Causality-Driven Disentangled Representation Learning in Multiplex Graphs

Saba Nasiri; Selin Aviyente; Dorina Thanou

cs.LGMar 25, 2026

Causality-Driven Disentangled Representation Learning in Multiplex Graphs

Authors: Saba Nasiri, Selin Aviyente, Dorina Thanou

Organizations: EPFL, Lausanne, Switzerland · Department of Electrical and Computer Engineering at Michigan State University, East Lansing, MI

Abstract

Learning representations from multiplex graphs, i.e., multi-layer networks where nodes interact through multiple relation types, is challenging due to the entanglement of shared (common) and layer-specific (private) information, which limits generalization and interpretability. In this work, we introduce a causal inference-based framework that disentangles common and private components in a self-supervised manner. CaDeM jointly (i) aligns shared embeddings across layers, (ii) enforces private embeddings to capture layer-specific signals, and (iii) applies backdoor adjustment to ensure that the common embeddings capture only global information while being separated from the private representations. Experiments on synthetic and real-world datasets demonstrate consistent improvements over existing baselines, highlighting the effectiveness of our approach for robust and interpretable multiplex graph representation learning.

Figures & tables

Panel of Figure 1 — Fig. 1 : CaDeM Network architecture for extracting common and private embeddings, with losses enforcing desired characteristics.

Fig. 2 : Causal diagram illustrating the roles of private and common embeddings in the layer-specific prediction task.

Embeddings	Final labels		Labels of layer 1		Labels of layer 2		Labels of layer 3
	Macro-F1	Micro-F1	Macro-F1	Micro-F1	Macro-F1	Micro-F1	Macro-F1	Micro-F1
Combined embeddings	0.8178 $\pm$ 0.0516	0.8200 $\pm$ 0.0510	0.9611 $\pm$ 0.0476	0.9600 $\pm$ 0.0490	0.9799 $\pm$ 0.0246	0.9800 $\pm$ 0.0245	0.9770 $\pm$ 0.0282	0.9800 $\pm$ 0.0245
Private embeddings of the 1st layer	0.8158 $\pm$ 0.0545	0.8202 $\pm$ 0.0511	0.9904 $\pm$ 0.0191	0.9900 $\pm$ 0.0200	0.4714 $\pm$ 0.0917	0.4900 $\pm$ 0.0663	0.2463 $\pm$ 0.0533	0.2600 $\pm$ 0.0374
Private embeddings of the 2nd layer	0.1946 $\pm$ 0.0684	0.2200 $\pm$ 0.0678	0.2502 $\pm$ 0.1208	0.2600 $\pm$ 0.1158	0.9476 $\pm$ 0.0032	0.9500 $\pm$ 0.0000	0.2774 $\pm$ 0.0659	0.3500 $\pm$ 0.0548
Private embeddings of the 3rd layer	0.3248 $\pm$ 0.0483	0.3500 $\pm$ 0.0316	0.3063 $\pm$ 0.0274	0.3200 $\pm$ 0.0245	0.3119 $\pm$ 0.0896	0.3500 $\pm$ 0.0949	0.9791 $\pm$ 0.0260	0.9800 $\pm$ 0.0245

TABLE I : Comparing performance (Macro- and Micro-F1) of embeddings across selected and layer-specific labels on Syn1.

Models	Macro F1	Micro F1
DeepWalk	$0.7709\pm 0.1075$	$0.7800\pm 0.0927$
Node2Vec	$0.3033\pm 0.0737$	$0.3300\pm 0.0510$
VGAE	$0.7962\pm 0.0294$	$0.8000\pm 0.0316$
DGI	$0.6782\pm 0.1954$	$0.6800\pm 0.1913$
MVGRL	$0.7779\pm 0.1098$	$0.7800\pm 0.1122$
GraphMAE	$0.7933\pm 0.0600$	$0.8000\pm 0.0548$

TABLE II : Comparison of Macro- and Micro-F1 scores for the final label prediction across different baselines on Syn1.

Fig. 3 : Clustering performance as a function of the trade-off parameter $\gamma$ on Syn2. This plot reports the ARI score obtained by applying K-means clustering to the learned embeddings.

Panel of Figure 6 — Fig. 4 : Visualization of predicted cluster assignments for Syn3. Panel (a) shows the clusters obtained from the common embeddings. Panels (b) and (c) show the clusters obtained from the private embeddings of layer 1 and layer 2, respectively, with the corresponding ground-truth center nodes overlaid.

