Learning Propagation Geometry from Message-Passing Feedback
Authors: Yingxu Wang, Kunyu Zhang, Xinwang Liu, Mengzhu Wang, Siyang Gao, Chang Tang, Nan Yin
Organizations: The Chinese University of Hong Kong · The Education University of Hong Kong · National University of Defense Technology · Hebei University of Technology · City University of Hong Kong · Huazhong University of Science and Technology
Learning local geometry enables graph neural networks (GNNs) to adapt how they compare and integrate neighborhood information. However, estimating geometry from aggregated representations can overlook variation among individual messages and dependencies across feature dimensions. We propose GeoF, a recurrent framework that jointly evolves node features and propagation geometry through message-passing feedback. Each node maintains a local symmetric positive-definite geometry, initialized from a structure-aware prototype atlas and parameterized in block log-triangular coordinates. At each step, the geometry determines neighborhood weights, while triangular frame transport maps transformed source messages into the target node's local coordinates before aggregation. Weighted second-order statistics of residuals between aligned messages and the transformed target state capture directional variation and within-block dependencies, yielding a geometric update target. A shared controller learns complementary corrections through task supervision. A bounded log-triangular update combines these corrections, the target, and the previous geometric state while preserving positive definiteness. The geometry governs subsequent propagation, closing the feedback loop. With parameters shared across recurrent steps, task-specific readouts support node classification, link prediction, and graph classification. Experiments on benchmark datasets show that GeoF consistently outperforms state-of-the-art GNN baselines.
Figures & tables
Model
CiteSeer
PubMed
CS
Physics
Photo
Computers
NC
LP
NC
LP
NC
LP
NC
LP
NC
LP
NC
LP
GCN
71.5 ±1.7
92.3 ±0.9
87.6 ±0.5
92.9 ±0.6
93.8 ±0.4
92.7 ±0.6
93.5 ±0.2
92.6 ±1.3
92.1 ±0.5
86.1 ±0.7
87.8 ±0.7
86.9 ±0.8
GIN
70.6 ±1.2
93.0 ±1.0
86.6 ±0.6
89.5 ±0.7
91.3 ±0.6
93.3 ±0.7
94.2 ±0.5
92.1 ±0.8
92.7 ±0.5
87.7 ±0.4
87.1 ±0.8
84.1 ±1.2
ML 2 -GCL
73.7 ±2.0
93.8 ±1.0
87.9 ±0.5
95.8 ±0.3
92.1 ±0.5
97.1 ±0.1
93.4 ±1.6
97.2 ±0.1
92.8 ±0.8
96.7 ±0.1
87.4 ±0.8
95.8 ±0.3
AMPs
74.4 ±1.8
93.5 ±0.9
88.9 ±0.3
96.6 ±0.3
94.6 ±0.3
97.1 ±0.2
96.1 ±0.1
96.6 ±0.6
94.6 ±0.8
96.8 ±0.2
90.2 ±0.6
96.0 ±0.1
WaveGC
75.4 ±1.9
92.8 ±0.7
87.6 ±0.5
97.5 ±0.3
94.4 ±0.3
97.1 ±0.2
96.2 ±0.1
97.2 ±0.2
94.6 ±0.8
97.7 ±0.4
90.2 ±0.8
97.5 ±0.4
Table 1: Performance comparisons (in %) between baselines and GeoF for Node Classification (NC) and Link Prediction (LP) on different datasets. Bold indicates the best performance.
Model
PROTEINS
Mutag
NCI1
FRANK
BBBP
molhiv
GCN
75.3 ±1.9
79.8 ±1.8
76.0 ±1.0
63.3 ±2.2
87.4 ±2.0
75.8 ±2.0
GIN
76.7 ±1.7
80.1 ±1.9
78.0 ±1.2
68.9 ±1.7
89.5 ±2.1
77.3 ±2.0
ML 2 -GCL
77.9 ±1.8
81.9 ±1.9
80.7 ±1.3
70.7 ±1.8
90.9 ±2.2
81.3 ±1.6
AMPs
78.8 ±2.0
83.4 ±1.6
82.0 ±1.1
72.4 ±1.8
92.5 ±2.3
82.0 ±2.4
WaveGC
79.0 ±1.8
82.3 ±2.3
81.6 ±1.8
72.6 ±1.9
92.6 ±1.8
82.0 ±1.9
SPARROW
78.7 ±1.5
83.7 ±1.9
80.9 ±1.2
72.3 ±2.2
91.8 ±1.9
81.5 ±2.3
Table 2: Performance comparisons (in %) between baselines and GeoF for Graph Classification.
Figure 1: (a) Ablations on Mutagenicity and NCI1. (b) Sensitivity to block count B and atlas size K on NCI1. (c), (d) Evolution depth and perturbation stability on PubMed, respectively.
