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If similarity graphs are the conceptual foundation of spectral clustering, then the graph Laplacian matrix is its computational engine. This matrix—seemingly just a simple difference between degree and adjacency matrices—possesses remarkable spectral properties that encode the graph's cluster structure in its eigenvalues and eigenvectors.
The Laplacian is not merely a convenient mathematical tool. It arises naturally in diverse contexts: heat diffusion on graphs, electrical resistance networks, random walks, and the continuous limit of differential operators on manifolds. These connections explain why the Laplacian's spectral properties reveal intrinsic geometric structure.
Mastering the Laplacian matrix is essential for understanding not just spectral clustering, but an entire family of graph-based machine learning methods including graph neural networks, Laplacian eigenmaps, and diffusion maps.
By the end of this page, you will understand: (1) The definition and construction of the unnormalized graph Laplacian, (2) Fundamental properties including positive semi-definiteness and the multiplicity of zero eigenvalues, (3) The normalized Laplacian variants and when to use each, (4) How the Laplacian connects to the continuous Laplace operator, and (5) Why the Laplacian's eigenvectors encode cluster structure.
Given an undirected weighted graph $G = (V, E, W)$ with $n$ vertices, we define three fundamental matrices:
Adjacency/Weight Matrix $W$: $$W_{ij} = \begin{cases} w_{ij} & \text{if edge } (i,j) \in E \ 0 & \text{otherwise} \end{cases}$$
For an undirected graph, $W = W^T$ (symmetric). Weights $w_{ij} \geq 0$ encode similarity.
Degree Matrix $D$: $$D = \text{diag}(d_1, d_2, \ldots, d_n)$$
where $d_i = \sum_{j=1}^{n} W_{ij}$ is the degree of vertex $i$—the total weight of edges incident to $i$.
Unnormalized Graph Laplacian $L$: $$L = D - W$$
This deceptively simple definition encodes profound structure. Let's examine what this matrix looks like and why it matters.
Matrix Structure:
The Laplacian has a specific pattern:
$$L_{ij} = \begin{cases} d_i & \text{if } i = j \ -W_{ij} & \text{if } i \neq j \text{ and } (i,j) \in E \ 0 & \text{otherwise} \end{cases}$$
Example: Simple Graph with 4 Nodes:
Consider an unweighted path graph: 1—2—3—4
$$W = \begin{pmatrix} 0 & 1 & 0 & 0 \ 1 & 0 & 1 & 0 \ 0 & 1 & 0 & 1 \ 0 & 0 & 1 & 0 \end{pmatrix}, \quad D = \begin{pmatrix} 1 & 0 & 0 & 0 \ 0 & 2 & 0 & 0 \ 0 & 0 & 2 & 0 \ 0 & 0 & 0 & 1 \end{pmatrix}$$
$$L = D - W = \begin{pmatrix} 1 & -1 & 0 & 0 \ -1 & 2 & -1 & 0 \ 0 & -1 & 2 & -1 \ 0 & 0 & -1 & 1 \end{pmatrix}$$
Notice: rows sum to zero, matrix is symmetric, diagonal entries are degrees.
