DIMPLE-SGRDPG: Multiplex Signed Graph Model

• DIMPLE-SGRDPG is a flexible multiplex network model that extends the generalized random dot product graph to incorporate both positive and negative edge weights.
• It integrates latent geometric embedding and tensor decomposition to enable efficient clustering and dimension reduction even in sparse, heterogeneous settings.
• The framework demonstrates actionable improvements in applications ranging from social and biological networks to cybersecurity, offering robust inference of latent group structures.

The DIverse MultiPLEx Signed Generalized Random Dot Product Graph (DIMPLE-SGRDPG) model defines a highly flexible and general framework for modeling, analyzing, and clustering multilayer networks in which the same set of entities (nodes) interact across multiple, possibly heterogeneous, layers. Each component layer can exhibit positive and negative edge weights, their probabilistic structure can be entirely distinct except for possible sharing of a latent subspace within groups, and edge formation is governed by a signed extension of the generalized random dot product graph principle. This approach integrates parameterized random edge generation, latent geometric embedding, multiplex structure, and group-wise heterogeneity within a common inferential and algorithmic paradigm.

1. Model Formulation and Parameterization

The DIMPLE-SGRDPG model extends the classic random dot product graph by enabling (a) edge weights from arbitrary parametric distributions; (b) both positive and negative edge signs; and (c) multiplex structure, where $L$ observed layers are defined on a common node set of $n$ vertices and can be partitioned into $M$ groups according to a (latent) clustering of the layers.

A typical parameterization for layer $l$, belonging to group $m$ with label $s(l) = m$, expresses the (possibly signed) $n \times n$ connection-probability matrix as:

$P(l) = X(m) B(l) X(m)^T,$

where $X(m) \in \mathbb{R}^{n \times K_m}$ is a latent position matrix (latent subspace or node embedding) for group $m$ and $B(l)$ is a symmetric block or "core" matrix encoding between- and within-community connectivity (possibly rescaled by a global sparsity parameter $\rho_n$). Alternative equivalent notations employ an orthonormal basis $U(m)$ and a block-specific matrix $Q(l)$:

$P(l) = U(m) Q(l) U(m)^T.$

The signature of the model is the relaxed domain of $P(l)$: entries may range in $[-1,1]$ (permitting both signs), as opposed to the strictly nonnegative domain of standard RDPG and its weighted variants (DeFord et al., 2016, Pensky, 14 Feb 2024). Edges are then observed as signed weighted random variables, frequently sampled according to:

$$A_{ij}^{(l)} \sim P\bigl(\langle X_1^{(i)}, X_1^{(j)}\rangle, \ldots \bigr),$$

where the number and nature of latent spaces and the choice of distribution capture the (possibly multimodal, signed, or weighted) attributes per edge.

The multiplex aspect emerges by allowing the $L$ network layers to have varying $B(l)$ and by partitioning the layers into mutually exclusive groups such that all layers in the same group share the same latent embedding $X(m)$ but may have arbitrary (and even highly heterogeneous) $B(l)$. This construction subsumes a broad class of previously studied multilayer, block, and signed models as particular cases (Pensky et al., 2021, Pensky, 14 Feb 2024).

2. Embedding and Dimension Reduction

At the core of inference for DIMPLE-SGRDPG is geometric embedding: the process of recovering latent node positions (or subspaces) from observed network data via matrix/tensor factorization and spectral methods. For a single layer, the classical approach is to extract $d$ leading eigenvectors and eigenvalues of the adjacency (or Laplacian) matrix, forming the adjacency spectral embedding (ASE) (Athreya et al., 2017, Rubin-Delanchy et al., 2017). In the generalized (signed) case, both positive and negative eigenvalues are retained, yielding consistent (and asymptotically Gaussian) estimators for the latent positions up to an indefinite orthogonal transformation (see O($p$,$q$) indeterminacy) (Rubin-Delanchy et al., 2017, Yan et al., 2023).

In the multiplex setting, multiple joint embedding algorithms are available. Key methodologies include:

• Unfolded Adjacency Spectral Embedding (UASE): Column-wise concatenation of the $L$ adjacency matrices followed by SVD, providing a low-rank representation for shared and layer-specific structure (Jones et al., 2020).
• Higher-Order Orthogonal Iterations (HOOI): Tensor decomposition of the $n \times n \times L$ adjacency or probability tensor, yielding orthonormal factor matrices for nodes and layers, capable of separating layer group structure even in extremely sparse regimes (Pensky, 25 Jul 2025).

Dimension reduction is controlled by minimizing a stress function or evaluating the Frobenius norm between reconstructed and observed graphs, with explicit criteria guiding the optimal choice of latent dimension (DeFord et al., 2016). The geometric structure (angles and lengths of embedding vectors) provides interpretable summaries for community structure and centrality.

