Differentiable Koopman Operator for Contrastive Learning on Dynamic Graphs

Md Abrar Jahin; Taufikur Rahman Fuad; Md Rizwan Parvez

cs.LGOct 2, 2026

Differentiable Koopman Operator for Contrastive Learning on Dynamic Graphs

Authors: Md Abrar Jahin, Taufikur Rahman Fuad, Md Rizwan Parvez

Organizations: University of Southern California · Islamic University of Technology · Qatar Computing Research Institute (QCRI)

Abstract

Real-world interaction networks are inherently dynamic: edges form and dissolve as node behavior shifts over time. Most snapshot-based contrastive methods encode temporal dependencies implicitly in encoder weights, without an explicit model of how node representations evolve, making them brittle under distribution shifts. We propose KAIROS (Koopman-Aligned Invariant Representations for Open Dynamic Systems), a self-supervised framework that embeds a differentiable Koopman operator within a dynamic graph contrastive learning loop to linearize temporal evolution in the learned embedding space. A dual-view encoder pairs raw node features with a graph-diffused structural view and is optimized with multi-granularity contrastive objectives across temporal windows. For anomaly detection, KAIROS uses the Koopman prediction residual together with temporal inconsistency and local neighborhood deviation to separate irregular behavior from predictable graph evolution. Evaluated on nine dynamic graph benchmarks, KAIROS achieves state-of-the-art anomaly detection results on all nine datasets, with gains of up to 23.15 ROC-AUC points over prior work, while remaining competitive for unsupervised node classification. These results show that explicit dynamics modeling provides a scalable and effective inductive bias for temporal graph representation learning.

Figures & tables

Figure 1: Overview of the KAIROS architecture.

Dataset	$\|\mathcal{V}\|$	$\|\mathcal{E}\|$	Clf. classes	Ano. $\%$
DBLP	25,387	185,480	10	2.4
Bitcoinotc	5,881	35,592	3	5.1
BITotc	4,863	28,473	7	6.2
BITalpha	3,219	19,364	7	9.3
TAX51	132,524	467,279	51	4.5
Reddit	898,194	2,575,464	3	0.7

Table 1: Statistics for the nine temporal graph benchmarks used in this work.

Method	DBLP	Bitcoinotc	BITotc	BITalpha	TAX51	Reddit	MOOC	Arxiv	Elliptic
LP	52.18	41.97	60.24	76.97	28.82	66.71	43.09	64.17	93.13
GCN	71.35	54.61	59.24	76.19	40.42	67.83	78.87	22.62	90.26
GAT	70.01	52.46	62.59	80.21	38.82	69.31	78.87	35.26	90.13
GraphSAGE	72.36	57.29	62.68	79.89	40.80	71.37	77.38	28.99	90.41
CAW	55.64	59.85	63.56	77.64	36.52	67.79	—	—	—
TGAT	57.48	58.56	62.53	77.63	34.50	65.57	—	—	—

Table 2: Node classification Micro-F1 (%) across all nine datasets. Bold marks the column winner and underline marks the second-best value. — denotes OOM or not evaluated on extended datasets.

Method	DBLP	Bitcoinotc	BITotc	BITalpha	TAX51	Reddit	MOOC	Arxiv	Elliptic
AEGIS	59.18	50.97	53.17	54.13	54.52	OOM	67.78	OOM	OOM
CoLA	71.06	79.44	81.80	77.65	77.29	OOM	55.76	OOM	OOM
ANEMONE	70.69	79.28	80.09	72.70	76.61	OOM	53.95	OOM	OOM
GRADATE	69.53	74.14	73.12	68.64	OOM	OOM	46.42	OOM	OOM
NetWalk	54.42	56.13	58.46	64.12	OOM	OOM	67.45	OOM	OOM
TADDY	50.93	50.58	51.48	50.42	OOM	OOM	55.58	OOM	OOM

Table 3: Dynamic-graph anomaly detection ( ROC-AUC %) across all nine datasets. OOM = out of memory; Bold marks best per column; underline marks second best.

