Correlated Clustering — BioNeuralNet 1.0.4 documentation

# Correlated Clustering

BioNeuralNet includes internal modules for performing correlated clustering on complex networks. These methods extend traditional community detection by integrating phenotype correlation, allowing users to extract biologically relevant, phenotype-associated modules from any network.

## Overview

Our framework supports three key correlated clustering approaches:

- Correlated PageRank:

  - A modified PageRank algorithm that prioritizes nodes based on their correlation with an external phenotype.

  - The personalization vector is computed using phenotype correlation, ensuring that biologically significant nodes receive more influence.

  - This method is ideal for identifying high-impact nodes within a given network.

- Correlated Louvain:

  - An adaptation of the Louvain community detection algorithm, modified to optimize for both network modularity and phenotype correlation.

  - The objective function for community detection is given by:

    \[Q^* = k_L \cdot Q + (1 - k_L) \cdot \overline{\lvert \rho \rvert},\]

    where:

    > - \(Q\) is the standard Newman-Girvan modularity, defined as:

    >   \[Q = \frac{1}{2m} \sum_{i,j} \bigl(A_{ij} - \frac{k_i k_j}{2m} \bigr) \delta(c_i, c_j),\]

    >   where \(A_{ij}\) represents the adjacency matrix, \(k_i\) and \(k_j\) are node degrees, and \(\delta(c_i, c_j)\) indicates whether nodes belong to the same community.

    > - \(\overline{\lvert \rho \rvert}\) is the mean absolute Pearson correlation between the first principal component (PC1) of the subgraph’s features and the phenotype.

    > - \(k_L\) is a user-defined weight (e.g., \(k_L = 0.2\)), balancing network modularity and phenotype correlation.

  - This method detects communities that are both structurally cohesive and strongly associated with phenotype.

- Hybrid Louvain:

  - A refinement approach that combines Correlated Louvain and Correlated PageRank in an iterative process.

  - The key steps are:

    1. Initial Community Detection:

       - The input network (adjacency matrix) is clustered using Correlated Louvain.

       - This identifies initial phenotype-associated modules.

    2. Iterative Refinement with Correlated PageRank:

       - In each iteration:

         - The most correlated module is expanded based on Correlated PageRank.

         - The refined network is re-clustered using Correlated Louvain.

         - This process continues until convergence.

    3. Final Cluster Extraction:

       - The final phenotype-optimized modules are extracted and returned.

       - The quality of the clustering is measured using both modularity and phenotype correlation metrics.

Hybrid Clustering: Precedure and steps for the hybrid clustering method.

## Mathematical Approach

Correlated PageRank:

> > - Correlated PageRank extends the traditional PageRank formulation by biasing the random walk towards phenotype-associated nodes.

> > - The ranking function is defined as:

> \[\mathbf{r} = \alpha \cdot \mathbf{M} \mathbf{r} + (1 - \alpha) \mathbf{p},\]

> where:

> - \(\mathbf{M}\) is the transition probability matrix, derived from the normalized adjacency matrix.

> - \(\mathbf{p}\) is the personalization vector, computed using phenotype correlation.

> - \(\alpha\) is the teleportation factor (default: \(\alpha = 0.85\)).

- Unlike standard PageRank, which assumes a uniform teleportation distribution, Correlated PageRank prioritizes phenotype-relevant nodes.

## Graphical Comparison

Below is an illustration of different clustering approaches on a sample network:

Figure 2: Comparison between SmCCNet generated clusters and Correlated Louvain clusters

## Integration with BioNeuralNet

Our correlated clustering methods seamlessly integrate into BioNeuralNet and can be applied to any network represented as an adjacency matrix.

Use cases include:

> - Multi-Omics Networks: Extracting biologically relevant subgraphs from gene expression, proteomics, or metabolomics data.

> - Brain Connectivity Graphs: Identifying functional modules associated with neurological disorders.

> - Social & Disease Networks: Detecting community structures in epidemiology and patient networks.

Our framework supports:

> - Graph Neural Network Embedding: Training GNNs on phenotype-optimized clusters.

> - Predictive Biomarker Discovery: Identifying key features associated with disease outcomes.

> - Customizable Modularity Optimization: Allowing users to adjust the trade-off between structure and phenotype correlation.

## Notes for Users

1. Input Requirements:

   - Any graph-based dataset can be used as input, provided as an adjacency matrix.

   - Phenotype data should be supplied in numerical format (e.g., disease severity scores, expression levels).

2. Cluster Comparison:

   - Correlated Louvain extracts phenotype-associated modules.

   - Hybrid Louvain iteratively refines clusters using Correlated PageRank.

   - Users can compare results using modularity scores and phenotype correlation metrics.

3. Method Selection:

   - Correlated PageRank is ideal for ranking high-impact nodes in a phenotype-aware manner.

   - Correlated Louvain is best for detecting phenotype-associated communities.

   - Hybrid Louvain provides the most refined, biologically meaningful clusters.

## Conclusion

The correlated clustering methods implemented in BioNeuralNet provide a powerful, flexible framework for extracting highly structured, phenotype-associated modules from any network. By integrating phenotype correlation directly into the clustering process, these methods enable more biologically relevant and disease-informative network analysis.

paper link: https://doi.org/10.3389/fdata.2022.894632

Return to BioNeuralNet - Multi-Omics Integration with Graph Neural Networks