcal{W}_0$ . Then
$$
\left\| \operatorname {A g g} (W, Q) - \operatorname {A g g} (W, V) \right\| _ {1} \leq \| Q - V \| _ {1}.
$$
$$
\operatorname {A g g} (W, Q) (x) - \operatorname {A g g} (W, V) (x) = \int_ {0} ^ {1} W (x, y) (Q (x, y) - V (x, y)) d y
$$
$$
\begin{array}{l} \| \operatorname {A g g} (W, Q) - \operatorname {A g g} (W, V) \| _ {1} = \int_ {0} ^ {1} \left| \int_ {0} ^ {1} W (x, y) (Q (x, y) - V (x, y)) d y \right| d x \\ \leq \int_ {0} ^ {1} \int_ {0} ^ {1} | W (x, y) (Q (x, y) - V (x, y)) | d y d x \\ \leq \int_ {0} ^ {1} \int_ {0} ^ {1} | (Q (x, y) - V (x, y)) | d y d x = \| Q - V \| _ {1}. \\ \end{array}
$$
As a result of Lemma F.2 and the product rule Lemma F.1, we have the following corollary, that computes the error in aggregating two message kernels with the same graphon.
Corollary F.3.
$$
\| \operatorname {A g g} (W, \Phi_ {f}) - \operatorname {A g g} (W, \Phi_ {g}) \| _ {1} \leq \sum_ {k = 1} ^ {K} \left(L _ {\xi_ {r} ^ {k}} \| \xi_ {t} ^ {k} \| _ {\infty} + \| \xi_ {r} ^ {k} \| _ {\infty} L _ {\xi_ {t} ^ {k}}\right) \| f - g \| _ {1}.
$$
Next we fix the message kernel, and bound the difference between the aggregation of the message kernel with respect to two different graphons. Let  $L^{+}[0,1]$  be the space of measurable function  $f:[0,1]\to [0,1]$ . The following lemma is a trivial extension of [24, Lemma 8.10] from  $\mathcal{K}_1$  to  $\mathcal{K}_r$ .
Lemma F.4. For any kernel  $Q \in \mathcal{K}_r$
$$
\| Q \| _ {\square} = \sup  _ {f, g \in L ^ {+} [ 0, 1 ]} \left| \int_ {[ 0, 1 ] ^ {2}} f (x) Q (x, y) g (y) d x d y \right|,
$$
where the supremum is attained for some  $f,g\in L^{+}[0,1]$
The following Lemma is proven as part of the proof of [24, Lemma 8.11].
Lemma F.5. For any kernel  $Q \in \mathcal{K}_r$
$$
\sup  _ {f, g \in L _ {1} ^ {\infty} [ 0, 1 ]} \left| \int_ {[ 0, 1 ] ^ {2}} f (x) Q (x, y) g (y) d x d y \right| \leq 4 \| Q \| _ {\square}.
$$
For completeness, we give here a self-contained proof.
Proof. Any function  $f \in L_1^\infty[0,1]$  can be written as  $f = f_+ - f_-$ , where  $f_+, f_- \in L^+[0,1]$ . Hence, by Lemma F.4,
$$
\begin{array}{l} \sup  _ {f, g \in L _ {1} ^ {\infty} [ 0, 1 ]} \left| \int_ {[ 0, 1 ] ^ {2}} f (x) Q (x, y) g (y) d x d y \right| \\ = \sup  _ {f _ {+}, f _ {-}, g _ {+}, g _ {-} \in L ^ {+} [ 0, 1 ]} \left| \int_ {[ 0, 1 ] ^ {2}} (f _ {+} (x) - f _ {-} (x)) Q (x, y) (g _ {+} (y) - g _ {-} (y)) d x d y \right| \\ \leq \sum_ {s \in \{+, - \}} \sup  _ {f _ {s}, g _ {s} \in L ^ {+} [ 0, 1 ]} \left| \int_ {[ 0, 1 ] ^ {2}} f _ {s} (x) Q (x, y) g _ {s} (y) d x d y \right| = 4 \| Q \| _ {\square}. \\ \end{array}
$$
Next we state a simple lemma.
