functional-connectivity-graph-neural-networks

name	functional-connectivity-graph-neural-networks
description	Functional Connectivity Graph Neural Networks methodology combining structural and functional connectivity with persistent graph homology for brain-inspired graph classification. Activation triggers: functional connectivity, graph neural network, persistent homology, brain network, multi-modal GNN.

Functional Connectivity Graph Neural Networks (FC-GNN)

Brain-inspired graph neural network framework that combines local structural and global functional connectivity through persistent graph homology for improved graph-level classification across diverse networks.

Metadata

• Source: arXiv:2508.05786 [cs.NE]
• Authors: Yang Li, Luopeiwen Yi, Tananun Songdechakraiwut
• Published: 2025-08-07

Core Methodology

Key Innovation

Traditional GNNs typically rely solely on structural connectivity (adjacency information), missing the rich global interaction patterns. FC-GNN introduces a functional connectivity block based on persistent graph homology to capture global topological features, inspired by multi-modal brain imaging where structural and functional connectivity offer complementary views.

Technical Framework

1. Structural Connectivity Component

• Standard message passing on the graph structure
• Captures local neighborhood information
• Node features propagate through edges

2. Functional Connectivity Block

• Persistent Graph Homology: Computes topological features across multiple scales
• Captures higher-order interactions beyond pairwise connections
• Identifies persistent topological structures (loops, voids, clusters)
• Provides global network organization information

3. Multi-Modal Fusion

• Combines structural and functional representations
• Joint learning from complementary connectivity views
• Enhanced graph-level embeddings for classification

Implementation Guide

Prerequisites

• Python 3.8+
• PyTorch Geometric or DGL
• GUDHI or Ripser for persistent homology
• NumPy, SciPy

Step-by-Step Implementation

import torch
import torch.nn as nn
from torch_geometric.nn import GCNConv, global_mean_pool
from gudhi import RipsComplex  # or alternative PH library

class FunctionalConnectivityGNN(nn.Module):
    """
    FC-GNN: Combining structural and functional connectivity
    """
    def __init__(self, in_channels, hidden_channels, out_channels):
        super().__init__()
        # Structural connectivity layers
        self.conv1 = GCNConv(in_channels, hidden_channels)
        self.conv2 = GCNConv(hidden_channels, hidden_channels)
        
        # Functional connectivity processing
        self.fc_encoder = FunctionalConnectivityEncoder(hidden_channels)
        
        # Fusion and classification
        self.fusion = nn.Linear(hidden_channels * 2, hidden_channels)
        self.classifier = nn.Linear(hidden_channels, out_channels)
    
    def forward(self, x, edge_index, batch, node_positions=None):
        # Structural connectivity path
        h_struct = self.conv1(x, edge_index).relu()
        h_struct = self.conv2(h_struct, edge_index)
        
        # Functional connectivity path
        h_func = self.fc_encoder(h_struct, node_positions)
        
        # Multi-modal fusion
        h_combined = torch.cat([h_struct, h_func], dim=-1)
        h_fused = self.fusion(h_combined).relu()
        
        
        h_graph = global_mean_pool(h_fused, batch)
         .classifier(h_graph)

 (nn.Module):
    
     ():
        ().__init__()
        .mlp = nn.Sequential(
            nn.Linear(, hidden_dim),  
            nn.ReLU(),
            nn.Linear(hidden_dim, hidden_dim)
        )
    
     ():
        
        
        dist_matrix = torch.cdist(node_features, node_features)
        
        
        
        persistence_pairs = ._compute_persistence(dist_matrix)
        
        
        features = ._extract_topo_features(persistence_pairs)
         features
    
     ():
        ph_features = .compute_persistence_features(node_features, positions)
         .mlp(ph_features)

Persistent Homology Integration

from gudhi import RipsComplex
import numpy as np

def compute_persistent_homology_features(distance_matrix, max_dim=2):
    """
    Compute persistent homology features from distance matrix
    
    Args:
        distance_matrix: Pairwise distance matrix (N x N)
        max_dim: Maximum homology dimension
    
    Returns:
        Dictionary of topological features
    """
    # Create Rips complex
    rips = RipsComplex(distance_matrix=distance_matrix, max_edge_length=2.0)
    simplex_tree = rips.create_simplex_tree(max_dimension=max_dim)
    
    # Compute persistence
    persistence = simplex_tree.persistence()
    
    # Extract features
    features = {
        'betti_0': count_betti_numbers(persistence, 0),
        'betti_1': count_betti_numbers(persistence, 1),
        'persistence_entropy': compute_persistence_entropy(persistence),
        'total_persistence': compute_total_persistence(persistence),
        'lifetime_statistics': compute_lifetime_stats(persistence)
    }
    
    return features

def functional_connectivity_from_activations(node_activations, threshold=0.5):
    """
    Construct functional connectivity from node activations
    
    Similar to fMRI functional connectivity: correlation between
    time series (here: node activations across layers)
    """
    # Correlation-based functional connectivity
    corr_matrix = torch.corrcoef(node_activations.T)
    
    # Threshold to create binary/adjacency representation
    func_adj = (corr_matrix.abs() > threshold).float()
    
    return func_adj

Applications

1. Brain Network Analysis

• fMRI Connectome Classification: Classify psychiatric disorders from functional connectivity
• Structural-Functional Integration: Combine DTI structural and fMRI functional connectivity
• Multi-Modal Brain Imaging: Leverage complementary information from different modalities

2. General Graph Classification

• Social Networks: Capture community structures via functional connectivity
• Molecular Graphs: Encode topological drug features
• Citation Networks: Model semantic relationships beyond direct citations

3. Network Neuroscience

• Connectome Comparison: Compare brain networks across populations
• Developmental Studies: Track connectivity changes over time
• Disease Biomarkers: Identify disrupted connectivity patterns

Pitfalls

1. Computational Cost: Persistent homology computation can be expensive for large graphs

   • Mitigation: Use approximations or subsampling for large-scale graphs

2. Feature Engineering: Choice of persistence features affects performance

   • Mitigation: Experiment with different topological descriptors

3. Fusion Balance: Combining structural and functional information requires careful weighting

   • Mitigation: Use attention mechanisms or learned fusion strategies

4. Interpretability: Topological features can be abstract

   • Mitigation: Visualize persistence diagrams and barcode representations

Related Skills

• brain-graph-neural: GNN methods for brain connectivity
• higher-order-brain-networks: Topological analysis of brain networks
• persistent-homology-brain: Persistent homology applications in neuroscience
• graph-laplacian-denoising: Graph-based denoising for brain networks

References

@article{li2025functional,
  title={Functional Connectivity Graph Neural Networks},
  author={Li, Yang and Yi, Luopeiwen and Songdechakraiwut, Tananun},
  journal={arXiv preprint arXiv:2508.05786},
  year={2025}
}

Further Reading

• Persistent Homology: Edelsbrunner & Harer, "Computational Topology"
• Brain Functional Connectivity: Fornito et al., "Fundamentals of Brain Network Analysis"
• Graph Neural Networks: Kipf & Welling, "Semi-Supervised Classification with GCN"