Graph-theoretic analysis methods for brain connectivity. Covers small-world networks, rich-club organization, modularity, hub detection, and connectome analysis. Activation: brain network, graph theory, connectome, small-world, rich-club.
Brain networks can be represented as graphs where nodes are brain regions and edges represent connections. Graph theory provides tools to characterize network topology including small-worldness, rich-club structure, and modularity.
Nodes: Brain regions, neurons, or recording channels Edges: Structural or functional connections
Balance local specialization with global integration:
Hub regions are more densely interconnected than expected.
import numpy as np
import networkx as nx
def compute_basic_metrics(adj_matrix):
"""Compute basic graph metrics for brain network."""
G = nx.from_numpy_array(adj_matrix)
metrics = {
'density': nx.density(G),
'clustering_coeff': nx.average_clustering(G),
'characteristic_path_length': nx.average_shortest_path_length(G)
if nx.is_connected(G) else np.inf,
'efficiency': nx.global_efficiency(G),
}
return metrics
def small_world_coefficient(adj_matrix, n_random=100):
"""Calculate small-world coefficient."""
G = nx.from_numpy_array(adj_matrix)
n_nodes = len(G)
n_edges = G.number_of_edges()
C = nx.average_clustering(G)
L = nx.average_shortest_path_length(G) if nx.is_connected(G) else np.inf
C_randoms = []
for _ in range(n_random):
G_random = nx.gnm_random_graph(n_nodes, n_edges)
C_randoms.append(nx.average_clustering(G_random))
C_random = np.mean(C_randoms)
sigma = C / C_random if C_random > 0 else np.inf
return sigma, C, L
def detect_communities(adj_matrix, resolution=1.0):
"""Detect network communities using Louvain algorithm."""
import community as community_louvain
G = nx.from_numpy_array(adj_matrix)
partition = community_louvain.best_partition(G, weight='weight',
resolution=resolution)
modularity = community_louvain.modularity(partition, G, weight='weight')
return partition, modularity
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