Lemma F.6. Let  $f = f_{+} - f_{-}$  be a signal, where  $f_{+}, f_{-} : [0,1] \to (0,\infty)$  are measurable. Then the supremum in the cut norm  $\| f \|_{\square} = \sup_{S \subset [0,1]} \left| \int_{S} f(x) dx \right|$  is attained as the support of either  $f_{+}$  or  $f_{-}$ .
Lemma F.7. Let  $f \in \mathcal{L}_r^\infty[0,1]$ ,  $W, V \in \mathcal{W}_0$ , and suppose that  $\left|\xi_{\mathrm{r}}^{k}(f(x))\right|, \left|\xi_{\mathrm{t}}^{k}(f(x))\right| \leq \rho$  for every  $x \in [0,1]$  and  $k = 1, \ldots, K$ . Then
$$
\left\| \operatorname {A g g} \left(W, \Phi_ {f}\right) - \operatorname {A g g} \left(V, \Phi_ {f}\right) \right\| _ {\square} \leq 4 K \rho^ {2} \| W - V \| _ {\square}.
$$
Moreover, if  $\xi_{\mathrm{r}}^{k}$  and  $\xi_{\mathrm{t}}^{k}$  are non-negatively valued for every  $k = 1,\dots ,K$  , then
$$
\left\| \operatorname {A g g} \left(W, \Phi_ {f}\right) - \operatorname {A g g} \left(V, \Phi_ {f}\right) \right\| _ {\square} \leq K \rho^ {2} \| W - V \| _ {\square}.
$$
Proof. Let  $T = W - V$ . Let  $S$  be the maximizer of the supremum underlying the cut norm of  $\operatorname{Agg}(T, \Phi_f)$ . Suppose without loss of generality that  $\int_S \operatorname{Agg}(T, \Phi_f)(x) dx > 0$ . Denote  $q_{\mathrm{r}}^k(x) = \xi_{\mathrm{r}}^k(f(x))$  and  $q_{\mathrm{t}}^k(x) = \xi_{\mathrm{t}}^k(f(x))$ . We have
$$
\begin{array}{l} \int_ {S} \left(\operatorname {A g g} (W, \Phi_ {f}) (x) - \operatorname {A g g} (V, \Phi_ {f}) (x)\right) d x = \int_ {S} \operatorname {A g g} (T, \Phi_ {f}) (x) d x \\ = \sum_ {k = 1} ^ {K} \int_ {S} \int_ {0} ^ {1} q _ {\mathrm {r}} ^ {k} (x) T (x, y) q _ {\mathrm {t}} ^ {k} (y) d y d x. \\ \end{array}
$$
$$
v _ {\mathrm {r}} ^ {k} (x) = \left\{ \begin{array}{c c} q _ {\mathrm {r}} ^ {k} (x) / \rho & x \in S \\ 0 & x \notin S. \end{array} \right. \tag {25}
$$
Moreover, define  $v_{\mathrm{t}}^{k} = q_{\mathrm{t}}^{k} / \rho$ , and note that  $v_{\mathrm{r}}^{k}, v_{\mathrm{t}}^{k} \in L_{1}^{\infty}[0,1]$ . We hence have, by Lemma F.5,
$$
\begin{array}{l} \int_ {S} \operatorname {A g g} (T, \Phi_ {f}) (x) d x = \sum_ {k = 1} ^ {K} \rho^ {2} \int_ {0} ^ {1} \int_ {0} ^ {1} v _ {\mathrm {r}} ^ {k} (x) T (x, y) v _ {\mathrm {t}} ^ {k} (y) d y d x \\ \leq \sum_ {k = 1} ^ {K} \rho^ {2} \left| \int_ {0} ^ {1} \int_ {0} ^ {1} v _ {\mathrm {r}} ^ {k} (x) T (x, y) v _ {\mathrm {t}} ^ {k} (y) d y d x \right| \\ \leq 4 K \rho^ {2} \| T \| _ {\square}. \\ \end{array}
$$
$$
\left\| \operatorname {A g g} \left(W, \Phi_ {f}\right) - \operatorname {A g g} \left(V, \Phi_ {f}\right) \right\| _ {\square} \leq 4 K \rho^ {2} \| T \| _ {\square}
$$
Lastly, in case  $\xi_{\mathrm{r}}^{k},\xi_{\mathrm{t}}^{k}$  are nonnegatively valued, so are  $q_{\mathrm{r}}^{k},q_{\mathrm{t}}^{k}$ , and hence by Lemma F.4,
$$
\int_ {S} \operatorname {A g g} (T, \Phi_ {f}) (x) d x \leq K \rho^ {2} \| T \| _ {\square}.
