Questions about the theory in the mausctript

[With-the-sun]

Hi! Thanks for the valuable research about Heteropyily and Oversmoothing in GCN. I am aware of some questions about the manuscript, could I communicate with you, for instance, about the metric's define of 'Relative Degree' at the initial layer, and the 'effective relative degree' . For the initial layer, all neighbor nodes j are considered. However, for deeper layer l, only the positively neighbor nodes j are considered. Why are negative contributing neighbors not considered in Deeper Layer, but considered in the initial layer?

[With-the-sun]

and the second question, at Section3.3.4 in the manuscript, about 'Relation between the problems' for '(2)-In homophilous graphs', why the homophilous graph have node's label is 'Class 2' ? Shouldn't the nodes of the homophilous graph all be of the same label 'Class 1' ?

[With-the-sun]

and the third question. as the layer be deeper, the i node's 'effective homophily' would get lower?
thanks

[With-the-sun]

and the fouth question. At the Equ(41), The symbol in the red box should be a minus sign？

[Yujun-Yan]

Thanks for your interest in our paper.

1. For both initial and deeper layers, we consider positive and negative contributing nodes. In our proofs (A.1 and A.2), we use whether a neighbor's representation at k-th layer contribute positively or negatively as the condition, and decompose the total expectation into the sum of conditional expectations, e.g. (Eq. 12 and second equality in Eq. 34).

[Yujun-Yan]

2. You are right that in heterophilous graphs, most nodes belong to class 1 (e.g., table 3, Texas, Wisconsin, Actor & Cornell). However, in some real datasets (e.g. table 3, Squirrel & Chameleon), we also found non-neglectable # of nodes in class 2. That's why we analyze the phenomenon in heterophilous graphs by considering both nodes in class 1 and class 2.

[Yujun-Yan]

3. yes, the effective homophily will get lower in heterophilous graphs. We have experiments provided in the appendix to show that: table B.3

[Yujun-Yan]

4. It should be a + here because the two negative signs offset each other in equation 40. And the total expectation (equation 41) should be a sum of equation 39 and equation 40. That's why we have a positive sign instead of negative sign there

[With-the-sun]

Thanks for your interest in our paper.

1. For both initial and deeper layers, we consider positive and negative contributing nodes. In our proofs (A.1 and A.2), we use whether a neighbor's representation at k-th layer contribute positively or negatively as the condition, and decompose the total expectation into the sum of conditional expectations, e.g. (Eq. 12 and second equality in Eq. 34).

Thanks for your response. And i have learned the appendices except A.4. The reply for my questions 2-4 are well understood. And i have learned the appendices except A.4) For the Question 1: 1.1- How to define 'positive contribution' or 'negative contribution'? Is it based on the label after classification? Is it mentioned in 'B.3 Measurement of effective homophily'?

1.2- the positive and negative contributing nodes are indeed considered in proofs (A.1 and A.2). However, for the definitions of 'relative degree' , the range of j is the i node's neighbor, but why in the 'effective relative degree' of 3.3.2 (2), the domain of j is only the positive neighborhood? What is the theoretical basis of the definition of 'effective relative degree'? Is it a subjective variable that matches proof A.2 ?

[With-the-sun]

I have a preliminary understanding of most theories about appendices (except A.4). But I still have some doubts. and I hope you could support me. Sorry to have troubled you. 5.1- In 3.3.4 "Oversmoothing problem", Case1 and Case2 trigger over-smoothing? What is the definition or quantification of over-smoothing? 5.2- In this article, over-smoothing means that 'effective homophily' is reduced to a lower limit(in other papers, the over-smoothing problem is more about the Dirichlet energy)? What is the connection or difference between 'effective homophily' and 'Dirichlet energy', namely, low h is low Dirichlet energy? Case1 is low h?

[With-the-sun]

6. In 3.3.4, The node of Case2 transforms into Case1 in deep layers, resulting in 'pseudo-heterophily'. However, if the homophily h=1 graph data is taken into account, namely only Class1 in graph data. Is there still 'pseudo-heterophily'? If not, why is GCN still too smooth on homophily h=1 graph data ?

[With-the-sun]

7.1- In 3.3.3, 'the movement of representations towards the other class' is it Case2? 7.2- In 3.3.3, What is the meaning of 'opposeite classes swap places', can there be A general, summative conclusion, rather than having to refer to the theoretical reasoning of A.4.

[With-the-sun]

8. In 4.1, formula 7 represents the weighting of messages on different nodes (i.e., giving edge weights τ), why was the definition of the relative degree of 'Thm 3.1' changed? Is this mathematically equivalent to putting τ in the formula 14?

[With-the-sun]

9.1- In 4.1, for the 'Strusture-based Edge Correction', since λ0 and λ1 are obtained by learning, how to ensure that formula (8) satisfies to 'Intuitively, when r is small, we would like to compensate for it via a larger τ '? Just because the r of formula (8) is in the denominator?

9.2- In 4.1, edge weights are learned at each layer. Moreover, GAT's edge weights are also learned at each layer, but GAT cannot learn edge symbols; So, is the edge symbols an important reason why GGCN is better than GAT?

[With-the-sun]

10.1- In A.1.Proof of Theorem 3.1, Is the symbol in the gray part of the figure a negative sign?

10.2- Formula (51), whether the gray part should have a u? 10.3- Formula (27), whether the gray part should be fj?

[With-the-sun]

I wonder whether I have the honor to have your contact information? e.g. chat app or others