GraphMMP: Dual Graph Models

• GraphMMP is a dual-use term representing both a generative graph Laplacian mixture model for unsupervised clustering and a supervised multimodal model for medical prognosis.
• The generative approach uses Gaussian mixtures with Laplacian constraints, EM estimation, and convex optimization to jointly infer clusters and latent graph structures from mixed data.
• The medical prognosis variant leverages mutual information to weight feature graphs and employs a Mamba-based global fusion block within a graph neural network to improve cancer and liver disease predictions.

GraphMMP is a graph-centric designation that appears in two distinct arXiv usages. In one usage, it denotes the Graph Laplacian Mixture Model, introduced by Petric Maretic and Frossard as a generative model for mixture data in which the mapping of data to graphs is not known and the objective is to jointly cluster a set of data and learn a graph for each of the clusters (Maretic et al., 2018). In a later and unrelated usage, GraphMMP denotes a two-stage multimodal prognosis model based on graph neural networks, mutual information, and a Mamba-based global fusion module for medical prognosis (Shan et al., 24 Aug 2025). The shared label masks a substantive difference in problem setting: the former is a latent-variable probabilistic model for multiple graph inference, whereas the latter is a supervised multimodal graph-learning pipeline.

1. Nomenclature and scope

In the supplied arXiv literature, the same name is attached to two non-equivalent model classes. The distinction is not merely terminological; it extends to the statistical assumptions, optimization procedures, and target tasks.

Usage	Problem setting	Core mechanism
Graph Laplacian Mixture Model	jointly clustering a set of data and learning a graph for each of the clusters	Gaussian mixtures with graph-structured precision matrices
GraphMMP	multimodal medical prognosis	feature graphs using Mutual Information and a global fusion module built on Mamba

The 2018 model is motivated by the observation that most graph learning methods assume that all available data relate to the same graph, whereas data is often available in mixed form (Maretic et al., 2018). The 2025 model is motivated by multimodal medical data analysis, where the challenge is to model complex interactions between heterogeneous data modalities while capturing both local and global dependencies across modalities (Shan et al., 24 Aug 2025). This suggests that “GraphMMP” is best treated as an overloaded term rather than a single standardized architecture.

2. Graph Laplacian Mixture Model: generative construction

The Graph Laplacian Mixture Model assumes $N$-dimensional signals $x_n$ for $n=1,\dots,M$, arising from a mixture of $K$ clusters. Each cluster $k$ has a prior weight $\pi_k$ with $\sum_k \pi_k = 1$ and $\pi_k > 0$, together with one-hot latent indicators $z_n \in \{e_1,\dots,e_K\}$, where $z_{nk}=1$ if and only if signal $x_n$0 belongs to cluster $x_n$1. The prior over assignments is

$x_n$2

Conditioned on cluster $x_n$3, $x_n$4 is drawn from a zero-mean Gaussian whose precision is tied to a graph Laplacian $x_n$5, up to a small regularizer $x_n$6: $x_n$7 The marginal over $x_n$8 is therefore

$x_n$9

By construction, each $n=1,\dots,M$0 must lie in the set $n=1,\dots,M$1 of valid combinatorial graph Laplacians: $n=1,\dots,M$2 These constraints encode symmetry, zero row-sums, and nonpositive off-diagonals. To avoid degenerate singular likelihoods and to promote structure, the model may add penalties $n=1,\dots,M$3, including a “log-degree” term $n=1,\dots,M$4 to encourage connectivity and an $n=1,\dots,M$5- or $n=1,\dots,M$6-norm penalty on off-diagonals $n=1,\dots,M$7 to promote sparsity (Maretic et al., 2018).

The model therefore marries Gaussian mixtures with graph-structured precision matrices. Its central assumption is that each latent subpopulation of signals naturally lives on a different graph, and that those graphs can be inferred jointly with cluster assignments.

3. Likelihood, EM estimation, and optimization

With $n=1,\dots,M$8 and $n=1,\dots,M$9, the complete-data log likelihood, ignoring additive constants, is

$K$0

Estimation proceeds in an EM framework, replacing $K$1 by the posterior expectation

$K$2

the responsibilities. The E-step is

$K$3

In the M-step, $K$4 admits the closed-form update

$K$5

For each cluster, the Laplacian update solves

$K$6

The quadratic term $K$7 arises from the Mahalanobis component, and $K$8 arises from the Gaussian normalizer (Maretic et al., 2018).

