Attribute-missing graph clustering model based on stacked joint optimization

doi:10.11772/j.issn.1001-9081.2025081008

Abstract

Abstract:

To address the inconsistency between attribute completion and clustering objectives in two-stage methods and the low efficiency of deep learning methods for attribute-missing graph clustering， an Attribute-Missing Graph Clustering model based on Stacked Joint Optimization （AMGC-SJO） was proposed. First， a Matrix Factorization-based Attribute-Missing Graph Clustering model （AMGC-MF） was constructed， which introduced an enhanced adjacency matrix to represent global node connection relations and employed graph regularization for attribute completion， and then a joint Non-negative Matrix Factorization （NMF） was applied to the completed the attribute matrix and enhanced adjacency matrix， so as to learn the node clustering degrees of membership. On this basis， an AMGC-SJO was further designed to update the attribute matrix and enhanced adjacency matrix dynamically during iteration， thereby enabling co-optimization of attribute completion and clustering tasks. Experimental results show that AMGC-SJO outperforms AMGC-MF on multiple clustering metrics， with better key metrics on five datasets. Compared to Attribute-Missing Graph Clustering （AMGC）， the proposed model achieves comparable clustering accuracy while reducing running time by at least 89.57% under the topological centrality-based missing mechanism. Furthermore， on the Cora dataset and attribute privacy missing mechanism， when the missing rate is increased from 10% to 90%， the clustering metrics of AMGC-SJO fluctuate within 2.08 percentage points， demonstrating strong robustness， whereas those of AMGC decline by over 30.88 percentage points. It can be seen that the proposed model provides an accurate， efficient， and robust solution for attribute-missing graph clustering.

Key words: attribute-missing graph, clustering, Non-negative Matrix Factorization (NMF), graph regularization, attribute completion

摘要：

针对属性缺失图聚类中属性补全和聚类两阶段方法的目标不一致及深度学习方法效率低的问题，提出一种基于堆叠式联合优化的属性缺失图聚类模型（AMGC-SJO）。首先，构建基于矩阵分解的属性缺失图聚类模型（AMGC-MF），通过增强邻接矩阵刻画节点间的全局连接关系，并借助图正则化实现属性补全，进而对补全后的属性矩阵与增强邻接矩阵进行联合非负矩阵分解（NMF），以学习节点聚类的隶属度。在此基础上，进一步设计AMGC-SJO，在迭代优化中动态更新属性矩阵与增强邻接矩阵，从而实现属性补全与聚类任务的协同优化。实验结果表明，AMGC-SJO的多项聚类评估指标显著优于AMGC-MF，在5个数据集上都取得了更优结果；与AMGC（Attribute-Missing Graph Clustering）相比，所提模型在保持相当聚类精度的同时，运行时间在拓扑中心性缺失机制下至少缩短了89.57%。此外，在Cora数据集和属性隐私缺失机制下，当缺失率从10%升至90%时，AMGC-SJO的聚类评估指标波动不超过2.08个百分点，表现出良好的鲁棒性，而AMGC则下降了超过30.88个百分点。可见，所提模型为属性缺失图聚类提供了一种高精度、高效率且强鲁棒的解决方案。

关键词: 属性缺失图, 聚类, 非负矩阵分解, 图正则, 属性补全

CLC Number:

TP391.4

Xiben LUO, Xiaoyun CHEN. Attribute-missing graph clustering model based on stacked joint optimization[J]. Journal of Computer Applications, 2026, 46(9): 2827-2837.

罗细奔, 陈晓云. 堆叠式联合优化属性缺失图聚类模型[J]. 《计算机应用》唯一官方网站, 2026, 46(9): 2827-2837.

Figures/Tables 10

Fig. 1 Attributed graph and attribute-missing graph

Tab. 1 Statistics of datasets

数据集	节点数	边数	特征数	类别数
Cora	2 708	5 278	1 433	7
Citeseer	3 327	4 228	3 703	6
PubMed	19 717	44 324	500	3
Amac	13 752	245 861	767	10
Amap	7 650	119 081	745	8

Tab. 2 Comparison of clustering performance under different missing data mechanisms

