Graph Representation Learning via Adaptive Expectile Aggregation

Tianyi Zhu; Wei Lan

stat.MLOct 27, 2026

Graph Representation Learning via Adaptive Expectile Aggregation

Authors: Tianyi Zhu, Wei Lan

Organizations: Department of Statistics, Colorado State University Fort Collins, CO 80523, USA · Laboratory for Data Intelligence and Innovation in Finance and Economics and Center of Statistical Research Southwestern University of Finance and Economics, Chengdu, China

Abstract

Graph neural networks (GNNs) rely critically on neighborhood aggregation, yet conventional mean aggregation may fail to capture class-relevant distributional information, particularly in heterophilic graphs. We propose ExpectileGCN, an adaptive graph aggregation framework that introduces a learnable expectile level to capture neighborhood information beyond the mean. The proposed operator continuously adjusts aggregation between minimum- and maximum-type behaviors, with mean aggregation as a special case. A differentiable approximation based on sigmoid smoothing and fixed-point iterations enables end-to-end optimization. Theoretically, we establish stability properties and transductive generalization guarantees for ExpectileGCN. We further investigate its classification risk under heterophilic neighborhood models. For informative neighborhoods, we show that expectile aggregation can exploit class-dependent scale differences to achieve a classification advantage over mean aggregation. Under a local asymptotic regime with diminishing per-neighbor label informativeness, we establish that this advantage can persist in terms of limiting classification risk. Experiments on homophilic and heterophilic benchmark graphs demonstrate strong predictive performance and improved robustness under feature corruption and structural perturbations.

Figures & tables

Figure 1: Illustration of the Expectile aggregation mechanism compared with conventional mean aggregation. Mean aggregation treats all neighboring signals uniformly, which may cause representation distortion under noisy or heterophilic neighborhoods. In contrast, ExpectileGCN adaptively emphasizes different parts of the neighborhood distribution through the expectile parameter $\tau$ , enabling more flexible aggregation across different neighborhood distributions.

Dataset	Nodes	Edges	Classes	Homophily	Avg Degree	Diameter	Node Features
pubmed	19,717	88,648	3	0.80	8.99	18	500
cora	2,708	13,264	7	0.81	9.80	19	1,433
minesweeper	10,000	39,402	2	0.68	7.88	99	7
questions	48,921	153,540	2	0.84	6.28	16	301
film	7,600	30,019	5	0.22	7.90	12	932
citeseer	2,277	62,742	6	0.74	55.11	28	3,703

Table 1: Summary statistics of benchmark datasets, including graph size, structural properties, and node feature dimensions.

Dataset	ExpectileGCN	GMGCN	MedianGCN	SoftMedoidGCN	GCN	GAT	APPNP	AGNN	JacobiConv
pubmed	90.12 $\pm$ 0.21	89.88 $\pm$ 0.32	89.34 $\pm$ 0.12	88.44 $\pm$ 0.05	88.65 $\pm$ 0.18	87.58 $\pm$ 0.62	88.20 $\pm$ 0.34	85.56 $\pm$ 8.13	89.60 $\pm$ 0.26
cora	85.89 $\pm$ 0.52	85.71 $\pm$ 0.60	86.34 $\pm$ 0.38	84.42 $\pm$ 0.38	85.75 $\pm$ 0.19	85.08 $\pm$ 0.53	86.85 $\pm$ 0.19	87.11 $\pm$ 0.45	85.75 $\pm$ 0.47
minesweeper	91.57 $\pm$ 0.05	88.38 $\pm$ 0.44	77.74 $\pm$ 0.38	54.55 $\pm$ 8.27	66.33 $\pm$ 8.52	68.41 $\pm$ 11.59	58.55 $\pm$ 1.68	72.96 $\pm$ 5.22	93.06 $\pm$ 1.60
questions	72.78 $\pm$ 2.13	OOM	OOM	35.93 $\pm$ 0.77	63.33 $\pm$ 2.25	71.73 $\pm$ 0.77	41.28 $\pm$ 2.43	62.78 $\pm$ 12.19	76.60 $\pm$ 1.02
citeseer	76.61 $\pm$ 1.66	76.01 $\pm$ 1.24	77.45 $\pm$ 0.90	77.90 $\pm$ 0.22	75.74 $\pm$ 0.32	76.43 $\pm$ 0.74	76.70 $\pm$ 0.35	76.25 $\pm$ 0.63	77.18 $\pm$ 0.73
film	35.82 $\pm$ 0.79	34.32 $\pm$ 0.78	34.92 $\pm$ 1.20	31.59 $\pm$ 0.41	35.03 $\pm$ 0.77	33.88 $\pm$ 0.77	34.63 $\pm$ 0.65	34.51 $\pm$ 0.53	33.39 $\pm$ 0.15

Table 2: Test accuracy (%) (or ROC-AUC (%) for minesweeper and questions) on thirteen real-world datasets. Best results are shown in bold; ‘OOM’ denotes out-of-memory. The dashed line separates homophilic and heterophilic datasets. All-rank and Heter-rank denote average ranks over all datasets and heterophilic datasets, respectively, with lower values indicating better performance.

Figure 2: Heatmap of the average training time (in seconds) for each method across different datasets. The reported time is the wall-clock training cost averaged over 5 runs.

