Embedding Learning on Multiplex Networks

• Embedding learning on multiplex networks converts nodes from multi-relational systems into low-dimensional vectors while preserving both intra-layer and inter-layer structures.
• Approaches include random-walk methods, optimization-based models, and graph neural networks, each tailored to address complex network geometries and layer imbalances.
• These techniques enable practical applications such as link prediction, node classification, and robustness analysis while ensuring scalability and fair evaluation.

Embedding learning on multiplex networks is the process of mapping nodes from a multilayered or multi-relational network into a vector space, aiming to jointly preserve both the complex intra-layer and inter-layer structures. Multiplex networks, defined by a common node set connected through multiple types of edges (or "layers"), are prevalent in modeling real complex systems such as social, biological, and technological systems. Modern embedding methods for multiplex networks seek not only to capture structural properties and node similarities, but also to facilitate downstream tasks such as link prediction, classification, robustness assessment, and network compression.

1. Foundations of Multiplex Network Embedding

A multiplex network is formally defined as $G = (V, \{A_\alpha\}_{\alpha=1}^L)$, where $V$ is the global node set and $A_\alpha$ is the adjacency matrix for the $\alpha$-th layer. Unlike single-layer graphs, multiplexes encode multiple, potentially heterogeneous relational patterns. Embedding learning, in this context, aims to compute low-dimensional representations for each node (and possibly for each node-layer pair), maintaining proximity and alignment across layers, while also being robust to missing data, layer imbalances, or layer-specific semantics (Trautmann et al., 2 Feb 2026).

Multiplex embeddings differ from monoplex embeddings along two axes: (i) the representation regime (unique vs. layer-specific vs. role-based vectors for nodes), and (ii) the algorithmic approach (shallow random-walk/factorization vs. graph neural network based; see Section 3). This dual taxonomy enables a systematic characterization and comparison of existing frameworks (Trautmann et al., 2 Feb 2026).

2. Core Methodologies and Architectures

2.1 Random Walk and Proximity-Preserving Models

Random-walk–based methods such as Multi-Net, MultiVERSE, and PMNE generalize skip-gram training by simulating walks that jump both within and across layers, constructing node contexts that reflect the multiplex topology. Multi-Net utilizes unbiased layer switching and uniform neighbor selection to ensure fair sampling over the multilayer structure, optimizing the skip-gram likelihood across walks that traverse all layers (Bagavathi et al., 2018). MultiVERSE enhances this by introducing Random Walk with Restart tailored for multiplex and multiplex-heterogeneous networks, generating node proximity distributions which are then approximated by low-dimensional embeddings via noise-contrastive estimation (Pio-Lopez et al., 2020).

Optimization-based models, such as MNE, GATNE, and MELL, extend LINE-style objectives to multiplexes, typically by learning common base embeddings and layer-specific offsets, preserving first- and second-order proximities within and across layers (Cen et al., 2019, Trautmann et al., 2 Feb 2026). Weighted variants (e.g., MulCEV) interpret edge weights as measures of relational strength, impacting the proximity and consistency in the latent space (Tang et al., 2020).

2.2 Deep Neural and Attention-based Approaches

Graph Neural Network (GNN) extensions constitute a powerful class of methods for multiplex embedding. MultiSAGE generalizes GraphSAGE by aggregating both intra-layer and inter-layer neighbors, learning representations via multiple rounds of neighborhood aggregation (Gallo et al., 2022). MPXGAT introduces a two-phase graph attention mechanism: a horizontal phase to learn per-layer node embeddings, and a vertical inter-layer attention phase to fuse representations of the same node across layers (Bongiovanni et al., 2024).

Relation-wise fusion with self-attention has emerged as a key mechanism for distilling multi-layer or multi-relation structure: GATNE deploys self-attention over edge embeddings, RAHMeN uses relational attention to aggregate per-relation embeddings, and DMGI/HDMI utilize attention to combine layer-specific GCN outputs (Cen et al., 2019, Melton et al., 2022, Park et al., 2019, Jing et al., 2021). Cross-layer transformer fusion, as in "Mind the Links" (Trans-SLE/Trans-GAT), applies multi-head self-attention to sequences of per-layer edge representations, learning to aggregate evidence for target-layer link prediction (Sharma et al., 27 Sep 2025).

2.3 Geometric and Manifold Embedding

High-dimensional multiplex graphs can induce curvature and manifold distortions that undermine downstream tasks. Hierarchical dimension aggregation coupled with hyperbolic graph neural networks (e.g., HYPER-MGE) addresses these geometric pathologies by learning hyperbolic embeddings on Riemannian manifolds, integrating both inter-layer hierarchy and the negative curvature structure suitable for hierarchical or tree-like data (Abdous et al., 29 Jan 2025, Sun, 2019). Hyperbolic multiplex embedding frameworks further unify geometric embedding with community detection, leveraging random walks and information-theoretic objectives to jointly optimize latent space geometry and multiplex modularity (Sun, 2019).

2.4 Partial View and Imbalance-Aware Models

Real-world multiplex data are often incomplete or exhibit strong imbalance across layers. DPMNE addresses incomplete views by coupling deep autoencoders for each layer with a shared latent subspace, leveraging masks to handle missing data in both reconstruction and alignment losses. Topological regularization via a multiplex graph Laplacian enforces consistency and smoothness across observed layers, while the orthogonal-invariance property facilitates both real and binary embeddings (Wang et al., 2022). LIAMNE targets layer-imbalance: by undersampling dense layers according to node similarity in the sparse target-layer's embedding space, it preserves salient auxiliary edges while preventing dominant layers from overwhelming training (Chen et al., 2022).