方法组合	指标	拓扑中心性缺失机制					邻居同质性缺失机制					属性隐私缺失机制
方法组合	指标	Cora	Citeseer	PubMed	Amac	Amap	Cora	Citeseer	PubMed	Amac	Amap	Cora	Citeseer	PubMed	Amac	Amap
1+A	NMI	14.14	15.31	18.44	13.48	15.51	14.24	12.37	19.41	16.37	21.81	11.29	10.97	19.68	11.58	11.32
	ACC	37.19	42.11	58.77	37.55	32.73	50.67	29.24	58.89	28.31	36.58	30.57	23.45	38.15	33.16	32.89
	F1	32.84	33.15	51.11	27.84	33.11	30.98	29.76	45.32	18.29	34.29	28.91	8.56	14.42	28.79	8.42
	ARI	13.52	16.45	23.33	11.41	2.36	15.33	10.56	20.88	15.21	7.23	6.43	9.82	7.95	6.92	5.76
1+B	NMI	23.39	19.54	11.78	21.63	20.71	13.98	11.34	12.78	9.16	18.42	27.86	9.13	17.32	14.73	19.97
	ACC	35.41	41.92	44.84	28.42	40.73	39.44	34.98	34.69	31.25	34.93	43.52	34.18	39.15	33.89	34.41
	F1	36.86	31.81	15.36	17.52	12.35	30.45	21.35	15.32	17.33	13.21	7.94	2.57	35.89	7.66	7.93
	ARI	2.56	13.21	30.33	2.13	1.23	15.64	1.80	9.96	6.82	9.58	2.18	2.64	33.61	2.47	2.95
1+C	NMI	16.00	8.84	17.00	2.20	4.31	7.96	22.39	13.98	9.75	12.68	16.76	8.01	9.85	5.22	4.51
	ACC	34.75	31.08	53.63	19.96	22.25	26.51	42.29	29.16	25.40	27.79	35.97	28.70	34.91	20.58	23.44
	F1	17.73	11.80	51.94	5.63	9.75	5.41	17.53	7.15	11.04	12.28	13.61	9.01	12.27	7.74	10.39
	ARI	4.63	3.09	8.57	2.58	2.59	2.43	15.69	2.48	5.76	5.28	4.46	2.83	11.32	3.39	2.81
2+A	NMI	10.27	14.24	17.42	24.32	30.71	12.16	14.12	11.58	24.83	23.15	35.41	12.47	12.68	10.58	7.32
	ACC	23.67	33.00	44.81	43.83	44.53	25.55	32.97	47.93	41.29	47.52	39.28	37.36	31.14	33.16	33.91
	F1	22.43	31.57	30.57	6.32	8.12	21.90	31.50	38.26	6.75	19.78	25.73	31.13	29.17	8.91	7.11
	ARI	10.02	9.55	9.25	13.21	22.13	1.50	9.43	16.41	12.38	5.36	5.69	10.63	5.32	6.94	6.44
2+B	NMI	18.54	10.91	12.87	20.94	21.33	14.32	19.84	18.41	25.18	30.24	34.27	15.53	18.76	17.43	15.48
	ACC	35.99	30.77	38.54	33.05	36.59	51.87	43.67	39.62	44.39	43.97	45.86	23.49	32.34	33.89	29.12
	F1	10.22	6.51	26.91	9.88	22.13	39.64	20.59	45.73	8.72	21.68	26.39	8.74	6.58	7.63	22.15
	ARI	1.36	3.21	15.58	8.55	10.66	8.59	9.15	22.98	9.54	19.46	4.52	1.54	3.29	2.47	11.67
2+C	NMI	41.23	17.34	11.18	18.48	34.33	42.99	28.12	13.27	29.95	52.35	35.92	16.85	11.23	20.11	29.05
	ACC	56.18	35.89	44.39	29.25	44.77	58.49	53.31	40.53	38.89	62.20	49.19	38.52	40.71	30.85	38.67
	F1	19.62	20.39	31.19	5.58	4.70	5.74	16.81	12.14	4.84	12.78	12.69	25.47	11.16	6.30	11.75
	ARI	33.46	7.05	7.90	10.69	21.78	38.20	23.56	12.47	18.47	40.71	24.72	7.82	14.96	11.63	18.28
3+A	NMI	10.95	11.33	17.55	19.99	29.30	11.97	16.83	15.79	18.55	31.27	23.18	11.88	15.42	15.73	11.76
	ACC	29.33	30.11	59.94	31.82	44.55	48.63	35.91	45.82	30.97	41.93	47.93	24.19	38.28	29.54	34.08
	F1	14.47	13.55	22.14	21.00	23.32	14.58	17.82	27.36	8.64	19.26	22.84	7.58	24.74	15.83	9.66
	ARI	10.22	8.87	18.88	10.03	15.51	13.29	9.34	28.45	7.29	14.85	11.97	8.73	5.01	0.82	6.89
3+B	NMI	24.41	8.51	6.77	20.44	21.12	25.23	9.28	10.92	19.41	28.64	24.57	9.68	13.39	14.71	17.15