$$
Theorem F.8. Let  $(W, f), (V, g) \in \mathcal{WL}_r$ , and suppose that  $|\xi_{\mathrm{r}}^k(f(x))|, |\xi_{\mathrm{t}}^k(f(x))| \leq \rho$  and  $L_{\xi_{\mathrm{t}}^k}, L_{\xi_{\mathrm{t}}^k} < L$  for every  $x \in [0,1]$  and  $k = 1, \ldots, K$ . Then,
$$
\left\| \operatorname {A g g} \left(W, \Phi_ {f}\right) - \operatorname {A g g} \left(V, \Phi_ {g}\right) \right\| _ {\square} \leq 4 K L \rho \| f - g \| _ {\square} + 4 K \rho^ {2} \| W - V \| _ {\square}.
$$
Proof. By Lemma F.1, Lemma F.2 and Lemma F.7,
$$
\begin{array}{l} \left\| \operatorname {A g g} \left(W, \Phi_ {f}\right) - \operatorname {A g g} \left(V, \Phi_ {g}\right) \right\| _ {\square} \\ \leq \left\| \operatorname {A g g} \left(W, \Phi_ {f}\right) - \operatorname {A g g} \left(W, \Phi_ {g}\right) \right\| _ {\square} + \left\| \operatorname {A g g} \left(W, \Phi_ {g}\right) - \operatorname {A g g} \left(V, \Phi_ {g}\right) \right\| _ {\square} \\ \leq \sum_ {k = 1} ^ {K} \left(L _ {\xi_ {r} ^ {k}} \| \xi_ {t} ^ {k} \| _ {\infty} + \| \xi_ {r} ^ {k} \| _ {\infty} L _ {\xi_ {t} ^ {k}}\right) \| f - g \| _ {1} + 4 K \rho^ {2} \| W - V \| _ {\square} \\ \leq 4 K L \rho \| f - g \| _ {\square} + 4 K \rho^ {2} \| W - V \| _ {\square}. \\ \end{array}
$$
Lastly, we show that update layers are Lipschitz continuous. Since the update function takes two functions  $f:[0,1]\to \mathbb{R}^{d_i}$  (for generally two different output dimensions  $d_{1},d_{2}$ ), we "concatenate" these two inputs and treat it as one input  $f:[0,1]\rightarrow \mathbb{R}^{d_1 + d_2}$ .
Lemma F.9. Let  $\eta : \mathbb{R}^{d + p} \to \mathbb{R}^s$  be Lipschitz with Lipschitz constant  $L_{\eta}$ , and let  $f, g \in \mathcal{L}_r^\infty[0,1]$  with values in  $\mathbb{R}^{d + p}$  for some  $d, p \in \mathbb{N}$ .
$$
\left\| \eta (f) - \eta (g) \right\| _ {1} \leq L _ {\eta} \| f - g \| _ {1}.