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import numpy as npfrom scipy.sparse import diags, csr_matrixfrom scipy.sparse.linalg import eigsh class GraphLaplacian: """ Comprehensive implementation of graph Laplacian matrices and their variants. The graph Laplacian is the cornerstone of spectral graph theory and spectral clustering. This class implements multiple variants and demonstrates their properties through eigendecomposition. """ def __init__(self, W): """ Initialize with a symmetric weight matrix W. Parameters: ----------- W : ndarray, shape (n, n) Symmetric weight matrix. W[i,j] >= 0 represents the edge weight between nodes i and j. Self-loops (diagonal) should typically be 0. """ self.W = np.asarray(W, dtype=np.float64) self.n = W.shape[0] # Verify symmetry if not np.allclose(self.W, self.W.T): raise ValueError("Weight matrix must be symmetric") # Compute degrees self.d = np.sum(self.W, axis=1) # Row sums = degrees self.D = np.diag(self.d) # Degree matrix def unnormalized(self): """ Compute the unnormalized graph Laplacian: L = D - W Properties: ----------- - Symmetric, positive semi-definite - Smallest eigenvalue is 0 with eigenvector = [1, 1, ..., 1]^T - Number of zero eigenvalues = number of connected components - For vector f: f^T L f = (1/2) * sum_{i,j} W_{ij}(f_i - f_j)^2 Returns: -------- L : ndarray, shape (n, n) Unnormalized Laplacian matrix """ L = self.D - self.W return L def verify_quadratic_form(self, f=None): """ Verify the Laplacian quadratic form identity: f^T L f = (1/2) * sum_{i,j} W_{ij} * (f_i - f_j)^2 This identity shows L measures "smoothness" of f over the graph. A function f is smooth if it varies slowly along edges. """ if f is None: f = np.random.randn(self.n) # Random test vector L = self.unnormalized() # Left side: quadratic form left_side = f @ L @ f # Right side: sum of squared differences right_side = 0.0 for i in range(self.n): for j in range(self.n): right_side += self.W[i, j] * (f[i] - f[j])**2 right_side *= 0.5 return { 'quadratic_form': left_side, 'sum_of_differences': right_side, 'are_equal': np.isclose(left_side, right_side) } def normalized_symmetric(self): """ Compute the symmetric normalized Laplacian: L_sym = D^{-1/2} L D^{-1/2} Also written as: L_sym = I - D^{-1/2} W D^{-1/2} Properties: ----------- - Symmetric, positive semi-definite - Eigenvalues in [0, 2] - Related to random walk normalized Laplacian - Better for graphs with varying degrees Returns: -------- L_sym : ndarray, shape (n, n) Symmetric normalized Laplacian """ # Handle zero degrees carefully d_inv_sqrt = np.zeros(self.n) nonzero = self.d > 0 d_inv_sqrt[nonzero] = 1.0 / np.sqrt(self.d[nonzero]) D_inv_sqrt = np.diag(d_inv_sqrt) # L_sym = I - D^{-1/2} W D^{-1/2} L_sym = np.eye(self.n) - D_inv_sqrt @ self.W @ D_inv_sqrt return L_sym def normalized_random_walk(self): """ Compute the random walk normalized Laplacian: L_rw = D^{-1} L = I - D^{-1} W Properties: ----------- - NOT symmetric (unless graph is regular) - Related to random walk transition matrix P = D^{-1} W - L_rw = I - P - Same eigenvalues as L_sym (similar matrices) - Eigenvectors related by: v_{L_sym} = D^{1/2} v_{L_rw} Returns: -------- L_rw : ndarray, shape (n, n) Random walk normalized Laplacian """ # Handle zero degrees d_inv = np.zeros(self.n) nonzero = self.d > 0 d_inv[nonzero] = 1.0 / self.d[nonzero] D_inv = np.diag(d_inv) L_rw = np.eye(self.n) - D_inv @ self.W return L_rw def transition_matrix(self): """ Compute the random walk transition matrix: P = D^{-1} W P[i,j] = probability of walking from i to j = W[i,j] / d_i Returns: -------- P : ndarray, shape (n, n) Row-stochastic transition matrix """ d_inv = np.zeros(self.n) nonzero = self.d > 0 d_inv[nonzero] = 1.0 / self.d[nonzero] P = np.diag(d_inv) @ self.W return P def eigendecomposition(self, which='unnormalized', k=None): """ Compute eigendecomposition of a Laplacian variant. Parameters: ----------- which : str 'unnormalized', 'symmetric', or 'random_walk' k : int or None Number of smallest eigenvalues/vectors to compute. If None, compute all. Returns: -------- eigenvalues : ndarray Sorted eigenvalues (ascending) eigenvectors : ndarray Corresponding eigenvectors as columns """ if which == 'unnormalized': L = self.unnormalized() elif which == 'symmetric': L = self.normalized_symmetric() elif which == 'random_walk': L = self.normalized_random_walk() else: raise ValueError(f"Unknown Laplacian type: {which}") if k is None: from scipy.linalg import eigh eigenvalues, eigenvectors = eigh(L) else: # For sparse matrices or when only few eigenvalues needed eigenvalues, eigenvectors = eigsh( csr_matrix(L), k=k, which='SM' # Smallest Magnitude ) # Sort by eigenvalue idx = np.argsort(eigenvalues) eigenvalues = eigenvalues[idx] eigenvectors = eigenvectors[:, idx] return eigenvalues, eigenvectors def demonstrate_laplacian_properties(): """ Demonstrate fundamental properties of the graph Laplacian. """ # Create a simple graph: two connected components (two triangles) # Triangle 1: nodes 0-1-2 # Triangle 2: nodes 3-4-5 W = np.zeros((6, 6)) # Triangle 1 W[0, 1] = W[1, 0] = 1 W[0, 2] = W[2, 0] = 1 W[1, 2] = W[2, 1] = 1 # Triangle 2 W[3, 4] = W[4, 3] = 1 W[3, 5] = W[5, 3] = 1 W[4, 5] = W[5, 4] = 1 gl = GraphLaplacian(W) print("Graph Laplacian Properties Demonstration") print("=" * 55) print("\nGraph: Two disconnected triangles (6 nodes)") print("\nWeight matrix W:") print(W) L = gl.unnormalized() print("\nUnnormalized Laplacian L = D - W:") print(L) # Verify row sums are zero print(f"\nRow sums: {L.sum(axis=1)}") # Eigendecomposition eigenvalues, eigenvectors = gl.eigendecomposition('unnormalized') print(f"\nEigenvalues: {eigenvalues}") print("\nNote: Two zero eigenvalues -> Two connected components!") # Eigenvectors corresponding to zero eigenvalue print("\nEigenvectors for λ=0 (span the nullspace):") print(eigenvectors[:, :2].round(3)) print("Note: Each eigenvector is constant within a connected component") # Verify quadratic form result = gl.verify_quadratic_form() print(f"\nQuadratic form identity verified: {result['are_equal']}") # Add a weak connection between components print("\n" + "=" * 55) print("Adding weak edge between triangles (W[2,3] = 0.1)...") W[2, 3] = W[3, 2] = 0.1 gl2 = GraphLaplacian(W) eigenvalues2, eigenvectors2 = gl2.eigendecomposition('unnormalized') print(f"\nNew eigenvalues: {eigenvalues2.round(4)}") print("Note: Now only ONE zero eigenvalue (graph is connected)") print(f"Second eigenvalue (algebraic connectivity): {eigenvalues2[1]:.6f}") print("This small value indicates weak connectivity between components") return gl, gl2 if __name__ == "__main__": demonstrate_laplacian_properties()The graph Laplacian possesses several remarkable properties that make it central to spectral graph theory. Understanding these properties is essential for understanding why spectral clustering works.
Property 1: Symmetry and Positive Semi-Definiteness
The unnormalized Laplacian $L = D - W$ is:
Positive semi-definiteness follows from the quadratic form (Property 2).
Property 2: The Quadratic Form Identity
For any vector $f \in \mathbb{R}^n$:
$$f^T L f = \frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n} W_{ij}(f_i - f_j)^2$$
This identity is profound. The left side is a quadratic form—a standard object in linear algebra. The right side measures how much $f$ varies across graph edges, weighted by edge strength.
Interpretation: $f^T L f$ measures the smoothness of the function $f$ over the graph. If $f$ assigns similar values to connected nodes, the sum is small. If $f$ differs greatly across edges, the sum is large.
We derive this crucial identity. Start with $f^T L f = f^T (D - W) f = f^T D f - f^T W f$.
The first term: $f^T D f = \sum_i d_i f_i^2 = \sum_i f_i^2 \sum_j W_{ij} = \sum_{i,j} W_{ij} f_i^2$.
The second term: $f^T W f = \sum_{i,j} W_{ij} f_i f_j$.
Thus: $f^T L f = \sum_{i,j} W_{ij} f_i^2 - \sum_{i,j} W_{ij} f_i f_j = \sum_{i,j} W_{ij} (f_i^2 - f_i f_j)$.
By symmetry of $W$, this equals $(1/2) \sum_{i,j} W_{ij} [(f_i^2 - f_i f_j) + (f_j^2 - f_j f_i)] = (1/2) \sum_{i,j} W_{ij} (f_i - f_j)^2$. QED.
Property 3: Zero Eigenvalue and the Constant Eigenvector
The vector $\mathbf{1} = (1, 1, \ldots, 1)^T$ is always an eigenvector of $L$ with eigenvalue 0:
$$L \mathbf{1} = (D - W) \mathbf{1} = D\mathbf{1} - W\mathbf{1}$$
$D\mathbf{1} = (d_1, d_2, \ldots, d_n)^T$ and $W\mathbf{1} = (\sum_j W_{1j}, \ldots)^T = (d_1, \ldots, d_n)^T$
Thus $L\mathbf{1} = \mathbf{0}$.