	ACC	38.54	30.70	42.63	28.86	36.55	39.87	21.56	43.65	30.44	32.91	38.91	28.18	46.14	33.26	29.28
	F1	12.55	10.25	22.14	6.81	9.52	35.42	23.78	37.98	7.68	7.53	22.84	1.83	15.48	6.84	4.11
	ARI	10.01	9.94	6.64	10.07	4.34	5.19	5.19	17.61	3.26	2.87	1.97	4.42	4.90	6.49	7.42
3+C	NMI	4.32	1.87	10.29	3.17	3.55	26.81	7.83	13.29	2.17	2.55	11.98	11.22	10.58	9.42	9.63
	ACC	31.61	21.85	40.20	27.88	25.21	42.45	25.62	46.85	27.84	24.30	30.65	21.55	40.79	28.75	32.88
	F1	3.32	2.61	21.26	5.42	4.14	38.76	11.97	39.71	5.42	8.33	13.22	12.61	13.75	12.67	2.59
	ARI	0.67	1.33	11.22	2.49	5.05	15.26	7.34	15.38	2.49	5.00	10.60	10.31	10.22	8.31	1.18
4+A	NMI	6.42	6.98	7.78	18.43	7.29	28.17	5.98	16.75	12.37	13.58	21.36	9.42	18.21	11.59	2.85
	ACC	28.35	26.51	45.61	39.27	28.41	44.39	28.37	45.29	26.48	35.32	43.94	24.68	33.58	33.83	21.36
	F1	29.61	12.54	23.14	27.15	19.57	40.15	3.61	51.36	6.15	15.71	19.58	2.21	17.46	4.63	9.24
	ARI	3.65	0.35	0.21	9.36	11.83	14.58	2.84	8.94	4.91	4.19	9.20	1.88	5.33	4.69	3.36
4+B	NMI	30.19	19.56	10.37	24.13	19.87	29.74	9.64	10.29	11.92	23.95	26.85	12.15	10.15	9.44	13.54
	ACC	49.48	37.91	58.64	39.06	37.18	46.83	11.82	48.34	26.34	37.83	39.37	21.89	43.64	17.91	26.21
	F1	39.55	18.43	25.92	21.28	16.55	44.38	7.53	35.17	2.57	13.18	25.62	9.43	28.49	14.46	5.77
	ARI	23.48	4.27	9.81	7.65	8.92	11.79	9.08	11.69	1.84	1.46	3.19	1.57	8.21	7.74	4.81
4+C	NMI	14.54	11.77	3.66	1.53	4.69	20.67	16.85	24.51	2.61	7.64	15.49	10.80	14.91	6.27	2.11
	ACC	21.64	31.55	40.85	31.54	22.37	30.80	36.76	56.13	26.85	24.44	23.45	31.31	49.60	29.53	21.33
	F1	21.48	16.79	29.57	27.94	21.35	29.58	36.36	50.29	24.70	24.24	22.32	19.28	41.25	5.79	19.63
	ARI	1.34	0.21	0.25	1.31	5.61	6.61	10.26	21.53	1.68	3.27	1.55	11.15	10.32	3.38	3.10
AMGC- MF	NMI	41.57	16.72	27.01	31.81	34.30	44.15	19.85	27.10	31.54	51.75	43.00	17.00	27.09	31.30	48.65
	ACC	56.09	34.42	65.59	46.10	49.54	57.50	35.71	64.52	48.41	56.41	58.01	32.52	66.52	43.97	61.74
	F1	57.78	36.34	61.53	35.65	44.71	54.37	32.04	60.02	32.27	55.24	59.14	31.89	61.32	32.45	61.14
	ARI	32.20	8.05	30.25	23.10	23.60	32.08	6.73	29.88	22.10	36.41	34.95	5.43	30.11	25.16	41.52
AMGC	NMI	43.94	25.61	19.75	45.02	58.59	46.55	19.16	27.93	47.67	60.32	43.66	20.80	17.32	43.78	62.62
	ACC	62.00	48.12	58.04	50.68	64.42	61.67	43.28	65.79	57.98	70.73	62.30	43.01	58.46	44.12	65.44
	F1	58.32	47.64	57.58	37.28	56.76	53.38	42.58	65.47	40.63	64.70	59.77	41.76	57.39	29.10	57.76
	ARI	37.98	19.89	15.98	34.63	46.08	38.49	15.70	26.82	37.60	48.92	36.91	16.38	16.80	27.47	50.92
AMGC- SJO	NMI	44.07	21.81	27.31	36.43	40.51	44.20	21.78	26.97	33.16	39.57	43.45	21.93	26.88	33.92	40.55
	ACC	64.03	45.00	69.48	49.75	58.27	62.85	44.57	69.58	50.76	57.18	63.77	44.51	69.54	50.64	57.84
	F1	62.73	38.83	67.45	42.52	51.68	61.25	41.84	67.57	37.03	53.01	61.87	41.78	67.50	38.92	54.31
	ARI	36.47	14.22	31.39	27.25	25.83	34.65	13.88	31.98	25.95	29.86	35.86	14.01	31.86	26.87	30.45

