Spectral Graph Theory

3.1 Graph Laplacian

Properties

• $\mathbf{L}$ is symmetric positive semidefinite
• Smallest eigenvalue $\lambda_1 = 0$ (eigenvector = $\mathbf{1}$)
• Multiplicity of 0 equals number of connected components
• $\mathbf{x}^T\mathbf{L}\mathbf{x} = \sum_{(i,j) \in E} (x_i - x_j)^2$

3.2 Spectral Clustering

python
Run Inline
from sklearn.cluster import KMeans
import numpy as np
# Affinity matrix
A = np.random.randn(100, 100)
A = A @ A.T  # make symmetric
# Graph Laplacian
D = np.diag(np.sum(A, axis=1))
L = D - A
# Eigendecomposition
eigvals, eigvecs = np.linalg.eigh(L)
# Use bottom k eigenvectors for clustering
k = 3
X = eigvecs[:, :k]
kmeans = KMeans(n_clusters=k).fit(X)
labels = kmeans.labels_

3.3 Cheeger's Inequality