Intuition: A constant function on the graph has zero "variation" across edges, so the quadratic form is zero. This corresponds to eigenvalue 0.
Property 4: Connected Components and Zero Eigenvalue Multiplicity
Theorem: The multiplicity of eigenvalue 0 equals the number of connected components in the graph.
This is fundamental: eigenvalue 0 detects disconnection. If we see two zero eigenvalues, we know the graph has (at least) two components, each of which is already a natural cluster.
Zero Eigenvalues | Graph Structure | Clustering Implication |
|---|---|---|
| 1 | Graph is connected | Look at higher eigenvectors for structure |
| k (k > 1) | Graph has k connected components | Components are automatic clusters |
| Near-zero eigenvalues | Weak connections between subgraphs | Natural cluster structure exists |
Property 5: The Algebraic Connectivity (Fiedler Value)
The second smallest eigenvalue $\lambda_2$ is called the algebraic connectivity or Fiedler value. For a connected graph:
$$\lambda_2 > 0$$
$\lambda_2$ measures how well-connected the graph is:
The corresponding eigenvector $v_2$ (Fiedler vector) provides the optimal partition direction for RatioCut.
The Cheeger Inequality relates algebraic connectivity to the minimum cut:
$$\frac{h^2}{2} \leq \lambda_2 \leq 2h$$
where $h$ is the Cheeger constant (minimum normalized cut). This inequality guarantees that spectral methods find cuts within a constant factor of optimal.
The unnormalized Laplacian has a bias: high-degree nodes contribute more to the quadratic form than low-degree nodes. For graphs with varying node degrees, normalization can lead to better clustering.
Symmetric Normalized Laplacian $L_{sym}$:
$$L_{sym} = D^{-1/2} L D^{-1/2} = I - D^{-1/2} W D^{-1/2}$$
where $D^{-1/2} = \text{diag}(d_1^{-1/2}, \ldots, d_n^{-1/2})$.
Properties of $L_{sym}$:
Random Walk Laplacian $L_{rw}$:
$$L_{rw} = D^{-1} L = I - D^{-1} W = I - P$$
where $P = D^{-1}W$ is the random walk transition matrix (row-stochastic).
Properties of $L_{rw}$:
Relationship Between Normalized Laplacians:
If $(\lambda, v)$ is an eigenpair of $L_{rw}$, then $(\lambda, D^{1/2}v)$ is an eigenpair of $L_{sym}$.
Conversely, if $(\lambda, u)$ is an eigenpair of $L_{sym}$, then $(\lambda, D^{-1/2}u)$ is an eigenpair of $L_{rw}$.
This relationship means we can work with the symmetric $L_{sym}$ (better numerical properties) and transform eigenvectors when needed.
When to Use Which Laplacian:
| Laplacian | Use Case | Advantages | Disadvantages |
|---|---|---|---|
| Unnormalized L | Uniform degree graphs; RatioCut objective | Simple; direct interpretation | Biased toward high-degree nodes |
| Symmetric L_sym | General graphs; NCut objective | Symmetric; normalized by degree | Requires degree normalization of results |
| Random Walk L_rw | Random walk interpretation; NCut | Direct cluster indicator interpretation | Not symmetric (complicates computation) |
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import numpy as npfrom scipy.linalg import eighfrom sklearn.datasets import make_moonsfrom sklearn.neighbors import kneighbors_graphfrom sklearn.cluster import KMeans def compare_laplacians_for_clustering(X, n_clusters=2, n_neighbors=10): """ Compare spectral clustering using different Laplacian variants. This demonstration shows how the choice of Laplacian affects the resulting clustering, particularly for graphs with varying node degrees. Parameters: ----------- X : ndarray, shape (n_samples, n_features) Data points to cluster n_clusters : int Number of clusters n_neighbors : int Number of neighbors for KNN graph construction Returns: -------- results : dict Clustering results for each Laplacian type """ n = X.shape[0] # Construct KNN graph connectivity = kneighbors_graph(X, n_neighbors, mode='connectivity') W = 0.5 * (connectivity + connectivity.T).toarray() # Compute degree