Tab. 2 Comparison of clustering performance under different missing data mechanisms

方法组合	指标	拓扑中心性缺失机制					邻居同质性缺失机制					属性隐私缺失机制
方法组合	指标	Cora	Citeseer	PubMed	Amac	Amap	Cora	Citeseer	PubMed	Amac	Amap	Cora	Citeseer	PubMed	Amac	Amap
1+A	NMI	14.14	15.31	18.44	13.48	15.51	14.24	12.37	19.41	16.37	21.81	11.29	10.97	19.68	11.58	11.32
	ACC	37.19	42.11	58.77	37.55	32.73	50.67	29.24	58.89	28.31	36.58	30.57	23.45	38.15	33.16	32.89
	F1	32.84	33.15	51.11	27.84	33.11	30.98	29.76	45.32	18.29	34.29	28.91	8.56	14.42	28.79	8.42
	ARI	13.52	16.45	23.33	11.41	2.36	15.33	10.56	20.88	15.21	7.23	6.43	9.82	7.95	6.92	5.76
1+B	NMI	23.39	19.54	11.78	21.63	20.71	13.98	11.34	12.78	9.16	18.42	27.86	9.13	17.32	14.73	19.97
	ACC	35.41	41.92	44.84	28.42	40.73	39.44	34.98	34.69	31.25	34.93	43.52	34.18	39.15	33.89	34.41
	F1	36.86	31.81	15.36	17.52	12.35	30.45	21.35	15.32	17.33	13.21	7.94	2.57	35.89	7.66	7.93
	ARI	2.56	13.21	30.33	2.13	1.23	15.64	1.80	9.96	6.82	9.58	2.18	2.64	33.61	2.47	2.95
1+C	NMI	16.00	8.84	17.00	2.20	4.31	7.96	22.39	13.98	9.75	12.68	16.76	8.01	9.85	5.22	4.51
	ACC	34.75	31.08	53.63	19.96	22.25	26.51	42.29	29.16	25.40	27.79	35.97	28.70	34.91	20.58	23.44
	F1	17.73	11.80	51.94	5.63	9.75	5.41	17.53	7.15	11.04	12.28	13.61	9.01	12.27	7.74	10.39
	ARI	4.63	3.09	8.57	2.58	2.59	2.43	15.69	2.48	5.76	5.28	4.46	2.83	11.32	3.39	2.81
2+A	NMI	10.27	14.24	17.42	24.32	30.71	12.16	14.12	11.58	24.83	23.15	35.41	12.47	12.68	10.58	7.32
	ACC	23.67	33.00	44.81	43.83	44.53	25.55	32.97	47.93	41.29	47.52	39.28	37.36	31.14	33.16	33.91
	F1	22.43	31.57	30.57	6.32	8.12	21.90	31.50	38.26	6.75	19.78	25.73	31.13	29.17	8.91	7.11
	ARI	10.02	9.55	9.25	13.21	22.13	1.50	9.43	16.41	12.38	5.36	5.69	10.63	5.32	6.94	6.44
2+B	NMI	18.54	10.91	12.87	20.94	21.33	14.32	19.84	18.41	25.18	30.24	34.27	15.53	18.76	17.43	15.48
	ACC	35.99	30.77	38.54	33.05	36.59	51.87	43.67	39.62	44.39	43.97	45.86	23.49	32.34	33.89	29.12
	F1	10.22	6.51	26.91	9.88	22.13	39.64	20.59	45.73	8.72	21.68	26.39	8.74	6.58	7.63	22.15
	ARI	1.36	3.21	15.58	8.55	10.66	8.59	9.15	22.98	9.54	19.46	4.52	1.54	3.29	2.47	11.67
2+C	NMI	41.23	17.34	11.18	18.48	34.33	42.99	28.12	13.27	29.95	52.35	35.92	16.85	11.23	20.11	29.05
	ACC	56.18	35.89	44.39	29.25	44.77	58.49	53.31	40.53	38.89	62.20	49.19	38.52	40.71	30.85	38.67
	F1	19.62	20.39	31.19	5.58	4.70	5.74	16.81	12.14	4.84	12.78	12.69	25.47	11.16	6.30	11.75
	ARI	33.46	7.05	7.90	10.69	21.78	38.20	23.56	12.47	18.47	40.71	24.72	7.82	14.96	11.63	18.28
3+A	NMI	10.95	11.33	17.55	19.99	29.30	11.97	16.83	15.79	18.55	31.27	23.18	11.88	15.42	15.73	11.76
	ACC	29.33	30.11	59.94	31.82	44.55	48.63	35.91	45.82	30.97	41.93	47.93	24.19	38.28	29.54	34.08
	F1	14.47	13.55	22.14	21.00	23.32	14.58	17.82	27.36	8.64	19.26	22.84	7.58	24.74	15.83	9.66
	ARI	10.22	8.87	18.88	10.03	15.51	13.29	9.34	28.45	7.29	14.85	11.97	8.73	5.01	0.82	6.89
3+B	NMI	24.41	8.51	6.77	20.44	21.12	25.23	9.28	10.92	19.41	28.64	24.57	9.68	13.39	14.71	17.15