Embeddings	Layer 1 clusters		Layer 2 clusters
Embeddings	ARI	NMI	ARI	NMI
Common embedding	0.0099 $\pm$ 0.0590	0.0077 $\pm$ 0.0227	0.0362 $\pm$ 0.1345	0.0261 $\pm$ 0.0910
Private embedding of the 1st layer	0.4935 $\pm$ 0.0286	0.4066 $\pm$ 0.0234	-0.0734 $\pm$ 0.0267	0.0348 $\pm$ 0.0174
Private embedding of the 2nd layer	-0.0911 $\pm$ 0.0911	0.0450 $\pm$ 0..0450	0.6441 $\pm$ 0.0644	0.5405 $\pm$ 0.0135

TABLE III : Clustering performance (ARI and NMI) of embeddings evaluated on the true clusters of each layer in Syn3.

Task	Embeddings	Macro-F1 score	Micro-F1 score
Multiplex graph classification (3 dynamical classes)	Combined graph embeddings	$0.9329\pm 0.0548$	$0.9333\pm 0.0544$
	Common graph embeddings	$0.3844\pm 0.0790$	$0.3889\pm 0.0786$
	Private graph embeddings (1st layer)	$0.9892\pm 0.0054$	$0.9893\pm 0.0053$
	Private graph embeddings (2nd layer)	$0.9888\pm 0.0224$	$0.9889\pm 0.0222$
Single-layer graph classification (6 dynamical subclasses)	Private graph embeddings	$0.9944\pm 0.0112$	$0.9944\pm 0.0111$

TABLE IV : Classification results on Syn4 for different tasks and embedding types.

Panel of Figure 9 — Fig. 5 : For each task, we display the top $10\%$ of nodes based on the mean co-activation (left) and anti-activation (right) layer embeddings across subjects. Nodes are projected onto the cortical surface and colored according to the Yeo network atlas, revealing distinct task-dependent patterns across functional brain networks.

Method	Macro F1	Micro F1
MVGRL	0.3338 $\pm$ 0.0529	0.3365 $\pm$ 0.0520
DGI	0.6009 $\pm$ 0.0208	0.6027 $\pm$ 0.0240
HDMI	0.4038 $\pm$ 0.0272	0.4188 $\pm$ 0.0256
DMG	0.3983 $\pm$ 0.0432	0.4032 $\pm$ 0.0443
Graph2Vec	0.6033 $\pm$ 0.0264	0.6192 $\pm$ 0.0325
CaDeM	0.6628 $\pm$ 0.0277	0.6659 $\pm$ 0.0287

TABLE V : Comparison of Macro- and Micro-F1 scores across baselines on the HCP dataset for classifying multiplex graphs into Rest, Emotion, and Working Memory tasks.

Method	DBLP		IMDB		ACM		Freebase
	Macro-F1	Micro-F1	Macro-F1	Micro-F1	Macro-F1	Micro-F1	Macro-F1	Micro-F1
DeepWalk	0.5488 $\pm$ 0.0130	0.5653 $\pm$ 0.0134	0.4651 $\pm$ 0.0283	0.4794 $\pm$ 0.0276	0.8505 $\pm$ 0.0167	0.8499 $\pm$ 0.0157	0.5993 $\pm$ 0.0203	0.6638 $\pm$ 0.0195
Node2Vec	0.5433 $\pm$ 0.0033	0.5681 $\pm$ 0.0037	0.5663 $\pm$ 0.0175	0.6000 $\pm$ 0.0123	0.7249 $\pm$ 0.0178	0.7226 $\pm$ 0.0177	0.4732 $\pm$ 0.0145	0.6254 $\pm$ 0.0156
VGAE	0.8164 $\pm$ 0.0105	0.8080 $\pm$ 0.0101	0.6278 $\pm$ 0.0142	0.6301 $\pm$ 0.0129	0.9093 $\pm$ 0.0084	0.9084 $\pm$ 0.0085	0.5523 $\pm$ 0.0112	0.6375 $\pm$ 0.0052
DGI	0.7283 $\pm$ 0.0130	0.7199 $\pm$ 0.0120	0.5213 $\pm$ 0.0658	0.5349 $\pm$ 0.0548	0.9107 $\pm$ 0.0128	0.9101 $\pm$ 0.0132	0.5426 $\pm$ 0.0116	0.6629 $\pm$ 0.0072
MVGRL	0.7455 $\pm$ 0.0063	0.7310 $\pm$ 0.0073	0.6094 $\pm$ 0.0120	0.6101 $\pm$ 0.0110	0.9105 $\pm$ 0.0032	0.9098 $\pm$ 0.0032	0.5102 $\pm$ 0.0210	0.6438 $\pm$ 0.0167

TABLE VI : Performance comparison of baseline methods for node classification on four multiplex datasets.