3. Layer Group Recovery and Clustering in Sparse Regimes

The principal inferential challenge is to recover the group assignment $s: [L] \to [M]$ which partitions the $L$ layers according to their shared latent subspace structure. Prior approaches relied on layer-per-layer spectral analyses followed by clustering on per-layer embeddings or affinity matrices, requiring relatively dense network observations for statistical consistency (Pensky et al., 2021, Noroozi et al., 2022).

The recent tensor-based methodology pools information across layers, dramatically relaxing sparsity constraints. By stacking all $L$ observed layers into a cubic tensor and performing a Tucker/HOOI decomposition, the factor matrix associated with the layer (third) mode is extracted. The pairwise scalar product matrix of its rows, after thresholding, forms a near block-diagonal matrix that recovers the true groupings with high probability, even when the average layer degree is low (e.g., $O(\log n)$ or below) (Pensky, 25 Jul 2025). Theoretical sharpness is established: the clustering error decays and perfect recovery is achieved under sparsity scaling that matches information-theoretic lower bounds up to logarithmic factors.

The error guarantees for layer group recovery typically hold in two-to-infinity norm or are quantified in terms of the sine of canonical angles between estimated and true subspaces, benefiting from strong concentration inequalities and perturbation theory for random tensors.

4. Assumptions, Theoretical Guarantees, and Lower Bounds

The DIMPLE-SGRDPG analysis is predicated on nonrestrictive, interpretable assumptions:

• Balanced group assignment: layer membership is distributed with positive lower and upper bounds.
• Node latent positions are distributed within a compact set and have covariance matrices bounded away from singularity.
• Block matrices $B(l)$ (or $Q(l)$) within a group must induce sufficient structural diversity (typically ensured by diversity in the vectorized blocks).
• The number of layers $L$ grows at least polylogarithmically in $n$.

Under these conditions, new results (Pensky, 25 Jul 2025) show that perfect clustering and sharp subspace estimation are achieved provided that relevant scaling quantities (such as $n^2 \rho_n \sqrt{L}$ for the two-to-infinity norm in HOOI) exceed polylogarithmic functions of $n$.

Critically, minimax lower bounds for latent position estimation and subspace recovery (measured in two-to-infinity norm) have been established for the broader GRDPG class via constructions based on Hadamard matrix packings, and the tensor-embedding approaches for DIMPLE-SGRDPG align (modulo log factors) with these fundamental limits (Yan et al., 2023, Pensky, 25 Jul 2025).

5. Algorithmic Innovations and Practical Implementation

DIMPLE-SGRDPG estimation algorithms build on higher-order spectral techniques:

• Regularized HOOI: Extracts node and layer factors, providing control over error even with highly heterogeneous and sparse tensors. Thresholding inner products of estimated layer factors segments the layers into groupings corresponding to shared latent subspaces (Pensky, 25 Jul 2025).
• Clustering by Sparse Subspace Clustering (SSC): For structured subspace models (e.g., SBMs), layers are mapped to points in high-dimensional vector space, and sparse linear representations reveal group structure via affinity and subsequent spectral clustering (Noroozi et al., 2022).

Gradient-based optimization algorithms, including manifold-constrained Riemannian gradient descent, have also been proposed for direct optimization of embedding objectives with signed and/or multiplex structure, offering scalability and adaptability to missing data or streaming settings (Fiori et al., 2023).

Practical considerations include the suitability of the approach for extremely large-scale networks, the benefit of exploiting parallel computing features of the algorithms (especially in the SSC and HOOI steps), and the robustness of the methods to noise, missing edges, and layer heterogeneity. The tensor-based methods remain effective for a variety of edge weight distributions, provided latent subspace structure is present.

6. Applications, Extensions, and Significance

DIMPLE-SGRDPG accommodates a broad spectrum of real-world scenarios:

• Social and Economic Multiplexes: Edge signs may represent friendship/enmity, trade surplus/deficit, or voting agreement/disagreement across multiple topics or commodities.
• Biological Networks: Layered signed connectivity among brain regions across time or conditions, with clustering revealing phenotypic or subject groupings (Pensky, 14 Feb 2024).
• Cybersecurity and Communication: Multilayer signed graphs encode port-specific interactions, anomaly detection leverages subspace clustering in dynamic sparse multiplexes (Jones et al., 2020).

Retaining edge sign information yields substantial estimation and clustering improvements, as rigorous simulations and real-data analyses confirm. Algorithms are shown to outperform prior layerwise or "averaging" approaches, particularly when layer sparsity or block matrix heterogeneity would otherwise challenge reliable inference.

A plausible implication is that, as multilayer relational data with complex positive/negative interactions becomes increasingly prevalent, tensor-based methods under the DIMPLE-SGRDPG paradigm offer a statistically and computationally optimal route for inferring latent structure with minimal assumptions on density, homogeneity, or identically-distributed layers.

This framework unifies and extends earlier multilayer, mixed membership, and signed SBM/RDPG approaches, providing a robust and theoretically grounded toolkit for high-dimensional, heterogeneous, sparse, signed multiplex network analysis.