	ACC	38.54	30.70	42.63	28.86	36.55	39.87	21.56	43.65	30.44	32.91	38.91	28.18	46.14	33.26	29.28
	F1	12.55	10.25	22.14	6.81	9.52	35.42	23.78	37.98	7.68	7.53	22.84	1.83	15.48	6.84	4.11
	ARI	10.01	9.94	6.64	10.07	4.34	5.19	5.19	17.61	3.26	2.87	1.97	4.42	4.90	6.49	7.42
3+C	NMI	4.32	1.87	10.29	3.17	3.55	26.81	7.83	13.29	2.17	2.55	11.98	11.22	10.58	9.42	9.63
	ACC	31.61	21.85	40.20	27.88	25.21	42.45	25.62	46.85	27.84	24.30	30.65	21.55	40.79	28.75	32.88
	F1	3.32	2.61	21.26	5.42	4.14	38.76	11.97	39.71	5.42	8.33	13.22	12.61	13.75	12.67	2.59
	ARI	0.67	1.33	11.22	2.49	5.05	15.26	7.34	15.38	2.49	5.00	10.60	10.31	10.22	8.31	1.18
4+A	NMI	6.42	6.98	7.78	18.43	7.29	28.17	5.98	16.75	12.37	13.58	21.36	9.42	18.21	11.59	2.85
	ACC	28.35	26.51	45.61	39.27	28.41	44.39	28.37	45.29	26.48	35.32	43.94	24.68	33.58	33.83	21.36
	F1	29.61	12.54	23.14	27.15	19.57	40.15	3.61	51.36	6.15	15.71	19.58	2.21	17.46	4.63	9.24
	ARI	3.65	0.35	0.21	9.36	11.83	14.58	2.84	8.94	4.91	4.19	9.20	1.88	5.33	4.69	3.36
4+B	NMI	30.19	19.56	10.37	24.13	19.87	29.74	9.64	10.29	11.92	23.95	26.85	12.15	10.15	9.44	13.54
	ACC	49.48	37.91	58.64	39.06	37.18	46.83	11.82	48.34	26.34	37.83	39.37	21.89	43.64	17.91	26.21
	F1	39.55	18.43	25.92	21.28	16.55	44.38	7.53	35.17	2.57	13.18	25.62	9.43	28.49	14.46	5.77
	ARI	23.48	4.27	9.81	7.65	8.92	11.79	9.08	11.69	1.84	1.46	3.19	1.57	8.21	7.74	4.81
4+C	NMI	14.54	11.77	3.66	1.53	4.69	20.67	16.85	24.51	2.61	7.64	15.49	10.80	14.91	6.27	2.11
	ACC	21.64	31.55	40.85	31.54	22.37	30.80	36.76	56.13	26.85	24.44	23.45	31.31	49.60	29.53	21.33
	F1	21.48	16.79	29.57	27.94	21.35	29.58	36.36	50.29	24.70	24.24	22.32	19.28	41.25	5.79	19.63
	ARI	1.34	0.21	0.25	1.31	5.61	6.61	10.26	21.53	1.68	3.27	1.55	11.15	10.32	3.38	3.10
AMGC- MF	NMI	41.57	16.72	27.01	31.81	34.30	44.15	19.85	27.10	31.54	51.75	43.00	17.00	27.09	31.30	48.65
	ACC	56.09	34.42	65.59	46.10	49.54	57.50	35.71	64.52	48.41	56.41	58.01	32.52	66.52	43.97	61.74
	F1	57.78	36.34	61.53	35.65	44.71	54.37	32.04	60.02	32.27	55.24	59.14	31.89	61.32	32.45	61.14
	ARI	32.20	8.05	30.25	23.10	23.60	32.08	6.73	29.88	22.10	36.41	34.95	5.43	30.11	25.16	41.52
AMGC	NMI	43.94	25.61	19.75	45.02	58.59	46.55	19.16	27.93	47.67	60.32	43.66	20.80	17.32	43.78	62.62
	ACC	62.00	48.12	58.04	50.68	64.42	61.67	43.28	65.79	57.98	70.73	62.30	43.01	58.46	44.12	65.44
	F1	58.32	47.64	57.58	37.28	56.76	53.38	42.58	65.47	40.63	64.70	59.77	41.76	57.39	29.10	57.76
	ARI	37.98	19.89	15.98	34.63	46.08	38.49	15.70	26.82	37.60	48.92	36.91	16.38	16.80	27.47	50.92
AMGC- SJO	NMI	44.07	21.81	27.31	36.43	40.51	44.20	21.78	26.97	33.16	39.57	43.45	21.93	26.88	33.92	40.55
	ACC	64.03	45.00	69.48	49.75	58.27	62.85	44.57	69.58	50.76	57.18	63.77	44.51	69.54	50.64	57.84
	F1	62.73	38.83	67.45	42.52	51.68	61.25	41.84	67.57	37.03	53.01	61.87	41.78	67.50	38.92	54.31
	ARI	36.47	14.22	31.39	27.25	25.83	34.65	13.88	31.98	25.95	29.86	35.86	14.01	31.86	26.87	30.45