$\mathcal{L}_{\textit{matching}}$	$\mathcal{L}_{\textit{self-supervised}}$	$\mathcal{L}_{\textit{causal}}$	Syn1		IMDB		ACM		Freebase
$\mathcal{L}_{\textit{matching}}$	$\mathcal{L}_{\textit{self-supervised}}$	$\mathcal{L}_{\textit{causal}}$	Macro	Micro	Macro	Micro	Macro	Micro	Macro	Micro
✓	✗	✗	0.7450 $\pm$ 0.0520	0.7500 $\pm$ 0.0447	0.6145 $\pm$ 0.0196	0.6187 $\pm$ 0.0180	0.8658 $\pm$ 0.0142	0.8651 $\pm$ 0.0142	0.5760 $\pm$ 0.0187	0.6607 $\pm$ 0.0136
✓	✓	✗	0.7673 $\pm$ 0.1500	0.7700 $\pm$ 0.1503	0.6599 $\pm$ 0.0244	0.6617 $\pm$ 0.0233	0.9152 $\pm$ 0.0106	0.9147 $\pm$ 0.0106	0.5807 $\pm$ 0.0188	0.6615 $\pm$ 0.0109
✓	✗	✓	0.7844 $\pm$ 0.0477	0.7800 $\pm$ 0.0510	0.6576 $\pm$ 0.0125	0.6580 $\pm$ 0.0109	0.9169 $\pm$ 0.0100	0.9164 $\pm$ 0.0101	0.6109 $\pm$ 0.0101	0.6793 $\pm$ 0.0035
✓	✓	✓	0.8178 $\pm$ 0.0516	0.8200 $\pm$ 0.0510	0.6667 $\pm$ 0.0158	0.6682 $\pm$ 0.0155	0.9287 $\pm$ 0.0084	0.9283 $\pm$ 0.0085	0.6224 $\pm$ 0.0121	0.6887 $\pm$ 0.0137

TABLE VII : Ablation study on different loss functions across one synthetic and three real-world datasets. Results are reported as Macro-F1 and Micro-F1 scores on the node classification task.

Dataset	$M$	$N$	Task	#Clusters	$N_{\text{aug}}$	$r$	$\sigma$	Epochs	Learning rates	Loss weights $(\mathcal{L}_{\text{match}},\mathcal{L}_{\text{self-sup}},\mathcal{L}_{\text{causal}})$
Syn1	$100$	$3$	Node classification	$K=3$	$5$	$0.6$	$0.1$	$140$	$(10^{-2},\,10^{-4})$	$(0.9,\,1.5,\,3.4)$
Syn2	$1000$	$3$	Node clustering	$K=3$	$20$	$0.6$	$0.1$	$200$	$(10^{-2},\,10^{-4})$	$(1,\,0.5,\,0.5)$
Syn3	$600$	$2$	Node clustering	$3$ centers/layer	$20$	$0.6$	$0.1$	$200$	$(10^{-2},\,10^{-4})$	$(1,\,0.05,\,0.05)$
Syn4	$100$	$2$	Graph classification	$3$ classes (or $6$ subclasses)	$20$	$0.6$	$0.1$	$200$	$(10^{-2},\,10^{-4})$	$(1,\,0.5,\,0.5)$

TABLE VIII : Summary of synthetic datasets and experimental configurations.

