higher-order-topological-ad-alzheimer

name	higher-order-topological-ad-alzheimer
description	Extracting interpretable higher-order topological features across multiple scales for Alzheimer's Disease classification using persistent homology. Captures connected components, cycles, and cavities from fMRI brain networks. Activation: higher-order topology, Alzheimer classification, persistent homology, brain network topology, topological features.

Higher-Order Topological Features for Alzheimer's Disease Classification

Multi-scale persistent homology framework extracting connected components, cycles, and cavities from fMRI brain networks for interpretable Alzheimer's Disease diagnosis.

Metadata

• Source: arXiv:2509.14634v1
• Authors: Dengyi Zhao, Shanyong Li, Yunping Wang, Chenfei Wang, Zhiheng Zhou
• Published: 2025-09-18
• Category: q-bio.NC, cs.LG

Core Methodology

Key Innovation

Traditional brain network analysis focuses on lower-order topological features (node degrees, clustering coefficients), missing critical higher-order structures. This methodology introduces multi-scale persistent homology to extract interpretable higher-order features (connected components, cycles, cavities) that capture complex neural organization patterns in Alzheimer's Disease.

Technical Framework

1. Brain Network Construction

• Input: Resting-state fMRI time series
• Nodes: Brain regions (atlas-based parcellation)
• Edges: Pearson correlation between regional time series
• Output: Weighted undirected graph G = (V, E, w)

2. Persistent Homology Pipeline

Step 1: Filtration Construction

For threshold ε from 0 to max_weight:
    G_ε = subgraph with edges w > ε
    Track birth/death of topological features

Step 2: Multi-Scale Feature Extraction

• H0 (0-dimensional): Connected components

  • Tracks network fragmentation
  • Betti-0: Number of connected components

• H1 (1-dimensional): Cycles/loops

  • Captures recurrent connectivity patterns
  • Betti-1: Number of independent cycles

• H2 (2-dimensional): Cavities/voids

  • Represents enclosed network spaces
  • Betti-2: Number of 3D cavities

3. Persistence Diagram Features

Extract summary statistics from persistence diagrams:

• Persistence entropy: Shannon entropy of persistence values
• Persistence landscape: Functional summaries for ML
• Betti curves: Evolution of Betti numbers across scales

4. Classification Architecture

fMRI → Preprocessing → Network Construction 
    → Persistent Homology → Feature Vector 
    → Random Forest/SVM → AD Diagnosis

Higher-Order Feature Interpretability

Feature Type	Neurobiological Meaning	AD Relevance
Connected Components	Network fragmentation	Disrupted integration
Cycles	Recurrent circuits	Altered feedback loops
Cavities	Enclosed information spaces	Compromised segregation

Implementation Guide

Prerequisites

# Required libraries
pip install gudhi          # Persistent homology
pip install networkx       # Graph construction
pip install nilearn        # fMRI preprocessing
pip install scikit-learn   # Classification

Step-by-Step Implementation

Step 1: fMRI Preprocessing

from nilearn import datasets, preprocessing
import numpy as np

# Load fMRI data and atlas
atlas = datasets.fetch_atlas_schaefer_2018(n_rois=100)
func_data = load_fmri_data(subject_path)

# Extract time series
from nilearn.maskers import NiftiLabelsMasker
masker = NiftiLabelsMasker(labels_img=atlas.maps, standardize=True)
time_series = masker.fit_transform(func_data)

Step 2: Network Construction

import numpy as np
from scipy.stats import pearsonr

# Compute correlation matrix
corr_matrix = np.corrcoef(time_series.T)

# Threshold and binarize (optional)
threshold = 0.3
adj_matrix = (np.abs(corr_matrix) > threshold).astype(int)
np.fill_diagonal(adj_matrix, 0)

Step 3: Persistent Homology Computation

import gudhi as gd

# Create Rips filtration
rips_complex = gd.RipsComplex(
    distance_matrix=1 - np.abs(corr_matrix),
    max_edge_length=1.0
)
simplex_tree = rips_complex.create_simplex_tree(max_dimension=2)

# Compute persistence
persistence = simplex_tree.persistence()

# Extract Betti numbers across filtration
betti_numbers = []
for epsilon in np.linspace(0, 1, 50):
    simplex_tree.compute_persistence()
    betti = simplex_tree.persistent_betti_numbers(epsilon, epsilon)
    betti_numbers.append(betti)

Step 4: Feature Extraction

def extract_topological_features(persistence):
    """Extract features from persistence diagram."""
    features = {}
    
    # Separate by dimension
    dim0 = [p for p in persistence if p[0] == 0]  # H0
    dim1 = [p for p in persistence if p[0] == 1]  # H1
    dim2 = [p for p in persistence if p[0] == 2]  # H2
    
    # Persistence statistics for each dimension
    for dim_name, dim_pers in [('h0', dim0), ('h1', dim1), ('h2', dim2)]:
        if dim_pers:
            persistences = [death - birth for _, (birth, death) in dim_pers if death != float('inf')]
            features[f'{dim_name}_mean_persistence'] = np.mean(persistences)
            features[f'{dim_name}_entropy'] = compute_entropy(persistences)
            features[f'{dim_name}_count'] = len(persistences)
    
    return features

def compute_entropy(persistence_values):
    """Compute persistence entropy."""
    if  persistence_values:
         
    probs = np.array(persistence_values) / (persistence_values)
     -np.(probs * np.log(probs + ))

Step 5: Classification

from sklearn.ensemble import RandomForestClassifier
from sklearn.model_selection import cross_val_score

# Build feature matrix
X = []
y = []
for subject in subjects:
    features = extract_subject_features(subject)
    X.append(features)
    y.append(subject.diagnosis)

# Train classifier
clf = RandomForestClassifier(n_estimators=100, random_state=42)
scores = cross_val_score(clf, X, y, cv=5)
print(f"Cross-validation accuracy: {scores.mean():.3f} ± {scores.std():.3f}")

Applications

• Alzheimer's Disease diagnosis from resting-state fMRI
• Mild Cognitive Impairment (MCI) detection
• Disease progression tracking through longitudinal analysis
• Treatment response prediction

Pitfalls

• Computational complexity: Persistent homology scales O(n³); use approximations for large networks
• Parameter sensitivity: Edge threshold selection affects results; consider multi-threshold approach
• Interpretation: Higher-order features need domain expertise for clinical interpretation
• Small sample sizes: Neuroimaging datasets often limited; use regularization

Related Skills

• homology-morphometry-brain-atrophy
• higher-order-brain-networks
• brain-graph-neural
• combinatorial-complex-brain-fmri

References

@article{zhao2025higherorder,
  title={Extracting Interpretable Higher-Order Topological Features across Multiple Scales for Alzheimer's Disease Classification},
  author={Zhao, Dengyi and Li, Shanyong and Wang, Yunping and Wang, Chenfei and Zhou, Zhiheng},
  journal={arXiv preprint arXiv:2509.14634},
  year={2025}
}