SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Table 5: Performance comparison of TEDGCN and a 2-layer SGC on synthetic datasets. (%)
Homophily Level h(G)	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
SGC	44.8	53.2	58.3	67.9	73.0	77.6	81.6	83.5	90.7
TEDGCN	67.3	68.5	60.3	72.3	78.3	78.6	82.7	84.0	91.5
Optimal d	-3.24	-0.75	-0.60	0.74	0.94	1.01	2.69	2.98	3.58
To better demonstrate the relation between the negative depth  $d$  and edge homophily/heterphily, we construct synthetic datasets with different homophily levels. The synthetic graphs are generated via scheme adopted in ACM-GCN [38]. Specifically, there are 5 classes, each with 400 nodes. For a fixed homophily level  $h(\mathcal{G})$  (e.g.,  $h(\mathcal{G}) = 0.1$ ), the node degree  $\delta_v$  is sampled for each node via multinomial distribution. With the sampled  $\delta_v$ ,  $h(\mathcal{G})\delta_v$  intra-class edges and  $(1 - h(\mathcal{G}))\delta_v$  inter-class edges are randomly generated. We compare TEDGCN with a two-layer SGC. The results on the synthetic datasets with different homophily levels are presented in Table 5. We can observe that (1) as the homophily level  $h(\mathcal{G})$  decreases, the optimal  $d$  turns from positive to negative; and (2) under low homophily levels, the performance of TEDGCN is significantly better than SGC.
A.3 More Baselines and Unified Setting
Table 6: Performance comparison (mean±std accuracy)(%) on benchmark datasets.
Datasets	Texas	Cornell	Wisconsin	Actor	Squirrel	Chameleon	Cora	Pubmed	Citeseer
BerNet	82.70±2.70	81.35±4.05	87.05±2.54	34.46±0.77	34.02±1.14	47.01±1.60	87.51±0.63	85.35±0.57	76.82±0.94
LINKX	74.60±8.37	77.84±5.81	75.49±5.72	36.10±1.55	61.81±1.80	68.42±1.38	84.64±1.13	87.86±0.77	73.19±0.99
GloGNN	84.32±4.15	83.51±4.26	87.06±3.53	37.35±1.30	57.54±1.39	69.78±2.42	88.31±1.13	89.62±0.35	77.41±1.65
ACM-GCN	87.84±4.40	85.14±6.07	88.43±3.22	36.63±0.84	55.19±1.49	69.14±1.91	87.91±0.95	90.00±0.52	77.32±1.70
TeDGCN-S	90.32±4.71	85.81±4.21	88.63±4.51	37.40±1.11	48.43±1.42	67.02±1.91	88.41±1.73	85.34±2.11	77.42±1.30
TeDGCN-D	91.41±3.62	86.53±3.80	91.42±4.22	29.03±1.03	62.02±3.03	67.33±1.71	87.90±1.31	84.84±1.21	77.81±1.72
Under fully-supervised setting, we compare TEDGCN with ACM-GCN, LINKX, BernNet and GloGNN with the standard split of  $48\% /32\% /20\%$ . We directly use the performances of ACM-GCN, LINKX and GloGNN reported in the ACM-GCN paper. For BernNet, we obtain its performance by running its open-sourced code. The results are shown in Table 6. It is observed that the latest
methods, such as ACM-GCN and GloGNN do perform strongly. However, our model achieves the best classification accuracy on 7 out of the 9 datasets, which demonstrates the effectiveness of TEDGCN.
We also compare all methods (15 baselines and TeDGCN) under a unified setting. Since some methods do not report their performances on some datasets in their original paper and have not made their code publicly available, we compare all methods' performances on 4 commonly used heterophilic graph datasets: Texas, Cornell, Wisconsin and Actor, under the same public  $48\% /32\% /20\%$  split for training, validation and testing.
Table 7: Performance comparison (mean) on benchmark datasets.
Datasets	Texas	Cornell	Wisconsin	Actor
GCN	54.05	53.78	50.39	28.78
SGC	45.80	43.72	45.13	27.52
GAT	57.30	54.59	54.31	28.99
APPNP	58.92	51.12	58.84	31.83
GPRGNN	81.35	78.11	82.55	35.16
FAGCN	76.49	76.76	79.61	34.82
H2GCN	79.73	78.38	82.55	36.71
ChebNet	82.14	72.16	78.82	36.04
BerNet	82.70	81.35	87.05	34.46
LINKX	74.60	77.84	75.49	36.10
GloGNN	84.32	83.51	87.06	37.35
ACM-GCN	87.84	85.14	88.43	36.63
GeomGCN	66.76	60.54	64.51	31.59
GBK-GNN	70.27	75.68	68.63	36.12
HOG-GCN	85.17	84.32	86.67	36.82
TeDGCN-S	90.32	85.81	88.63	37.40
TeDGCN-D	91.41	86.53	91.42	29.03
From Table 7, we can observe that TeDGCN-S/TeDGCN-D outperforms all the remaining baselines.
We set the learning rate to 0.005 or 0.01, the decaying weight for the learning rate to  $5e^{-4}$ , the number of training epochs to 500. The  $K$  largest/smallest eigenvalues for TEDGCN-D is  $\max\{1000, 0.1n\}$ , where  $n$  is the number of nodes in the input graph. For all baselines, the parameters are set according to their original papers. All experiments are run on a Tesla-V100 GPU. Since this work is conducted during internship in Visa Research and the code can not be made public according to the policy of Visa Research. If you have any questions about the implementation, please email yucheny5@illinois.edu to get some help.
A.5 Visualization of Node Embeddings w.r.t. Negative Depth  $d$
We conduct a visualization of node embedding generated by TEDGCN under different depths for homophilic (Cora) and heterophilic (Texas) graphs. As shown in Figure 7, for Cora, when  $d = 5$ , the nodes in different classes form clearly distinguishable clusters (ACC=82.4%). When the depth decreases from a positive value  $d = 5$  (ACC=82.4%) to a negative value  $d = -2$  (ACC=29.5%), the nodes belonging to different classes/colors mix with each other, and the clearly clustered structure does not exist anymore. However, for Texas shown in Figure 8, when decreasing the depth from  $d = 4$  (ACC=50%) to  $d = -0.35$  (ACC=80%), the clusters of different classes/colors become well-separated. Especially, the purple nodes spread randomly all over the embedding space when  $d = 4$  but concentrate in one cluster when  $d = -0.35$ . The spatial distribution of the node embeddings reveals that TEDGCN with positive and negative depths respectively models graph homophily and heterophily in a proper way.
(a)  $d = -2$ $\mathrm{ACC} = 29.5\%$
(b)  $d = 0$ $\mathrm{ACC} = 46.6\%$
(d) GCN, ACC=81.0%
Figure 7: Visualization of node embeddings by t-SNE on Cora (homophilic).
(c)  $d = 5$ $\mathrm{ACC} = 82.4\%$
(a)  $d = -0.35$ $\mathrm{ACC} = 80.0\%$
Figure 8: Visualization of node embeddings by t-SNE on Texas (heterophilic).
(b)  $d = 2$ $\mathrm{ACC} = 59.0\%$
(c)  $d = 4$ $\mathrm{ACC} = 50.0\%$
(d) GCN, ACC=55.0%