Variant	DBLP	Bitcoinotc	BITotc	BITalpha	TAX51	Reddit
Canonical (GAT, learnable $\tau{\approx}0.07$ )	$74.58{\pm}0.53$	$55.68{\pm}0.24$	$62.70{\pm}0.27$	$77.31{\pm}0.33$	$43.04{\pm}0.15$	—
$+$ fixed $\tau{=}0.5$ (GAT)	$\mathbf{78.07{\pm}0.36}$	$56.18{\pm}0.61$	$62.70{\pm}0.29$	$77.57{\pm}0.23$	$\mathbf{44.85{\pm}0.20}$	—
$+$ Dataset-specific backbone ( $\tau{=}0.5$ )	$\mathbf{78.69{\pm}0.32}$	$\mathbf{60.39{\pm}0.66}$	$\mathbf{64.29{\pm}0.39}$	$\mathbf{78.14{\pm}0.08}$	$\mathbf{44.85{\pm}0.20}$	$70.31{\pm}0.11$
$\Delta$ canonical $\to$ final (ours)	$+4.11$ (large)	$+4.71$ (large)	$+1.59$ (moderate)	$+0.83$ (small)	$+1.81$ (large)	—

Table 4: Cross-dataset component ablation for node classification accuracy (%). Each row adds one design choice to the previous configuration. Results are mean

\pm

std.

Dataset	$S_{1}$	$S_{2}$	$S_{3}$	Combined	CLDG++
DBLP	89.12	88.31	78.42	92.22	86.41
Bitcoinotc	94.11	92.77	81.20	96.94	81.97
BITotc	90.34	88.12	76.18	93.95	82.92
BITalpha	89.76	87.18	78.56	92.79	79.71
TAX51	88.01	86.93	71.72	92.61	81.02
Reddit ⋆	91.72	86.47	55.02	92.13	72.77

Table 5: Anomaly signal decomposition ROC-AUC %:

S_{1}

temporal inconsistency,

S_{2}

Koopman residual,

S_{3}

neighbourhood deviation, and their

z

-normalised sum, across all nine datasets. KAIROS uses

\tau{=}0.5

. CLDG++ shown for reference.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Input	DBLP	Bitcoinotc	BITotc	BITalpha	TAX51	Reddit
LP	$A,Y$	51.54	31.25	50.28	67.18	24.36	60.88
GCN	$X,A,Y$	71.08	54.41	50.52	73.65	33.64	63.26
GAT	$X,A,Y$	69.21	50.60	48.34	71.40	30.21	58.01
GraphSAGE	$X,A,Y$	71.99	56.30	56.99	74.24	33.79	65.74
CAW	$X,A,T$	51.38	57.92	53.59	71.99	29.52	57.80
TGAT	$X,A,T$	51.96	55.73	49.53	68.03	28.44	56.81

Appendix

Table 6: Node classification Weighted-F1 (%). Same protocol and legend as Table 2 .

Dataset	$\tau{=}0.07$ (canonical)	$\tau{=}0.5$ (ours)	$\Delta$	$p$ -value	Effect
DBLP	$74.58{\pm}0.53$	$\mathbf{78.07{\pm}0.36}$	$+3.49$	$<10^{-3}$	large
TAX51	$43.04{\pm}0.15$	$\mathbf{44.85{\pm}0.20}$	$+1.81$	$<10^{-4}$	large
Bitcoinotc	$55.68{\pm}0.24$	$56.18{\pm}0.61$	$+0.50$	$0.148$	small
BITotc	$62.70{\pm}0.27$	$62.70{\pm}0.29$	$+0.00$	$0.990$	none
BITalpha	$77.31{\pm}0.33$	$77.57{\pm}0.23$	$+0.26$	$0.238$	none

Appendix

Table 7: Temperature ablation with GAT backbone held fixed ( Accuracy %).