Tab. 3 Running time comparison of various methods under topological centrality-based missing mechanism

方法组合	Cora	Citeseer	PubMed	Amac	Amap
1+A	60.69	88.39	483.37	345.31	343.25
1+B	209.15	72.77	764.93	626.32	372.48
1+C	51.23	66.91	1 677.99	305.76	456.49
2+A	35.08	93.31	1 146.45	345.31	320.87
2+B	184.37	136.96	1 068.36	427.33	448.32
2+C	30.65	65.32	399.49	769.35	245.66
3+A	159.34	24.69	386.74	149.16	15.17
3+B	158.95	68.07	467.64	230.09	74.40
3+C	2.11	3.21	3.71	216.20	2.99
4+A	95.07	143.26	422.36	252.45	103.44
4+B	244.71	186.64	503.26	333.37	132.66
4+C	87.87	121.78	138.33	607.14	63.15
AMGC	3 052.85	3 398.19	7 893.50	15 981.10	5 661.31
AMGC-MF	130.21	336.93	1 755.19	10 798.33	2 516.52
AMGC-SJO	88.09	139.39	602.82	602.82	590.41

Tab. 4 Comparison of clustering performance between AMGC-SJO and AMGC under different missing mechanisms and missing rates

数据集	缺失机制	方法	指标	缺失率为10%	缺失率为30%	缺失率为50%	缺失率为70%	缺失率为90%
Cora	拓扑中心性缺失	AMGC	NMI	49.71	44.86	43.94	23.39	16.95
			ACC	66.06	63.00	62.00	39.99	41.24
			F1	64.99	59.14	58.32	40.51	26.51
			ARI	41.30	38.88	37.98	13.31	10.88
		AMGC-SJO	NMI	44.36	44.51	44.20	44.10	44.19
			ACC	63.57	63.58	62.85	61.05	65.40
			F1	63.58	62.64	61.25	61.16	63.31
			ARI	35.49	35.39	34.65	34.55	38.50
	邻居同质性缺失	AMGC	NMI	48.70	47.14	46.55	35.30	23.84
			ACC	64.92	62.42	61.67	57.13	36.78
			F1	63.91	58.80	53.38	53.71	35.86
			ARI	40.46	38.70	38.49	29.95	11.99
		AMGC-SJO	NMI	44.20	44.20	44.20	44.20	44.20
			ACC	62.85	62.85	62.85	62.85	62.85
			F1	61.25	61.25	61.25	61.25	61.25
			ARI	34.65	34.65	34.65	34.65	34.65
	属性隐私缺失	AMGC	NMI	46.79	44.25	43.66	23.52	10.87
			ACC	63.01	62.42	62.30	37.48	32.13
			F1	61.22	60.50	59.77	36.89	19.42
			ARI	39.10	37.45	36.91	11.99	2.41
		AMGC-SJO	NMI	44.13	43.95	43.45	43.14	42.92
			ACC	64.09	64.20	63.77	62.94	62.01
			F1	62.10	62.10	61.87	61.76	61.70
			ARI	35.89	35.65	35.86	35.71	35.60
Citeseer	拓扑中心性缺失	AMGC	NMI	34.28	27.37	19.16	11.46	6.21