Model
CiteSeer
Photo
PROTEINS
BBBP
GCN
71.5 ±1.7
92.1 ±0.5
75.3 ±1.9
87.4 ±2.0
GCN w/ SO
72.3 ±1.9
92.4 ±1.1
77.0 ±2.4
86.7 ±2.5
GIN
70.6 ±1.2
92.7 ±0.5
76.7 ±1.7
89.5 ±2.1
GIN w/ SO
73.3 ±1.0
93.6 ±1.4
78.5 ±1.5
91.3 ±2.2
ARGNN
75.6 ±1.2
94.9 ±0.5
78.0 ±1.8
92.3 ±1.7
ARGNN w/ SO
74.4 ±2.3
95.3 ±0.6
78.3 ±2.2
91.9 ±2.1
Table 3: Performance comparison (in %) between baselines and their second-order (SO) variants. Bold indicates the best performance.
Feedback
Photo
CS
Mutag.
NCI1
Mean-only
94.9 ±0.6
95.1 ±0.3
83.7 ±1.7
81.9 ±1.6
Diagonal
95.1 ±0.3
95.2 ±0.4
83.2 ±2.0
80.4 ±1.0
One-shot
94.6 ±0.5
94.8 ±0.3
83.1 ±1.8
79.8 ±2.7
Full
96.0 ±0.4
95.8 ±0.3
84.9 ±1.7
83.0 ±1.5
Table 4: Performance comparison (in %) among different residual feedback statistics. Bold indicates the best performance.
Appendix figures & tables7 assets
Supplementary material from the paper’s appendix.
Appendix
Symbol
Description
G=(V,E,X),A
Attributed graph with node set V , edge set E , feature matrix X , and adjacency matrix A .
n,F0,C
Numbers of nodes, input features, and classes, respectively.
ui,U,q
Fixed structural signature of node i , signature matrix, and signature dimension.
N(i),N(i)
Neighborhood of node i and its extension with a self-loop.
d,B,m
Hidden dimension, number of geometric blocks, and block size, with d=Bm .
K,L
Numbers of geometric prototypes and recurrent evolution steps.
Appendix
Table 5: Summary of key notations.
Datasets
Graphs
Avg. Nodes
Avg. Edges
Classes
CiteSeer
-
3,327
9,104
6
PubMed
-
19,717
88,648
3
CS
-
18,333
163,788
15
Physics
-
34,493
495,924
5
Photo
-
7,650
238,162
8
Computers
-
13,752
491,722
10
Appendix
Table 6: Statistics of the experimental datasets.
Type
Model
NC Avg.
LP Avg.
GC Avg.
Overall Avg.
Avg. Rank
Avg. Gain
pHolm
General GNNs
GCN
87.7
90.6
76.3
84.9
13.2
+5.7
2.91×10−15
GIN
87.1
90.0
78.4
85.2
13.3
+5.4
1.82×10−15
ML 2 -GCL
87.9
96.1
80.6
88.2
9.4
+2.4
1.83×10−7
AMPs
89.8
96.1
81.9
89.3
5.0
+1.3
2.07×10−2
WaveGC
89.7
96.6
81.7
89.4
4.4
+1.2
4.17×10−2
SPARROW
87.8
96.5
81.5
88.6
7.6
+1.9
6.45×10−5
Appendix
Table 7: Cross-task aggregate comparison across all 18 reported dataset–task settings. Bold indicates the best result.
Figure 2: Ablation studies on BBBP and ogbg-molhiv in (a), and PubMed and Computers in (b); sensitivity to geometric blocks B and atlas prototypes K on PubMed in (c) and Mutagenicity in (d).
Methods
PubMed
CS
Computers
NCI1
Mutagenicity
ogbg-molhiv
GCN
0.0086
0.0167
0.0153
0.2094
0.2312
2.1633
AMPs
0.0393
0.0478
0.0761
0.3170
0.3173
2.8410
G 2 Former
0.0637
0.0739
0.0640
0.2011
0.2173
2.2996
SPDGNN
0.0277
0.0321
0.0283
0.2073
0.2287
2.3973
ARGNN
0.0902
0.1573
0.4328
0.2437
0.3383
2.5373
GeoF
0.0683
0.1207
0.1810
0.4391
0.5357
6.0456
Appendix
Table 8: Time consumption of different methods in the training stage for each epoch (in seconds).
Methods
PubMed
CS
Computers
NCI1
Mutagenicity
ogbg-molhiv
GCN
0.7
1.3
1.3
0.8
0.8
0.9
AMPs
2.7
4.9
5.0
1.3
1.6
2.0
G 2 Former
3.3
3.2
3.0
1.5
1.5
1.4
SPDGNN
1.3
2.9
1.9
0.9
1.0
0.9
ARGNN
4.8
8.3
21.3
2.7
2.6
3.8
GeoF
6.4
10.9
15.2
3.5
3.0
4.8
Appendix
Table 9: GPU memory consumption of different methods in the training stage (in GB).
Figure 3: (a), (b) show t-SNE visualizations of node representations learned by ARGNN and GeoF. (c), (d) compare representations with the learned geometric correction disabled and enabled.