SM-netFusion: Supervised Cross Diffusion

• The paper introduces SM-netFusion, which integrates multiple centrality measures and supervised MKL to yield representative and discriminative brain network atlases.
• SM-netFusion combines degree, closeness, and eigenvector centrality through diffusion kernels and cluster-aware weighting to capture class-specific neural connectivity.
• By leveraging iterative cross-diffusion and graph feature selection, SM-netFusion outperforms traditional methods with improved classification accuracy and reduced Frobenius distance.

Supervised Cross Diffusion (SM-netFusion) is a methodological framework for estimating class-specific, representative, and discriminative Brain Network Atlases (BNAs) by leveraging supervised multi-topology network cross-diffusion. It is designed to address limitations in prior brain network fusion methods that rely solely on single topological measures and unsupervised similarity network diffusion, thereby enhancing both the representativeness and discriminative capacity of learned BMAs, particularly for population-level neuroimaging analysis (Mhiri et al., 2020).

1. Motivation and Conceptual Overview

SM-netFusion is constructed to estimate a population-driven BNA that is (i) representative—capturing traits shared among subjects, (ii) centered—optimally close, in a metric sense, to all subjects in a class, and (iii) discriminative—easily identifying connections distinguishing between clinical and control populations. Classical BNA estimation methods employ similarity network diffusion and fusion, typically using only node degree as a topological descriptor and operating in an entirely unsupervised manner. Such approaches can obscure informative centrality structure and lack discriminative supervision, reducing performance in downstream classification tasks, especially for clinical prediction.

To address these gaps, SM-netFusion introduces (i) multiple complementary graph centrality measures—degree, closeness, and eigenvector centrality—into the kernelization and diffusion processes, and (ii) a supervised, cluster-aware multiple kernel learning (MKL) scheme that guides the fusion toward discriminative network features.

2. Mathematical Foundations

Let $\{\mathbf{X}^c_i\}_{i=1}^{N^c}$, $\mathbf{X}^c_i\in\mathbb{R}^{r\times r}$, denote the set of symmetric connectivity matrices for $N^c$ subjects in class $c$. For each subject $i$, SM-netFusion constructs three diffusion kernels corresponding to:

• Degree centrality: $$\mathbf{D}^c_i = \mathrm{diag}(\mathbf{X}^c_i\mathbf{1})$$, producing kernel

$$K^{(\mathrm{deg}),c}_i(k,l) = \exp\left(-\frac{|D^c_i(k,k)-D^c_i(l,l)|}{\sigma_{\mathrm{deg}}}\right).$$

• Closeness centrality: with node $n$,

$C^c_i(n)=\frac{r-1}{\sum_{k\neq n}d_{nk}},$

forming $$\mathbf{C}^c_i=\mathrm{diag}(C^c_i(1),...,C^c_i(r))$$ and

$\mathbf{X}^c_i\in\mathbb{R}^{r\times r}$0

• Eigenvector centrality: $\mathbf{X}^c_i\in\mathbb{R}^{r\times r}$1, $\mathbf{X}^c_i\in\mathbb{R}^{r\times r}$2,

$\mathbf{X}^c_i\in\mathbb{R}^{r\times r}$3

These centrality-based kernels are concatenated into a third-order tensor per subject,

$\mathbf{X}^c_i\in\mathbb{R}^{r\times r}$4

and fused through arithmetic averaging:

$\mathbf{X}^c_i\in\mathbb{R}^{r\times r}$5

3. Supervised Learning and Kernel Weight Optimization

To incorporate class structure and population heterogeneity, SM-netFusion initially clusters the training data $\mathbf{X}^c_i\in\mathbb{R}^{r\times r}$6 into $\mathbf{X}^c_i\in\mathbb{R}^{r\times r}$7 subspaces using a method such as SIMLR. For MKL, nonnegative weights $\mathbf{X}^c_i\in\mathbb{R}^{r\times r}$8 are learned to optimally map the subjectwise multi-topology kernels to the cluster labels. Using a diagonal label matrix $\mathbf{X}^c_i\in\mathbb{R}^{r\times r}$9 and label vector $N^c$0, the EasyMKL-style objective is:

$N^c$1

with

$N^c$2

Closed-form expressions exist for the optimal weights:

$N^c$3

where $N^c$4 is the regularization parameter.