			ACC	58.88	55.06	43.28	32.34	21.85
			F1	57.45	50.53	42.58	31.08	9.51
			ARI	31.91	24.31	15.70	3.29	1.17
		AMGC-SJO	NMI	21.69	21.43	21.81	21.60	21.91
			ACC	44.91	44.97	45.00	44.88	45.12
			F1	38.77	38.73	38.83	38.75	38.81
			ARI	14.16	14.04	14.22	14.17	14.33
	邻居同质性缺失	AMGC	NMI	33.75	33.09	26.68	19.76	11.60
			ACC	60.67	59.93	54.79	40.46	30.30
			F1	54.52	53.67	50.99	39.36	29.18
			ARI	33.21	30.33	24.45	15.13	4.05
		AMGC-SJO	NMI	21.78	21.78	21.78	21.78	21.78
			ACC	44.57	44.57	44.57	44.57	44.57
			F1	41.84	41.84	41.84	41.84	41.84
			ARI	13.88	13.88	13.88	13.88	13.88
	属性隐私缺失	AMGC	NMI	32.90	26.75	20.80	12.21	4.51
			ACC	59.12	55.40	43.01	33.66	23.11
			F1	56.16	51.18	41.76	31.15	12.20
			ARI	32.00	24.30	16.38	4.33	0.23
		AMGC-SJO	NMI	21.78	21.79	21.93	21.72	21.59
			ACC	44.60	44.48	44.51	44.88	44.33
			F1	41.87	41.79	41.78	41.79	41.71
			ARI	13.91	13.98	14.01	13.87	13.76

Tab. 5 Ablation experiment results

算法	Cora		Citeseer		PubMed		Amac		Amap
算法	ACC	NMI	ACC	NMI	ACC	NMI	ACC	NMI	ACC	NMI
AMGC-MF-MS	45.86	33.09	28.07	12.87	45.35	10.13	23.72	17.53	31.20	23.09
AMGC-MF-S	57.64	40.27	32.34	18.65	47.03	13.24	31.80	21.39	42.94	27.73
AMGC-MF	58.68	41.51	43.55	18.25	65.58	27.73	37.94	28.00	49.20	35.31
AMGC-MF+stackM	50.30	44.23	41.51	19.26	67.58	25.63	37.16	29.70	50.92	35.62
AMGC-MF+stackX	59.27	42.58	40.73	21.63	68.11	27.15	32.78	28.43	50.24	36.99
AMGC-SJO	64.03	44.07	45.00	21.81	69.48	27.31	49.75	36.43	58.27	40.51

Tab. 6 Optimal parameters of AMGC-SJO algorithm on different datasets

参数	Cora	Citeseer	PubMed	Amac	Amap
i	55	4	2	3	30
$α$	1.0	4.0	2.0	1.0	0.5
$λ$	10²	10^-3	10^-1	10¹	10⁰

Tab. 6 Optimal parameters of AMGC-SJO algorithm on different datasets

参数	Cora	Citeseer	PubMed	Amac	Amap
i	55	4	2	3	30
$α$	1.0	4.0	2.0	1.0	0.5
$λ$	10²	10^-3	10^-1	10¹	10⁰

Fig. 2 Variation of ACC and NMI with iteration count i

Fig. 3 Variation of ACC and NMI with balance parameter α and regularization parameter λ