Each M-step for $K$9 is a convex program over the cone $k$0 with smooth plus nonsmooth terms. Two solution techniques are explicitly identified: projected gradient or proximal gradient, and ADMM. The former handles the smooth part by gradient steps and then projects onto $k$1 while applying the proximal operator of $k$2. The latter splits $k$3 into convenient variables and alternates minimization with closed-form eigenvalue updates for the log-det term and soft-thresholding for sparsity. Complexity is described in terms of graph size $k$4 and number of signals $k$5: forming the precomputed weighted sample scatter $k$6 costs $k$7, while each gradient or ADMM iteration costs $k$8 in the worst case, with reductions to $k$9 when exploiting structure such as sparsity or partial eigenupdates (Maretic et al., 2018).

4. Empirical behavior of the Graph Laplacian Mixture Model

The reported experiments emphasize both clustering quality and graph interpretability. On synthetic data, signals are drawn from two- or three-cluster Laplacian-based Gaussians of dimension $\pi_k$0. In that setting, GraphMMP recovers both the correct clusters, described as low NMSE in $\pi_k$1, and the true edge support, described as high F-measure, better than standard GMM or K-means plus separate graph-learning (Maretic et al., 2018). A second synthetic regime considers heat-kernel mixtures with

$\pi_k$2

for which the variant using the known heat-kernel likelihood, denoted GHMM, outperforms the smooth-signal model as $\pi_k$3 grows.

Three real-world examples are highlighted. On Molène weather with $\pi_k$4, jointly clustering temperature versus wind time-series leads to approximately $\pi_k$5 clustering error versus $\pi_k$6 for GMM, while also yielding geographically sensible graphs in which neighbouring stations are strongly linked (Maretic et al., 2018). On Uber pickup patterns with $\pi_k$7 zones, the model finds five daily patterns, including rush hours and night, with higher day-to-day consistency than competing methods; the inferred graphs change over time in interpretable ways, including airport links at night and intra-city links by day. On MNIST “0” versus “1” with $\pi_k$8 pixels, the model reduces dimensionality implicitly, yielding approximately $\pi_k$9 clustering error versus approximately $\sum_k \pi_k = 1$0 for GMM, and returns pixel graphs that trace the digit shape; a masked version allowing only 2-hop pixel edges further improves the clustering error to approximately $\sum_k \pi_k = 1$1.

Across these experiments, the stated advantages are clustering robustness in high dimensions and highly interpretable graph structures that reveal the latent network each subpopulation of signals lives on. A plausible implication is that the Laplacian constraint functions not only as a structural prior but also as a form of statistical regularization.

5. GraphMMP for multimodal medical prognosis

The later model named GraphMMP addresses a different task: multimodal medical prognosis. Its stated primary goals are to construct a feature graph per patient whose edges encode pairwise dependencies between features across all modalities using Mutual Information (MI), to learn node representations via a lightweight Graph Neural Network equipped with a global fusion block based on Mamba, and to demonstrate superior prognosis performance on a private liver-disease cohort and the public METABRIC breast-cancer cohort (Shan et al., 24 Aug 2025).

The architecture is explicitly two-stage. In Stage 1, fixed dimension-reduced feature vectors are extracted using pre-trained encoders. For patient $\sum_k \pi_k = 1$2 with modalities $\sum_k \pi_k = 1$3, the modality-specific features are denoted

$\sum_k \pi_k = 1$4

These are concatenated into a node feature matrix

$\sum_k \pi_k = 1$5

where $\sum_k \pi_k = 1$6 is the total node count. For any two feature vectors $\sum_k \pi_k = 1$7 and $\sum_k \pi_k = 1$8, empirical MI is defined as

$\sum_k \pi_k = 1$9

The edge weight is then generated through a sigmoid,

$\pi_k > 0$0

and clipped at a threshold $\pi_k > 0$1 through

$\pi_k > 0$2

Within-modality connections are fully linked, while cross-modality edges are sampled sparsely but weighted by $\pi_k > 0$3. The final feature graph is

$\pi_k > 0$4

In Stage 2, GraphMMP uses a compact GNN with two graph layers, followed by a Mamba-based Global Fusion (MGF) block and a final classification head. The graph layers are inspired by the Heterogeneous Graph Transformer and use multi-head attention that fuses node features and edge weights. The MGF block is introduced because GNN layers capture local and medium-range graph structure but can miss long-range, cross-modal context. It operates by flattening all node embeddings into a sequence $\pi_k > 0$5, feeding $\pi_k > 0$6 into a Mamba block to produce $\pi_k > 0$7, applying residual fusion

$\pi_k > 0$8

and reshaping the fused sequence back into graph node form. Training minimizes cross-entropy,

$\pi_k > 0$9

where $z_n \in \{e_1,\dots,e_K\}$0 is the predicted class-probability vector and $z_n \in \{e_1,\dots,e_K\}$1 is the one-hot true label (Shan et al., 24 Aug 2025).