Dynamics	Equation
Population dynamics	$\displaystyle\frac{dX_{i}^{\ell}(t)}{dt}=-\bigl(X_{i}^{\ell}(t)\bigr)^{3}+\sum_{j=1}^{M}A_{ij}\,\bigl(X_{j}^{\ell}(t)\bigr)^{2}$
Regulatory dynamics (degree-driven)	$\displaystyle\frac{dX_{i}^{\ell}(t)}{dt}=-X_{i}^{\ell}(t)+\sum_{j=1}^{M}A_{ij}\,\frac{\bigl(X_{j}^{\ell}(t)\bigr)^{1/3}}{1+\bigl(X_{j}^{\ell}(t)\bigr)^{1/3}}$
Epidemic dynamics	$\displaystyle\frac{dX_{i}^{\ell}(t)}{dt}=-X_{i}^{\ell}(t)+\sum_{j=1}^{M}A_{ij}\,\bigl(1-X_{i}^{\ell}(t)\bigr)\,X_{j}^{\ell}(t)$
Biochemical dynamics	$\displaystyle\frac{dX_{i}^{\ell}(t)}{dt}=1-X_{i}^{\ell}(t)-\sum_{j=1}^{M}A_{ij}\,X_{i}^{\ell}(t)\,X_{j}^{\ell}(t)$
Mutualistic dynamics	$\displaystyle\frac{dX_{i}^{\ell}(t)}{dt}=X_{i}^{\ell}(t)\bigl(1-X_{i}^{\ell}(t)\bigr)+\sum_{j=1}^{M}A_{ij}\,X_{i}^{\ell}(t)\,\frac{\bigl(X_{j}^{\ell}(t)\bigr)^{2}}{1+\bigl(X_{j}^{\ell}(t)\bigr)^{2}}$
Regulatory dynamics (degree-avert)	$\displaystyle\frac{dX^{\ell}_{i}(t)}{dt}=-X^{\ell}_{i}(t)+\sum_{j=1}^{M}A_{ij}\,\frac{(X^{\ell}_{j}(t))^{2}}{1+(X^{\ell}_{j}(t))^{2}}$

TABLE IX : Summary of dynamical systems used to generate layer-specific signals in Syn4.

Panel of Figure 15 — Fig. 6 : t-SNE visualizations of common and private graph embeddings across the $3$ dynamical classes and $6$ subclasses.

Parameter	HCP	DBLP	IMDB	ACM	Freebase
$M$	360	7,907	3,550	3,025	3,492
$N$	2	3	2	2	3
Downstream task	Graph classification	Node classification	Node classification	Node classification	Node classification
Number of classes	3	4	3	3	3
$N_{\mathrm{aug}}$	20	20	20	20	30
$r$	0.6	0.6	0.6	0.6	0.6

TABLE X : Summary of real-world multiplex datasets and experimental configurations.

	Without augmentation		With augmentation
Dataset	Macro F1	Micro F1	Macro F1	Micro F1
IMDB	$0.6571\pm 0.0172$	$0.6588\pm 0.0159$	$\mathbf{0.6667\pm 0.0158}$	$\mathbf{0.6682\pm 0.0155}$
ACM	$0.9173\pm 0.0100$	$0.9167\pm 0.0100$	$\mathbf{0.9287\pm 0.0084}$	$\mathbf{0.9283\pm 0.0085}$
DBLP	$0.8339\pm 0.0158$	$0.8255\pm 0.0143$	$\mathbf{0.8497\pm 0.0111}$	$\mathbf{0.8414\pm 0.0108}$
Freebase	$0.6122\pm 0.0126$	$0.6766\pm 0.0098$	$\mathbf{0.6224\pm 0.0121}$	$\mathbf{0.6887\pm 0.0137}$
First synthetic data	$0.7843\pm 0.0654$	$0.7900\pm 0.0678$	$\mathbf{0.8178\pm 0.0516}$	$\mathbf{0.8200\pm 0.0510}$

TABLE XI : Node classification performance of CaDeM with and without graph augmentation.

Graph Representation Learning

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A Closer Look at the Application of Causal Inference in Graph Representation Learning

Hang Gao, Kunyu Li, Huang Hong, Baoquan Cui+1

Institute of Software Chinese Academy of Science

Graph Representation Learning Causal Inference

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Modeling causal relationships in graph representation learning remains a fundamental challenge. Existing approaches often draw on theories and methods from causal inference to identify causal subgraphs or mitigate confounders. However, due to the inherent complexity of graph-structured data, these approaches frequently aggregate diverse graph elements into single causal variables, an operation that risks violating the core assumptions of causal inference. In this work, we prove that such aggregation compromises causal validity. Building on this conclusion, we propose a theoretical model grounded in the smallest indivisible units of graph data to ensure that the causal validity is guaranteed. With this model, we further analyze the costs of achieving precise causal modeling in graph representation learning and identify the conditions under which the problem can be simplified. To empirically support our theory, we construct a controllable synthetic dataset that reflects realworld causal structures and conduct extensive experiments for validation. Finally, we develop a causal modeling enhancement module that can be seamlessly integrated into existing graph learning pipelines, and we demonstrate its effectiveness through comprehensive comparative experiments.

Abstract

Graph Representation Learning

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