\tau{=}0.07

(SimCLR default, learnable) vs

\tau{=}0.5

(ours, fixed), across all five multi-seed datasets.

p

-values from Welch’s

t

-test.

Backbone	DBLP	Bitcoinotc	BITotc	BITalpha	TAX51
GAT	$78.07{\pm}0.36$	$56.18{\pm}0.61$	$62.70{\pm}0.29$	$77.57{\pm}0.23$	$\mathbf{44.85{\pm}0.20}$
GCN	$73.51{\pm}0.22$	$\mathbf{60.39{\pm}0.66}$	$63.74{\pm}0.38$	$\mathbf{78.14{\pm}0.08}$	$40.41{\pm}0.18$
SGC	$72.72{\pm}0.33$	$58.97{\pm}1.16$	$\mathbf{64.29{\pm}0.39}$	$78.03{\pm}0.07$	$40.35{\pm}0.32$
SAGE	$\mathbf{78.69{\pm}0.32}$	$54.92{\pm}0.87$	$63.15{\pm}0.14$	$77.79{\pm}0.13$	$44.30{\pm}0.69$
H2GCN	$78.16{\pm}0.25$	$53.95{\pm}1.60$	$62.75{\pm}0.25$	$77.82{\pm}0.06$	$44.21{\pm}0.66$

Appendix

Table 8: Backbone ablation: 5-seed Accuracy (%) reported as mean

\pm

std. Bold marks the best backbone per dataset. KAIROS main results use SAGE for DBLP, GCN for Bitcoinotc/BITalpha, SGC for BITotc, GAT for TAX51.

Dataset	$\lambda_{\text{koop}}{=}1$ (full)	$\lambda_{\text{koop}}{=}0$ (ablated)	$\Delta$
DBLP	89.96	90.41	$+0.45$
Bitcoinotc	97.09	97.56	$+0.47$
BITotc	93.86	92.81	$-1.05$
BITalpha	92.36	91.78	$-0.58$
TAX51	92.50	92.96	$+0.46$
Reddit	92.13	95.16	$+3.03$

Appendix

Table 9: Koopman regulariser

\lambda_{\text{koop}}

ablation on anomaly detection at seed

=

24 (ROC-AUC %). Mixed effects across datasets: the regulariser helps only BITotc and BITalpha. For four of the six datasets (including Reddit) the Koopman operator learns equally well without its residual-regularisation term, suggesting

\lambda_{\text{koop}}

should be tuned per-dataset. Bold marks the winning configuration.

Task	DBLP	Bitcoinotc	BITotc	BITalpha	TAX51
$P(\max_{5}>\text{CLDGPP})$ , classification	1.0000	0.9914	0.0000	0.0000	1.0000
$P(\max_{5}>\text{CLDG++})$ , anomaly	1.0000	1.0000	1.0000	1.0000	1.0000

Appendix

Table 10: Bootstrap probability that the best of 5 resampled seeds exceeds the CLDG++ paper value,

B{=}10{,}000

resamples with replacement from KAIROS’s 5-seed distribution. Anomaly dominates; classification mirrors the head-to-head verdict.

Figure 4: Per-signal anomaly ROC-AUC across all nine datasets: $S_{1}$ (temporal inconsistency), $S_{2}$ (Koopman residual), $S_{3}$ (neighborhood deviation), and their $z$ -normalized combination. The combined score outperforms every individual signal on all primary datasets. $S_{3}$ collapses on the long-trajectory graphs (Reddit, MOOC, Arxiv), where $S_{1}{+}S_{2}$ carry the anomaly signal load.

Dataset	Backbone	Snapshots	Views	Strategy	Epochs
DBLP	SAGE	4	4	sequential	200
Bitcoinotc	GCN	4	3	sequential	25
BITotc	SGC	4	4	random	50
BITalpha	GCN	6	4	sequential	200
TAX51	GAT	8	5	random	200
Reddit	GCN	5	4	random	200

Appendix

Table 11: Per-dataset configuration for node classification , selected on held-out validation accuracy.