Fig. 4 MAE of reconstructed nodes’ membership matrix during training

References 26

[1]	Li Y， Sha C， Huang Xin， Zhang Y. Community detection in attributed graphs： an embedding approach ［J］. Proceedings of the AAAI Conference on Artificial Intelligence， 2018， 32（1）： 338-345.
[2]	Chunaev P. Community detection in node-attributed social networks： a survey ［J］. Computer Science Review， 2020， 37： No.100286.
[3]	Chen X， Chen S， Yao J， et al. Learning on attribute-missing graphs［J］. IEEE Transactions on Pattern Analysis and Machine Intelligence， 2022， 44（2）： 740-757.
[4]	Li M， Chen J， Yu C， et al. Topology-driven attribute recovery for attribute missing graph learning in social internet of things ［J］. IEEE Internet of Things Journal， 2025， 12（11）： 16298-16313.
[5]	Tu W， Guan R， Zhou S， et al. Attribute-missing graph clustering network ［J］. Proceedings of the AAAI Conference on Artificial Intelligence， 2024， 38（14）： 15392-15401.
[6]	Mazumder R， Hastie T， Tibshirani R. Spectral regularization algorithms for learning large incomplete matrices ［J］. Journal of Machine Learning Research， 2010， 11： 2287-2322.
[7]	Srebro N， Rennie J D， Jaakkola T S. Maximum-margin matrix factorization ［C］// NeurIPS 2004. Cambridge： MIT Press， 2004： 1329-1336.
[8]	Taguchi H， Liu X， Murata T. Graph convolutional networks for graphs containing missing features ［J］. Future Generation Computer Systems， 2021， 117： 155-168.
[9]	Vrahatis A G， Lazaros K， Kotsiantis S. Graph attention networks： a comprehensive review of methods and applications ［J］. Future Internet， 2024， 16（9）： No.318.
[10]	宋辉，代杰杰，张卫东，等. 基于变分贝叶斯自编码器的局部放电数据匹配方法［J］. 中国电机工程学报， 2018， 38（19）： 5869-5877.
	Song Hui， Dai Jiejie， Zhang Weidong， et al. A data matching method of partial discharge data based on auto-encoding variational Bayes ［J］. Proceedings of the CSEE， 2018， 38（19）： 5869-5877.
[11]	Şimşek Ö， Jensen D. Navigating networks by using homophily and degree ［J］. Proceedings of the National Academy of Sciences of the United States of America， 2008， 105（35）： 12758-12762.
[12]	Tu W， Zhou S， Liu X， et al. Initializing then refining： a simple graph attribute imputation network ［C］// IJCAI 2022. California： IJCAI， 2022： 3494-3500.
[13]	Xia R， Zhang C， Li A， et al. Attribute imputation autoencoders for attribute-missing graphs ［J］. Knowledge-Based Systems， 2024， 291： No.111583.
[14]	Li M， Zhang Y， Zhang W， et al. CSAT： contrastive sampling-aggregating Transformer for community detection in attribute-missing networks ［J］. IEEE Transactions on Computational Social Systems， 2024， 11（2）： 2277-2290.
[15]	Jin D， Wang R， Wang T， et al. AMER： a new attribute-missing network embedding approach ［J］. IEEE Transactions on Cybernetics， 2023， 53（7）： 4306-4319.
[16]	Lee D D， Seung H S. Learning the parts of objects by non-negative matrix factorization ［J］. Nature， 1999， 401（6755）： 788-791.
[17]	Ding C H Q， Li T， Jordan M I. Convex and semi-nonnegative matrix factorizations ［J］. IEEE Transactions on Pattern Analysis and Machine Intelligence， 2010， 32（1）： 45-55.
[18]	Kuang D， Yun S， Park H. SymNMF： nonnegative low-rank approximation of a similarity matrix for graph clustering ［J］. Journal of Global Optimization， 2015， 62（3）： 545-574.
[19]	Chapelle O， Schölkopf B， Zien A. Semi-supervised learning ［M］. Cambridge： MIT Press， 2006： 23-45.
[20]	Zhu X， Goldberg A B. Introduction to semi-supervised learning， SLAIML ［M］. Cham： Springer， 2009： 78-79.
[21]	Golub G H， van Loan C F. Matrix computations ［M］. 4th ed. Baltimore， MD： Johns Hopkins University Press， 2013： 257-263.
[22]	Boyd S， Vandenberghe L. Convex optimization ［M］. Cambridge： Cambridge University Press， 2004： 1-730.
[23]	吴雪莎. 矩阵Frobenius范数的几个不等式［J］. 西南师范大学学报（自然科学版）， 2014， 39（6）： 5-8.
	Wu Xuesha. Several inequalities about Frobenius norm of matrices ［J］. Journal of Southwest China Normal University （Natural Science Edition）， 2014， 39（6）： 5-8.
[24]	Pham T M， Farrell R， Tran L N. Revisiting the MIMO capacity with per-antenna power constraint： fixed-point iteration and alternating optimization ［J］. IEEE Transactions on Wireless Communications， 2019， 18（1）： 388-401.
[25]	Zheng Z， Chen X， Lin X. Kernel based dual-channel attributed graph community detection ［J］. IEEE Transactions on Network Science and Engineering， 2024， 11（1）： 592-603.
[26]	郭琳. 基于图嵌入的深度属性图聚类模型研究［D］. 南京：南京航空航天大学， 2022： 1-120.
	Guo Lin. Research on deep models of attributed graph clustering via graph embedding ［D］. Nanjing： Nanjing University of Aeronautics and Astronautics， 2022： 1-120.