Dataset	ExpectileGCN	GMGCN	JacobiConv	MedianGCN	GCN	AGNN
pubmed	1.19	2.88	3.25	4.81	5.59	3.28
cora	1.91	3.03	1.16	3.91	5.27	5.73
citeseer	1.78	3.50	1.31	3.41	5.00	6.00
minesweeper	2.66	4.38	1.06	3.22	4.88	4.81
questions	1.84	5.50	1.72	5.50	3.44	3.00
film	1.56	2.88	2.19	3.38	5.66	5.34

Table 3: Average rank of each method across datasets. Methods are ranked by test performance under each of the 32 perturbation settings, and the reported values are averaged across settings. Lower ranks indicate better performance. Best results are shown in bold. The dashed line separates homophilic and heterophilic datasets. All-rank and Heter-rank denote average ranks over all datasets and heterophilic datasets, respectively.

Figure 3: Robustness comparison of different GNN models on the pubmed dataset. Within each subplot, each bar shows the mean test accuracy under the corresponding added edge proportion

p=\{0,0.2,0.4,0.6\}

, marked as 1, 2, 3, and 4 below the bars. Black error bars denote the standard deviation across 5 independent runs.

Figure 4: Robustness comparison of different GNN models on the chameleon dataset. Within each subplot, each bar shows the mean test accuracy under the corresponding added edge proportion

p=\{0,0.2,0.4,0.6\}

, marked as 1, 2, 3, and 4 below the bars. Black error bars denote the standard deviation across 5 independent runs.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Figure A1: Robustness comparison of different GNN models on the cora dataset. Within each subplot, each bar shows the mean test accuracy under the corresponding value of added-edge proportion

p=\{0,0.2,0.4,0.6\}

, marked as 1, 2, 3, and 4 on the bottom of the bars. Black error bars denote the standard deviation across 5 independent runs.

Figure A2: Robustness comparison of different GNN models on the squirrel dataset. Within each subplot, each bar shows the mean test accuracy under the corresponding value of added-edge proportion

p=\{0,0.2,0.4,0.6\}

, marked as 1, 2, 3, and 4 on the bottom of the bars. Black error bars denote the standard deviation across 5 independent runs.

Figure A3: Robustness comparison of different GNN models on the texas dataset. Within each subplot, each bar shows the mean test accuracy under the corresponding value of added-edge proportion

p=\{0,0.2,0.4,0.6\}

, marked as 1, 2, 3, and 4 on the bottom of the bars. Black error bars denote the standard deviation across 5 independent runs.

Figure A4: Robustness comparison of different GNN models on the wisconsin dataset. Within each subplot, each bar shows the mean test accuracy under the corresponding value of added-edge proportion

p=\{0,0.2,0.4,0.6\}

, marked as 1, 2, 3, and 4 on the bottom of the bars. Black error bars denote the standard deviation across 5 independent runs.

$T$	$s$	pubmed	cora	film	citeseer	texas	cornell	wisconsin	chameleon	squirrel
1	1	90.01 $\pm$ 0.18	85.91 $\pm$ 0.41	35.68 $\pm$ 0.70	76.94 $\pm$ 1.28	87.03 $\pm$ 4.80	71.08 $\pm$ 3.21	84.12 $\pm$ 1.85	63.42 $\pm$ 1.13	45.73 $\pm$ 0.69
	3	89.97 $\pm$ 0.25	85.86 $\pm$ 0.46	35.75 $\pm$ 0.63	76.98 $\pm$ 1.25	85.68 $\pm$ 6.51	71.62 $\pm$ 3.02	82.75 $\pm$ 2.60	63.68 $\pm$ 1.36	45.86 $\pm$ 0.98
	5	90.03 $\pm$ 0.26	85.89 $\pm$ 0.43	35.66 $\pm$ 0.55	76.73 $\pm$ 1.48	85.14 $\pm$ 6.42	71.62 $\pm$ 3.47	83.73 $\pm$ 2.78	63.82 $\pm$ 1.39	45.97 $\pm$ 0.72
	10	89.96 $\pm$ 0.21	85.86 $\pm$ 0.46	35.69 $\pm$ 0.70	76.38 $\pm$ 1.80	84.59 $\pm$ 6.05	71.62 $\pm$ 3.47	84.71 $\pm$ 2.45	63.62 $\pm$ 0.94	46.11 $\pm$ 0.87
2	1	90.05 $\pm$ 0.21	85.91 $\pm$ 0.41	35.68 $\pm$ 0.70	76.71 $\pm$ 1.53	86.76 $\pm$ 3.30	71.08 $\pm$ 3.21	86.08 $\pm$ 2.05	63.42 $\pm$ 1.13	45.86 $\pm$ 0.84
	3	90.13 $\pm$ 0.18	85.84 $\pm$ 0.49	35.73 $\pm$ 0.65	76.98 $\pm$ 1.25	85.68 $\pm$ 6.63	71.62 $\pm$ 3.02	83.73 $\pm$ 3.40	63.68 $\pm$ 1.36	45.98 $\pm$ 0.97

Appendix

Table A1: Sensitivity analysis of ExpectileGCN with respect to the number of fixed-point iterations

T

and smoothing sharpness

s

. Results are reported as mean accuracy (%)

\pm

standard deviation over 5 runs.

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