# Universal Local Attractors on Graphs
## Appendix B Theoretical Proofs  
**Lemma A1**. : _For a positive definite graph filter \(F(\hat{M})\) and differentiable loss \(\mathcal{L}(r)\) over graph signal domain \(\mathcal{R}\subseteq\mathbb{R}^{|\mathcal{V}|}\), update graph signals over time \(t\in[0,\infty)\) as per_  
\[\frac{\partial h(t)}{\partial t}=f(r(t))\] \[r(t)=F(\hat{M})h(t)\] \[\text{s.t.}\ \|f(r)+\
abla_{r}\mathcal{L}(r)\|<\tfrac{\lambda_{1}}{ \lambda_{\max}}\|\
abla_{r}\mathcal{L}(r)\|\text{ for all }\|\
abla_{r}\mathcal{L}(r)\|>0,r\in \mathcal{R}\]  
_where \(\lambda_{1}\), \(\lambda_{\max}>0\) are the smallest positive and largest eigenvalues of \(F(\hat{M})\), respectively. This leads to \(\lim_{t\to\infty}\|\
abla_{r}\mathcal{L}(r(t))\|=0\) if posteriors remain closed in the domain, i.e., if \(r(t)\in\mathcal{R}\)._Proof. : For non-negative posteriors \(r=F(\hat{M})h(t)\), and non-zero loss gradients, the Cauchy-Schwartz inequality in the bilinear space \(\langle x,y\rangle=x^{T}F(\hat{M})y\) determined by the positive definite graph filter \(F(\hat{M})\) yields  
\[\frac{d\mathcal{L}\big{(}r(t)\big{)}}{dt}=\Big{(}\
abla_{r} \mathcal{L}\big{(}r(t)\big{)}\Big{)}^{T}F(\hat{M})\frac{dh(t)}{dt}=\Big{(} \
abla_{r}\mathcal{L}\big{(}r(t)\big{)}\Big{)}^{T}F(\hat{M})f\big{(}r(t)\big{)}\] \[\quad=\Big{(}\
abla_{r}\mathcal{L}\big{(}r(t)\big{)}\Big{)}^{T}F( \hat{M})\big{(}-\
abla_{r}\mathcal{L}\big{(}r(t)\big{)}+(f\big{(}r(t)\big{)}+ \
abla_{r}\mathcal{L}\big{(}r(t)\big{)})\big{)}\] \[\quad\leq-\Big{(}\
abla_{r}\mathcal{L}\big{(}r(t)\big{)}\Big{)}^{ T}F(\hat{M})\
abla_{r}\mathcal{L}\big{(}r(t)\big{)}+(\
abla_{r}\mathcal{L} \big{(}r(t)\big{)}^{T}F(\hat{M})\Big{(}f\big{(}r(t)\big{)}+\
abla_{r}\mathcal{ L}\big{(}r(t)\big{)}\Big{)}\] \[\quad\leq-\lambda_{1}\big{\|}\
abla_{r}\mathcal{L}\big{(}r(t) \big{)}\big{\|}^{2}+\lambda_{\max}\big{\|}f\big{(}r(t)\big{)}+\
abla\mathcal{L }\big{(}r(t)\big{)}\big{\|}\big{\|}\
abla\mathcal{L}\big{(}r(t)\big{)}\big{\|}\] \[\quad=0\]  
where \({}^{T}\) is the matrix transposition symbol. Therefore, the loss asymptotically converges to a locally minimum point. **Lemma A2**. : _Let a differentiable graph signal generation function \(\mathcal{H}_{\theta}:\Theta\to\mathbb{R}^{|\
abla|}\) on compact connected domain \(\Theta\) admit some parameters \(\theta_{0}\in\Theta\) such that \(\mathcal{H}_{\theta_{0}}=h_{0}\). If, for any parameters \(\theta\in\Theta\), the Jacobian \(\mathbb{J}_{\mathcal{H}_{\theta}}(\theta)\) has linearly independent rows (each row corresponds to a node) and satisfies_  
\[\big{\|}Err(\theta)\
abla_{r}\mathcal{L}\big{(}r\big{)}\big{\|} <\frac{\lambda_{1}}{\lambda_{\max}}\big{\|}\
abla_{r}\mathcal{L}(r)\big{\|} \quad\text{ for all }\
abla\mathcal{L}(r)\
eq\mathbf{0}\] \[Err(\theta)=\mathcal{I}-\mathbb{J}_{\mathcal{H}_{\theta}}(\theta )(\mathbb{J}_{\mathcal{H}_{\theta}}^{T}(\theta)\mathbb{J}_{\mathcal{H}_{ \theta}}(\theta))^{-1}\mathbb{J}_{\mathcal{H}_{\theta}}^{T}(\theta)\] \[r(t)=F(\hat{M})h(t)\]0  
_then there exist parameters \(\theta_{\infty}\in\Theta\) such that \(\big{\|}\
abla_{r}\mathcal{L}\big{(}r(\theta_{\infty})\big{)}\big{\|}=0\)._  
Proof. : Let us consider a differentiable trajectory for parameters \(\theta(t)\) for times \(t\in[0,\infty)\) that starts from \(\theta(0)=\theta_{0}\) and (asymptotically) arrives at some \(\theta_{\infty}=\lim_{t\to\infty}\theta(t)\). For this trajectory, it holds that \(\frac{d\mathcal{H}_{\theta(t)}}{dt}=\mathbb{J}_{\mathcal{H}_{\theta}}\big{(} \theta(t)\big{)}\frac{d\theta(t)}{dt}\).