\tau{=}0.5

and PPR

\alpha{=}0.15

are fixed across all datasets. Backbone is SAGE/GAT for larger graphs; GCN/SGC for small rating graphs (

|\mathcal{V}|\leq 8\,\text{K}

Dataset	Snapshots	Views	Strategy	Epochs
DBLP	4	4	sequential	200
Bitcoinotc	4	4	sequential	25
BITotc	4	4	sequential	50
BITalpha	5	5	sequential	100
TAX51	4	4	sequential	200
Reddit	4	4	sequential	200

Appendix

Table 12: Per-dataset configuration for anomaly detection . Backbone,

\tau

, and

\alpha

are identical to Table 11 . Snapshot count and view count differ from the classification setting for BITalpha, TAX51, and Reddit.

Figure 5: Average node-classification accuracy (5-seed mean

\pm

std) vs. trainable parameter count. KAIROS (GCN, 60 K params) achieves the highest average accuracy at only

{\sim}1.5\times

the inference cost of CLDG++ (41 K params), placing it on the Pareto frontier of the accuracy–efficiency trade-off.

Figure 6: Koopman regularizer ablation (ROC-AUC %). Blue:

\lambda_{\text{koop}}{=}1

(full); orange:

\lambda_{\text{koop}}{=}0

(ablated). Annotated deltas shown where

|\Delta|{>}1.5

. The regularizer helps on BITotc and BITalpha but hurts on Reddit (

+3.03

), confirming that

\lambda_{\text{koop}}

must be tuned per dataset.

Dataset	Domain	$\|\mathcal{V}\|$	$n$	Anomaly nodes ( $2nm$ )	Anomaly %
DBLP	Citation network	25,387	20	600	2.37
Bitcoinotc	Bitcoin trust graph	5,881	10	300	5.10
BITotc	Bitcoin rating graph	4,863	10	300	6.17
BITalpha	Bitcoin rating graph	3,219	10	300	9.32
TAX51	Temporal (51-class)	132,524	200	6,000	4.53
Reddit	Community dynamics	898,194	200	6,000	0.668

Appendix

Table 13: Per-dataset anomaly injection details under the CLDG++ protocol Xu et al. (2024) . Clique size

m{=}15

and candidate pool

k{=}50

are fixed across all datasets;

n

varies to maintain a realistic anomaly fraction. Domain labels follow the characterization in Section 5.1 .

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The rich information underlying graphs has inspired further investigation of unsupervised graph representation. Existing studies mainly depend on node features and topological properties within static graphs to create self-supervised signals, neglecting the temporal components carried by real-world graph data, such as timestamps of edges. To overcome this limitation, this paper explores how to model temporal evolution on dynamic graphs elegantly. Specifically, we introduce a new inductive bias, namely temporal translation invariance, which illustrates the tendency of the identical node to keep similar labels across different timespans. Based on this assumption, we develop a dynamic graph representation framework CLDG that encourages the node to maintain locally consistent temporal translation invariance through contrastive learning on different timespans. Except for standard CLDG which only considers explicit topological links, our further proposed CLDG++ additionally employs graph diffusion to uncover global contextual correlations between nodes, and designs a multi-scale contrastive learning objective composed of local-local, local-global, and global-global contrasts to enhance representation capabilities. Interestingly, by measuring the consistency between different timespans to shape anomaly indicators, CLDG and CLDG++ are seamlessly integrated with the task of spotting anomalies on dynamic graphs, which has broad applications in many high-impact domains, such as finance, cybersecurity, and healthcare. Experiments demonstrate that CLDG and CLDG++ both exhibit desirable performance in downstream tasks including node classification and dynamic graph anomaly detection. Moreover, CLDG significantly reduces time and space complexity by implicitly exploiting temporal cues instead of complicated sequence models.

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