6. Datasets, results, limitations, and conceptual comparison

The medical GraphMMP is evaluated on two datasets. The liver prognosis dataset contains 185 patients and five modalities: CT image, radiomics features, biochemical labs, fat analysis, and demographics. Images are passed through a frozen ResNet backbone, and tabular features are standardized to zero mean and unit variance. The METABRIC dataset contains 1,981 breast-cancer samples with three modalities: gene-expression, copy-number variation, and clinical; the 5-year survival label is binarized as the target, omics data are log-scaled and normalized, and clinical variables are one-hot or z-scored (Shan et al., 24 Aug 2025).

Training uses 5-fold stratified cross-validation, Adam with learning rate $z_n \in \{e_1,\dots,e_K\}$2, 200 epochs, and early stopping if validation loss does not improve for 50 epochs. The implementation stack is NVIDIA RTX 3090Ti, PyTorch 2.0, PyG 2.6.1, and CUDA 11.8. The GNN has two graph-attention layers, each with $z_n \in \{e_1,\dots,e_K\}$3 hidden units and 4 attention heads, LeakyReLU(0.2) activations, and dropout 0.3 on edges. The MGF block consists of one Mamba layer with state-space dimension 128 and linear-time complexity. Optimization uses Adam$z_n \in \{e_1,\dots,e_K\}$4, weight decay $z_n \in \{e_1,\dots,e_K\}$5, and batch size $z_n \in \{e_1,\dots,e_K\}$6 graphs. Metrics are Accuracy, Precision, Recall, F1-score, and AUC; baselines include GraphConv, UniMP, vanilla GCN, ChoqFuzGCN, MOGAT, SimpleFF, HFBSurv, MMD, and TMI-CLNet (Shan et al., 24 Aug 2025).

Quantitatively, on liver prognosis with full modalities, TMI-CLNet is reported as the best prior method with ACC $z_n \in \{e_1,\dots,e_K\}$7, F1 $z_n \in \{e_1,\dots,e_K\}$8, and AUC $z_n \in \{e_1,\dots,e_K\}$9, while GraphMMP attains ACC $z_{nk}=1$0 $z_{nk}=1$1, F1 $z_{nk}=1$2 $z_{nk}=1$3, and AUC $z_{nk}=1$4 $z_{nk}=1$5. On METABRIC, UniMP is reported as the best prior method with ACC $z_{nk}=1$6, F1 $z_{nk}=1$7, and AUC $z_{nk}=1$8, while GraphMMP reaches ACC $z_{nk}=1$9 $x_n$00, F1 $x_n$01 $x_n$02, and AUC $x_n$03 $x_n$04. Ablations show that removing MI-based edges decreases liver ACC from $x_n$05 to $x_n$06 and METABRIC ACC from $x_n$07 to $x_n$08, while removing the MGF block decreases liver ACC from $x_n$09 to $x_n$10 and METABRIC ACC from $x_n$11 to $x_n$12. The paper interprets these ablations as confirming that MI-based edge weighting is critical for capturing latent inter-modal information and that the MGF block provides indispensable global context (Shan et al., 24 Aug 2025).

The limitations identified for the medical model are that the private liver dataset is relatively small at 185 cases, MI estimation can be noisy for very high-dimensional or sparse features, and the Mamba block adds extra sequence overhead that may become a bottleneck for extremely large graphs. Proposed future directions are extension to larger multimodal cohorts, adaptive MI estimation, hierarchical or multi-scale graph constructions, and uncertainty quantification (Shan et al., 24 Aug 2025).

Taken together, the two GraphMMP usages occupy different methodological positions. The 2018 Graph Laplacian Mixture Model is an unsupervised generative model with EM inference and Laplacian-constrained precision estimation (Maretic et al., 2018). The 2025 GraphMMP is a supervised prognosis model that unifies feature-graph construction via MI and global fusion via Mamba (Shan et al., 24 Aug 2025). The commonality lies in their use of graphs as latent or constructed relational structure; the divergence lies in whether the graph is treated as a cluster-specific precision matrix in a Gaussian mixture or as a patient-specific feature graph in a graph neural network pipeline.