3. Interlayer Similarity, Clustering, and Reducibility

Measuring and utilizing interlayer similarity is critical for robustness analysis, network reducibility, and layer aggregation. EATSim proposes a principled, embedding-aided similarity metric by combining two losses: pairwise Euclidean Distance (PED), which assesses intralayer geometric consistency, and Aligned Euclidean Distance (AED), which aligns the entire embedding via optimal orthogonal transformation to test cross-layer anchor alignment. The overall dissimilarity $$D(G^{(\alpha)},G^{(\beta)}) = \omega L_{\text{PED}} + (1-\omega) L_{\text{AED}}$$ is tuned to maximize correspondence with robustness measures. EATSim achieves strong performance—Pearson's correlation $r > 0.97$ with ground-truth similarities in synthetic BA and GMM benchmarks, and state-of-the-art performance in predicting robustness and guiding hierarchical layer-merging for reducibility (Nan et al., 11 May 2025).

Hierarchical clustering of layer embeddings, using similarity matrices output by EATSim or related embedding-based metrics, enables effective layer aggregation and quantification of reducibility via von Neumann entropy. Sensitivity analyses on the weighting parameter $\omega$ and embedding dimension reveal clear regimes where local and global alignment are best balanced, further supporting the geometric interpretation of the framework.

4. Evaluation Protocols, Fairness, and Scalability

Evaluation in multiplex embedding presents unique challenges: negative sampling must respect the multiplex union set, leakage avoidance protocols must ensure that test-layer information is hidden during training, and both network-wide and layer-specific metrics must be reported to reflect task-specific and holistic performance (Trautmann et al., 2 Feb 2026, Sharma et al., 27 Sep 2025). Leakage-free evaluation, as formalized in "Mind the Links," combines cross-layer and inductive subgraph generalization protocols, enforcing that no labels or nodes from the target layer are visible at training. Vertex-centric metrics (e.g., VCMPR@k) and class-weighting balance evaluation in sparse, class-imbalanced scenarios.

Scalability considerations are paramount for large multiplex and multiplex-heterogeneous networks. Sampling-based random walk approaches (e.g., Multi-Net, MultiVERSE), embedding fusion through efficient transformer layers (Trans-SLE), and layer-candidate reduction schemes (union-set negative sampling) all enable linear or near-linear scaling in the number of nodes and edges (Bagavathi et al., 2018, Pio-Lopez et al., 2020, Sharma et al., 27 Sep 2025).

5. Downstream Applications and Open Problems

Multiplex network embeddings underpin a range of downstream tasks, including intra- and inter-layer link prediction, node and layer classification, community detection, robustness assessment, and network reconstruction (Tang et al., 2020, Kaiser et al., 2023, Nan et al., 11 May 2025). Dedicated frameworks have been developed for bipartite, attributed, or temporal multiplexes, with advanced architectures (e.g., DualHGCN) translating multiplex structures into hypergraph domains to support additional heterogeneity (Xue et al., 2021). Embedding-based approaches have been successfully deployed in large-scale production settings, including recommendation systems at Alibaba (Cen et al., 2019).

Open challenges include extending current models to dynamic or evolving multiplexes, further integrating temporal, attribute, or motif-based information, adapting to highly imbalanced or partially observed layers, learning curvature or dimensionality adaptively per layer or node, and establishing community standards for fair evaluation and benchmark data (Abdous et al., 29 Jan 2025, Chen et al., 2022, Trautmann et al., 2 Feb 2026). Advanced neural architectures (autoencoder- and transformer-based multiplex models) and theoretical analyses (e.g., spectral consistency in continuous-time Poisson multiplexes) represent active research directions (Corneck et al., 23 Jan 2026, Sharma et al., 27 Sep 2025).

6. Summary of Representative Embedding Methods

Method	Core Technique	Multiplex Handling
Multi-Net	Random walks + SG	Layer-aware traversal
MultiVERSE	RWR-M + VERSE	Proximity distributions
GATNE	Edge/base emb. + attn	Self-attn, inductive
MPXGAT	GAT-intra + interla.	Horizontal + vertical
DMGI/HDMI	DGI + attn fusion	Mutual info, attention
HYPER-MGE	Hierarchical HGNN	Hyperbolic manifolds
LIAMNE	Similarity sampling	Layer imbalance
DualHGCN	Hypergraph GCN	Multiplex bipartite
EATSim	PED + AED losses	Interlayer similarity
Mind the Links	X-layer transformer	Cross-view fusion

This table summarizes the primary approaches and their multiplex-specific mechanisms. Each method is evaluated on real or synthetic benchmarks, reporting metrics tailored to the target application and reflecting recent trends in the field.

• "Assessing the Robustness and Reducibility of Multiplex Networks with Embedding-Aided Interlayer Similarities" (Nan et al., 11 May 2025)
• "MPXGAT: An Attention based Deep Learning Model for Multiplex Graphs Embedding" (Bongiovanni et al., 2024)
• "Deep Partial Multiplex Network Embedding" (Wang et al., 2022)
• "A Geometric Perspective for High-Dimensional Multiplex Graphs" (Abdous et al., 29 Jan 2025)
• "Embedding Learning on Multiplex Networks for Link Prediction" (Trautmann et al., 2 Feb 2026)
• "Layer Imbalance Aware Multiplex Network Embedding" (Chen et al., 2022)
• "Reconstruction of multiplex networks via graph embeddings" (Kaiser et al., 2023)
• "Hyperbolic Multiplex Network Embedding with Maps of Random Walk" (Sun, 2019)
• "Mind the Links: Cross-Layer Attention for Link Prediction in Multiplex Networks" (Sharma et al., 27 Sep 2025)