Graph Neural Networks (GNNs) are widely used for representation learning on graphs, but most methods assume static topologies, making them inefficient on evolving networks where edges change over time. Existing dynamic approaches either model graph evolution through temporal GNN architectures without focusing on efficient dynamic maintenance, or are restricted to linear propagation models based on Personalized PageRank. In this work, we study how to efficiently maintain node representations for non-linear GNN propagation under edge insertions and deletions. The propagation has no learned parameters, and only a classifier applied afterward is trained. For a broad class of standard activation functions, we develop a residual-based dynamic algorithm that selectively propagates local errors via push operations, maintaining an approximation to the evolving fixed point without full recomputation. We prove that our method achieves amortized O(1/ε) update time per graph change under a degree-normalized error guarantee. Our approach uses a potential-based analysis in a degree-scaled norm and, in contrast to prior work on the linear case, requires no randomness assumptions on either the update sequence or the input vector. For the linear special case, we additionally provide an exact dynamic algorithm via low-rank matrix inverse updates. Experiments on benchmark datasets show that incorporating non-linearity improves accuracy while preserving efficient update performance, yielding a scalable and theoretically grounded method for maintaining this propagation on dynamic graphs.
Figures & tables
Figure 1: Results on the dynamic arxiv dataset. Accuracy is similar for the linear and hard-tanh models. Since this is the smallest dynamic experiment, recomputation from scratch is still faster than the “single” strategy, which handles the harder online setting of propagating after each edge update. Batched updates are much faster than both.
Figure 2: Results on the dynamic products dataset. Accuracy peaks early and then slowly decreases across later snapshots. Unlike on arxiv , recomputation from scratch becomes much more expensive on this larger graph, while dynamic updates keep the propagation cost substantially lower.
Figure 3: Results on the dynamic SBM-500K dataset. The dynamic methods remain efficient even with both edge insertions and deletions. After the first snapshot, their per-snapshot cost stays low and stable while preserving the accuracy behavior of the linear and hard-tanh models.
Appendix figures & tables7 assets
Supplementary material from the paper’s appendix.
Appendix
Symbol
Meaning
n
Number of nodes.
A
Adjacency matrix of the current graph, with a self-loop at every node.
D
Diagonal degree matrix of the current graph.
d(i)
Degree of node i . Since every node has a self-loop, d(i)≥1 .
β∈[0,1]
Normalization parameter in W=D−βADβ−1 .
W
Normalized weight matrix.
Appendix
Table 2: Parameters used in the paper.
Dataset
Nodes ( n )
Edges
Feature dimension
Classes
Cornell
183
164
1,703
5
Texas
183
167
1,703
5
Wisconsin
251
231
1,703
5
Squirrel
5,201
159,248
2,089
5
Actor
7,600
14,992
932
5
Chameleon
2,277
23,370
2,325
5
Appendix
Table 3: Statistics for datasets used in Section 4.1 .
Hyperparameter
Values
α
{0.01,0.1,0.2,0.3,0.5,0.7}
β
{0.5,1.0}
cfeat
{1,3,5}
Classifier
{linear,MLP}
Learning rate
{10−3,10−2,10−1,0.5}
Weight decay
5×10−4
Appendix
Table 4: Hyperparameter search grid used for each activation–dataset pair.
Method
Cham. (filt.)
Squi. (filt.)
GCN
35.6 ± 0.8
34.8 ± 0.4
GAT
35.3 ± 0.8
34.9 ± 0.4
APPNP
36.1 ± 0.9
34.9 ± 0.4
ACM-GCN
38.9 ± 0.7
35.9 ± 0.5
GREAD
44.0 ± 1.0
41.0 ± 0.5
Linear
41.1 ± 1.1
40.4 ± 0.6
Appendix
Table 6: Test accuracy on the filtered Chameleon and Squirrel datasets of [ 20 ] . The best method is underlined .
Activation
Cham. (filt.)
Squi. (filt.)
Linear
0.01/0.5/1/M/0.1
0.01/0.5/1/M/0.1
Sh Tanh -0.5
0.7/0.5/3/M/0.001
0.5/0.5/1/M/0.01
Sh Tanh -1.2
0.5/0.5/5/M/0.01
0.7/0.5/1/M/0.001
Appendix
Table 7: Configuration of (α,β,cfeat,classifier,lr) selected by validation accuracy for each (activation, dataset) cell of Table 6 . “L” denotes the linear classifier and “M” denotes the MLP.
Figure 4: Total propagation time of varying ϵ values. The propagation time scales inversely with ϵ .
Figure 5: Total propagation time of varying α values. Our theoretical bounds show that the propagation time scales with 1/α2 .