$$
$$
\begin{array}{l} \left\| \eta (f) - \eta (g) \right\| _ {1} = \int_ {0} ^ {1} \left| \eta (f (x)) - \eta (g (x)) \right| d x \\ \leq \int_ {0} ^ {1} L _ {\eta} | f (x) - g (x) | d x = L _ {\eta} \| f - g \| _ {1}. \\ \end{array}
$$
F.2 Bounds of signals and MPLs with Lipschitz message and update functions
We will consider three settings for the MPNN Lipschitz bounds. In all settings, the transmitter, receiver, and update functions are Lipschitz. In the first setting all message and update functions are assumed to be bounded. In the second setting, there is no additional assumption over Lipschitzness of the transmitter, receiver, and update functions. In the third setting, we assume that the message function  $\Phi$  is also Lipschitz with Lipschitz bound  $L_{\Phi}$ , and that all receiver and transmitter functions
are non-negatively bounded (e.g., via an application of ReLU or sigmoid in their implementation). Note that in case  $K = 1$  and all functions are differentiable, by the product rule,  $\Phi$  can be Lipschitz only in two cases: if both  $\xi_{\mathrm{r}}$  and  $\xi_{\mathrm{t}}$  are bounded and Lipschitz, or if either  $\xi_{\mathrm{r}}$  or  $\xi_{\mathrm{t}}$  is constant, and the other function is Lipschitz. When  $K > 1$ , we can have combinations of these cases.
We next derive bounds for the different settings. A bound for setting 1 is given in Theorem F.8. Moreover, When the receiver and transmitter message functions and the update functions are bounded, so is the signal at each layer.
Bounds for setting 2.
Next we show boundedness when the receiver and transmitter message and update functions are only assumed to be Lipschitz.
Define the formal bias  $B_{\xi}$  of a function  $\xi :\mathbb{R}^{d_1}\to \mathbb{R}^{d_2}$  to be  $\xi (0)$  [26]. We note that the formal bias of an affine-linear operator is its classical bias.
Lemma F.10. Let  $(W, f) \in \mathcal{WL}_r$ , and suppose that for every  $y \in \{\mathrm{r}, \mathrm{t}\}$  and  $k = 1, \ldots, K$
$$
\left| \xi_ {\mathrm {y}} ^ {k} (0) \right| \leq B, \quad L _ {\xi_ {\mathrm {y}} ^ {k}} <   L.
$$
$$
\left\| \xi_ {\mathrm {y}} ^ {k} \circ f \right\| _ {\infty} \leq L r + B
$$
$$
\left\| \operatorname {A g g} \left(W, \Phi_ {f}\right) \right\| _ {\infty} \leq K (L r + B) ^ {2}.
$$
Proof. Let  $\mathrm{y} \in \{\mathrm{r}, \mathrm{t}\}$ . We have
$$
\left| \xi_ {\mathrm {y}} ^ {k} (f (x)) \right| \leq \left| \xi_ {\mathrm {y}} ^ {k} (f (x)) - \xi_ {\mathrm {y}} ^ {k} (0) \right| + B \leq L _ {\xi_ {\mathrm {y}} ^ {k}} | f (x) | + B \leq L r + B,
$$
$$
\begin{array}{l} | \operatorname {A g g} (W, \Phi_ {f}) (x) | = \left| \sum_ {k = 1} ^ {K} \int_ {0} ^ {1} \xi_ {\mathrm {r}} ^ {k} (f (x)) W (x, y) \xi_ {\mathrm {t}} ^ {k} (f (y)) d y \right| \\ \leq K (L r + B) ^ {2}. \\ \end{array}
$$
Next, we have a direct result of Theorem F.8.
Corollary F.11. Suppose that for every  $\mathrm{y} \in \{\mathrm{r}, \mathrm{t}\}$  and  $k = 1, \dots, K$
$$
\left| \xi_ {y} ^ {k} (0) \right| \leq B, \quad L _ {\xi_ {y} ^ {k}} <   L.
$$
Then, for every  $(W,f),(V,g)\in \mathcal{WL}_r$
$$
\| \operatorname {A g g} (W, \Phi_ {f}) - \operatorname {A g g} (V, \Phi_ {g}) \| _ {\square} \leq 4 K \left(L ^ {2} r + L B\right) \| f - g \| _ {\square} + 4 K (L r + B) ^ {2} \| W - V \| _ {\square}.
$$
Bound for setting 3.
