Characterizing the underlying mechanism of graph topological evolution from a source graph to a target graph has attracted fast increasing attention in the deep graph learning domain. However, there lacks expressive and efficient that can handle global and local evolution patterns between source and target graphs. On the other hand, graph topological evolution has been investigated in the graph signal processing domain historically, but it involves intensive labors to manually determine suitable prescribed spectral models and prohibitive difficulty to fit their potential combinations and compositions. To address these challenges, this paper proposes the deep Graph Spectral Evolution Network (GSEN) for modeling the graph topology evolution problem by the composition of newly-developed generalized graph kernels. GSEN can effectively fit a wide range of existing graph kernels and their combinations and compositions with the theoretical guarantee and experimental verification. GSEN has outstanding efficiency in terms of time complexity (O(n)) and parameter complexity (O(1)), where n is the number of nodes of the graph. Extensive experiments on multiple synthetic and real-world datasets have demonstrated outstanding performance.
The paper proposed a method to model the graph topological evolution from the spectral domain by developing a new generalized graph kernel. The new graph kernels cover many existing graph kernels as well as their combination and composition as special cases. The idea of spectral graph translation and its integration with deep learning is interesting, especially considering that most previous spectral graph neural networks only transform the graph signal instead of graph structures. However, I do have some concerns of papers.
1.	My major concern is the soundness of keeping eigenvectors unchanged in the evolution. Although the authors claim that in previous studies eigenvectors are found stable in evolution, it is very counter-intuitive, and I am not sure it is the case for all types of graphs. Let us look at the proof of Lemma 3.1, obviously $L^\prime$ does not necessarily have the same eigenvectors, and $U^TL^\prime U$ is not a diagonal matrix, so this loss is actually very large in many cases. That is to say, the evolution model does not have enough expressive power to recover the $L^\prime$.
2.	Spectral graph translation looks interesting, but the main idea comes from Kunegis et al. (2010). Despite of a new designed graph kernel and adding nonlinear activations, the contributions seem not so significant.
3.	In Kunegis et al. (2010), they consider the evolution of adjacency matrix $A$, but in this paper the authors use the Laplacian matrix $L$. If there any reason to make this choice? Also, I think some of the conclusions (e.g. stable eigenvectors) in Kunegis et al. (2010) may not work since $L$ is used instesd of $A$.
4.    The correlation metric is acceptable, but it will be much better if the authors can do more analysis. For example, why not add the link prediction task as in Kunegis et al. (2010)? BTW, is correlation analysis used before in previous graph evolution papers? (if so, please add a reference)