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List of updates to the manuscript in light of the reviewers’ comments:
Added a sentence at the beginning of page 2 to highlight that Q.1 and Q.2 are indeed related and clarify how they look at different aspects of the same problem (frequency response the former, limit behaviour the latter). To compensate for space, we have slightly rephrased the second bullet point of the contributions.
At page 7 above the statement of Theorem 4.3, we have added a paragraph “Similarly to..” where we clarify that the exact spectral analysis in the time continuous and time discrete cases is performed for a simplified version of the gradient flow system. We have also stated more visibly the assumption used throughout the rest of Section 4 and Section 5 to help the reader. We note that the results can be extended to versions of the more general gradient flow equations of Proposition 1 quite trivially (e.g. when $\boldsymbol{\Omega}$ and $\mathbf{W}$ commute), however this could deviate too much from our story. In fact, the purpose of our work is showing that there exist simple graph convolutions that can enhance the high frequencies and induce behaviours other than over-smoothing. On the other hand, we highlight that the results in Section 6 for the non-linear layers hold in the generality of Eq. 1.
In line with the previous point, we have modified Eq. (10) at the beginning of Section 5 to be the simplified gradient flow system we consider for Theorem 5.1.
We have added in Proposition 4.1 that the graph is assumed to have at least one non-trivial edge to avoid corner cases. We have also extended the proof of Proposition 4.1 in the Appendix B.1 to have more details.
Added a sentence below Eq. (12) to clarify that if (12) holds, then $\mathbf{W}$ must have negative eigenvalues, i.e. $\mu_0 < 0$.
Added a clarification to the statement of Theorem 6.1 to specify that if there are no positive eigenvalues for the matrix $\boldsymbol{\Omega}\otimes\mathbf{I} - \mathbf{W}\otimes\boldsymbol{\mathsf{A}}$, then we can simply take c to be zero.
Added a derivation of Eq. (6) in Appendix B.1.
We have removed item (i) from page 7 and instead added an equivalent comment at page 8 below eq. 11, starting “In fact we note..”.
Inverted the ratio of the normalized energies at the last row of Table 2 to improve readability.
Added reference to “A critical look at the evaluation of GNNs under heterophily: are we really making progress?” in the last paragraph of page 1.
Added references to “Magnet: A neural network for directed graphs”, “ACMP: Allen-Cahn Message Passing for Graph Neural Networks with Particle Phase Transition”, Revisiting heterophily for graph neural networks", "Improving Graph Neural Networks with Learnable Propagation Operators” in the Related work Section.
We have changed the titles of Subsection 4.1 and 4.2 to “Non-learnable case” and “learnable case”, respectively to avoid any confusion.
Clarified differences between results in Table 1 and those in Table 3 in the appendix via an additional paragraph below eq. (35)
Removed reference to Cornell_old and clarified the instance of the dataset used in experiments