Lemma F.12. Let  $(W, f) \in \mathcal{WL}_r$ , and suppose that
$$
| \Phi (0, 0) | <   B, \quad L _ {\Phi} <   L.
$$
$$
\left\| \Phi_ {f} \right\| _ {\infty} \leq L r + B
$$
$$
\left\| \operatorname {A g g} \left(W, \Phi_ {f}\right) \right\| _ {\infty} \leq L r + B.
$$
$$
\left| \Phi (f (x), f (y)) \right| \leq \left| \Phi (f (x), f (y)) - \Phi (0, 0) \right| + B \leq L _ {\Phi} | (f (x), f (y)) | + B \leq L r + B,
$$
$$
\begin{array}{l} | \operatorname {A g g} (W, \Phi_ {f}) (x) | = \left| \int_ {0} ^ {1} W (x, y) \Phi (f (x), f (y)) d y \right| \\ \leq L r + B. \\ \end{array}
$$
Additional bounds.
Lemma F.13. Let  $f$  be a signal,  $W, V \in \mathcal{W}_0$ , and suppose that  $\| \Phi_f \|_{\infty} \leq \rho$  for every  $k = 1, \ldots, K$ , and that  $\xi_{\mathrm{r}}^{k}$  and  $\xi_{\mathrm{t}}^{k}$  are non-negatively valued. Then
$$
\| \operatorname {A g g} (W, \Phi_ {f}) - \operatorname {A g g} (V, \Phi_ {f}) \| _ {\square} \leq K \rho \| W - V \| _ {\square}.
$$
Proof. The proof follows the steps of Lemma F.7 until (25), from where we proceed differently. Since all of the functions  $q_{\mathrm{r}}^{k}$  and  $q_{\mathrm{t}}^{k}$ ,  $k \in [K]$ , and since  $\| \Phi_f \|_{\infty} \leq \rho$ , the product of each  $q_{\mathrm{r}}^{k}(x) q_{\mathrm{t}}^{k}(y)$  must be also bounded by  $\rho$  for every  $x \in [0,1]$  and  $k \in [K]$ . Hence, we may replace the normalization in (25) with
$$
v _ {\mathrm {r}} ^ {k} (x) = \left\{ \begin{array}{c c} q _ {\mathrm {r}} ^ {k} (x) / \rho_ {\mathrm {r}} ^ {k} & x \in S \\ 0 & x \notin S \end{array} \right., v _ {\mathrm {t}} ^ {k} (y) = \left\{ \begin{array}{c c} q _ {\mathrm {t}} ^ {k} (y) / \rho_ {\mathrm {t}} ^ {k} & y \in S \\ 0 & y \notin S, \end{array} \right.
$$
where for every  $k \in [K]$ ,  $\rho_{\mathrm{r}}^{k} \rho_{\mathrm{t}}^{k} = \rho$ . This guarantees that  $v_{\mathrm{r}}^{k}, v_{\mathrm{t}}^{k} \in L_{1}^{\infty}[0,1]$ . Hence,
$$
\begin{array}{l} \int_ {S} \operatorname {A g g} (T, \Phi_ {f}) (x) d x = \sum_ {k = 1} ^ {K} \int_ {0} ^ {1} \int_ {0} ^ {1} \rho_ {\mathrm {r}} ^ {k} v _ {\mathrm {r}} ^ {k} (x) T (x, y) \rho_ {\mathrm {t}} ^ {k} v _ {\mathrm {t}} ^ {k} (y) d y d x \\ \leq \sum_ {k = 1} ^ {K} \rho \left| \int_ {0} ^ {1} \int_ {0} ^ {1} v _ {\mathrm {r}} ^ {k} (x) T (x, y) v _ {\mathrm {t}} ^ {k} (y) d y d x \right| \leq K \rho \| T \| _ {\square}. \\ \end{array}
$$