4. Cross-Diffusion and Atlas Construction

After learning $N^c$5, each subject’s normalization kernel is constructed as

$N^c$6

with inverse $N^c$7. The procedure is as follows:

1. Initialization: Multi-topology normalization yields

$N^c$8

2. Local Kernel Computation: For each $N^c$9,

$c$0

where $c$1 denotes the set of $c$2 nearest neighbors of ROI $c$3.

3. Iterative Cross-Diffusion (for $c$4):

$c$5

4. Atlas Fusion: The class-level atlas is

$c$6

5. Representativeness and Discriminative Power

Representativeness and centrality of $c$7 are measured by average Frobenius distance:

$c$8

SM-netFusion achieves the lowest $c$9 among all evaluated baselines, as detailed below.

Method	$i$0(ASD)	$i$1(NC)
D-SNF (deg)	1.23	1.18
C-SNF (clo)	1.21	1.16
E-SNF (eig)	1.22	1.17
netNorm	1.19	1.14
NAGFS	1.17	1.12
SM-netFusion	1.09	1.04

Discriminativeness is assessed by forming the residual

$i$2

and using the top $i$3 features to train a linear SVM. On the ABIDE dataset (505 subjects: 266 ASD, 239 NC), SM-netFusion surpasses all baselines, with accuracy improvements from 5–15% ($i$4, paired two-tailed $i$5–test):

Method	Accuracy (%)
IFS+SVM	72.4
netNorm+SVM	74.1
RF-RFE	75.0
LLCFS+SVM	75.8
SM-netFusion+SVM	82.3 (+5–15%)

6. Experimental Protocol and Parameterization

• Dataset: ABIDE preprocessed resting-state fMRI, $i$6 regions of interest (ROIs).
• Cross-validation: 5-fold stratified; feature selection nested within each training split.
• Parameter Choices: $i$7 clusters via SIMLR; $i$8 diffusion iterations; $i$9 neighbors; MKL regularizer $$\mathbf{D}^c_i = \mathrm{diag}(\mathbf{X}^c_i\mathbf{1})$$0.
• Metrics: Mean Frobenius distance $$\mathbf{D}^c_i = \mathrm{diag}(\mathbf{X}^c_i\mathbf{1})$$1, classification accuracy, AUC, and statistical significance via paired two-tailed $$\mathbf{D}^c_i = \mathrm{diag}(\mathbf{X}^c_i\mathbf{1})$$2-tests ($$\mathbf{D}^c_i = \mathrm{diag}(\mathbf{X}^c_i\mathbf{1})$$3).

7. Extension to Graph Feature Selection

By producing class-specific, centrally located atlases $$\mathbf{D}^c_i = \mathrm{diag}(\mathbf{X}^c_i\mathbf{1})$$4, SM-netFusion natively enables scalable, supervised graph feature selection:

• Compute $$\mathbf{D}^c_i = \mathrm{diag}(\mathbf{X}^c_i\mathbf{1})$$5 per class.
• Form $$\mathbf{D}^c_i = \mathrm{diag}(\mathbf{X}^c_i\mathbf{1})$$6.
• Rank off-diagonal elements $$\mathbf{D}^c_i = \mathrm{diag}(\mathbf{X}^c_i\mathbf{1})$$7; select the top $$\mathbf{D}^c_i = \mathrm{diag}(\mathbf{X}^c_i\mathbf{1})$$8 features.
• Employ these features in downstream predictive learners.

This operationalizes SM-netFusion not only as an atlas estimator but as a systematic framework for identifying discriminative graph features relevant to network neuroscience and clinical prediction.

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In summary, SM-netFusion enriches conventional similarity network fusion by integrating degree, closeness, and eigenvector centrality measures; employing supervised, cluster-aware kernel weight learning; and providing a high-fidelity, discriminative cross-diffusion process. The result is a methodologically rigorous approach that produces state-of-the-art representative and discriminative BNAs, yielding marked improvements in clinical group classification and offering a principled pipeline for graph-based feature selection in population neuroscience (Mhiri et al., 2020).