# Universal Local Attractors on Graphs
## Appendix B Theoretical Proofs
By Theorem A2, the ULA1D architecture of Equation (3) admits parameters \(\theta\in\Theta_{\theta_{\infty}}\) that reproduce gradient norm up to \(\epsilon_{\infty}\) for any twice-differentiable loss, including the loss \(\mathcal{L}_{all}(r)\) such that  
\[\mathcal{L}_{all}(\mathcal{A}_{\theta}(M,H))=\mathcal{L}(M_{1},H_{1}, \mathcal{A}_{\theta}(M_{1},H_{1}))+\mathcal{L}(M_{2},H_{2},\mathcal{A}_{ \theta}(M_{2},H_{2}))+\dots\]  
It also holds that  
\[F(\hat{M})= \begin{bmatrix}F(\hat{M}_{1})&\mathbf{0}&\mathbf{0}&\dots\\ \mathbf{0}&F(\hat{M}_{2})&\mathbf{0}&\dots\\ \mathbf{0}&\mathbf{0}&F(\hat{M}_{3})&\dots\\ \dots&\dots&\dots&\dots\end{bmatrix} \]  
given that adjacency normalization considers only row and column degrees for each node, and potentially adding self-loops that add the unit matrix on the diagonal. Given this result, we now concatenate vertically all parts of our predictive pipeline to obtain that \(\mathcal{A}_{\theta}(M,H)=[\mathcal{A}_{\theta}(M_{1},H_{1});\mathcal{A}_{ \theta}(M_{2},H_{2});\mathcal{A}_{\theta}(M_{3},H_{3});\dots]\). Thus, for any element \(i\) of \(\mathcal{M}\), we obtain  
\[\|\
abla_{r}\mathcal{L}(M_{i},H_{i},\mathcal{A}_{\theta}(M_{i},H_{i}))\|\leq \|\
abla_{r}\mathcal{L}_{all}(r)\|\leq\epsilon_{\infty}\]  
As a final remark, we stress that previous theoretical results were obtained for a finite number of nodes, and for this reason this proof only works on a set \(\mathcal{M}\) with a finite number of elements, which set up \(M\) as the adjacency matrix of a graph with a finite number of nodes (the sum of all nodes in the set of graphs). Handling an infinite number of graphs is tackled in the proof of Theorem A3 that follows. **Theorem A3**. : _Consider infinite sets \(\mathcal{M}\), bounded node feature values, Lipschitz loss gradients with respect to both \(h\) and \(r\), and \(MLP_{\theta}\) that has activation functions that are almost everywhere Lipschitz with Lipschitz derivatives (e.g., relu). Then, Lemma A3 holds for a dense subset of the neighborhood._Proof. : Our chosen \(\mathcal{A}_{\theta}(M,H)\) is \(c(\theta)\)-Lipschitz with respect to \(H\) for some value \(c(\theta)>0\), as it is composed from Lipschitz functions (activations and linear matrix multiplications and additions). Consider that \(\mathcal{L}\) has a \(c_{\mathcal{L}}\)-Lipschitz derivative. We can now write  
\[\|\mathcal{A}_{\theta}(M,H_{i})-\mathcal{A}_{\theta}(M,H_{j})\|_{ \infty}\leq c(\theta)\|H_{i}-H_{j}\|_{\infty}\] \[\Rightarrow\|\
abla_{r}\mathcal{L}(M,H_{i},\mathcal{A}_{\theta} (M,H_{i}))-\
abla_{r}\mathcal{L}(M,H_{i},\mathcal{A}_{\theta}(M,H_{j}))\|_{ \infty}\leq c_{\mathcal{L}}c(\theta)\|H_{i}-H_{j}\|_{\infty}\]  
where \(\|\cdot\|_{\infty}\) obtains the maximum absolute value of vector elements. Similarly, the quantity  
\[errdif(\theta)=\
abla_{r}\mathcal{L}(M,H_{i},\mathcal{A}_{\theta}(M,H_{i}))- \
abla_{r}\mathcal{L}(M,H_{i},\mathcal{A}_{\theta}(M,H_{j}))\]  
is \(c_{ge}\)-Lipschitz with respect to \(\theta\) for some constant \(c_{ge}>0\), i.e., \(\|errdif(\theta_{i})-errdif(\theta_{j})\|\leq c_{ge}\|\theta_{i}-\theta_{j}\|_ {\infty}\). Note that above all loses have \(H_{i}\) as the second argument. We also have that the quantity  
\[apprfeats(H_{i})=\
abla_{r}\mathcal{L}(M,H_{i},\mathcal{A}_{\theta}(M,H_{j}))\]  
is \(c_{fe}\)-Lipschitz with respect to \(H_{i}\) for some constant \(c_{fe}>0\), i.e.,  
\(\|apprfeats(H_{i})-apprfeats(H_{j})\|\leq c_{fe}\|H_{i}-H_{j}\|_{\infty}\).