and degree matrix d = np.sum(W, axis=1) D = np.diag(d) results = {} # 1. Unnormalized Laplacian L = D - W eigenvalues_L, eigenvectors_L = eigh(L) embedding_L = eigenvectors_L[:, :n_clusters] # Cluster in embedding space kmeans_L = KMeans(n_clusters=n_clusters, random_state=42, n_init=10) labels_L = kmeans_L.fit_predict(embedding_L) results['unnormalized'] = { 'labels': labels_L, 'eigenvalues': eigenvalues_L[:n_clusters+2], 'embedding': embedding_L } # 2. Symmetric Normalized Laplacian d_inv_sqrt = np.zeros(n) d_inv_sqrt[d > 0] = 1.0 / np.sqrt(d[d > 0]) D_inv_sqrt = np.diag(d_inv_sqrt) L_sym = np.eye(n) - D_inv_sqrt @ W @ D_inv_sqrt eigenvalues_sym, eigenvectors_sym = eigh(L_sym) embedding_sym = eigenvectors_sym[:, :n_clusters] # Row-normalize embedding (important for normalized spectral clustering!) row_norms = np.linalg.norm(embedding_sym, axis=1, keepdims=True) row_norms[row_norms == 0] = 1 embedding_sym_normalized = embedding_sym / row_norms kmeans_sym = KMeans(n_clusters=n_clusters, random_state=42, n_init=10) labels_sym = kmeans_sym.fit_predict(embedding_sym_normalized) results['symmetric'] = { 'labels': labels_sym, 'eigenvalues': eigenvalues_sym[:n_clusters+2], 'embedding': embedding_sym_normalized } # 3. Random Walk Laplacian (use symmetric for computation, transform) # For clustering, we use the same eigenvectors (they have same eigenvalues) # The difference is in interpretation and possible row normalization L_rw = np.eye(n) - np.diag(1.0/np.maximum(d, 1e-10)) @ W eigenvalues_rw, eigenvectors_rw = eigh( L_sym # Use symmetric for computation (same eigenvalues) ) # Transform eigenvectors: v_rw = D^{-1/2} v_sym eigenvectors_rw = D_inv_sqrt @ eigenvectors_sym embedding_rw = eigenvectors_rw[:, :n_clusters] kmeans_rw = KMeans(n_clusters=n_clusters, random_state=42, n_init=10) labels_rw = kmeans_rw.fit_predict(embedding_rw) results['random_walk'] = { 'labels': labels_rw, 'eigenvalues': eigenvalues_rw[:n_clusters+2], 'embedding': embedding_rw } return results, W, d def analyze_degree_effect(): """ Analyze how node degree variation affects different Laplacians. """ # Create two moons with varying noise (creates varying local density) X, y_true = make_moons(n_samples=200, noise=0.08, random_state=42) results, W, degrees = compare_laplacians_for_clustering(X) print("Spectral Clustering: Laplacian Comparison") print("=" * 55) print(f"\nDegree statistics: min={degrees.min():.1f}, max={degrees.max():.1f}, " f"mean={degrees.mean():.1f}") for name, res in results.items(): acc1 = np.mean(res['labels'] == y_true) acc2 = np.mean(res['labels'] != y_true) accuracy = max(acc1, acc2) print(f"\n{name.upper()} Laplacian:") print(f" Eigenvalues: {res['eigenvalues'][:4].round(4)}") print(f" Accuracy: {accuracy:.1%}") print("\n" + "=" * 55) print("Note: For two moons with uniform density, results are similar.") print("Differences emerge with varying density or graph degree distribution.") return results, X, y_true if __name__ == "__main__": analyze_degree_effect()In practice, the symmetric normalized Laplacian (L_sym) combined with row-normalization of eigenvectors (the Ng-Jordan-Weiss algorithm) is most commonly used. It handles varying node degrees well and connects directly to the normalized cut objective. The unnormalized Laplacian is simpler but can be biased toward isolating high-degree regions.
The graph Laplacian is not merely a convenient construction—it is the discrete analog of the Laplace operator (or Laplacian) from calculus, which is fundamental in physics and mathematics.
The Continuous Laplacian:
For a function $f: \mathbb{R}^d \to \mathbb{R}$, the Laplace operator is:
$$\Delta f = \nabla^2 f = \sum_{i=1}^{d} \frac{\partial^2 f}{\partial x_i^2}$$
This operator appears in:
Discretization on a Grid:
Consider a 1D grid with spacing $h$. The second derivative is approximated by:
$$f''(x) \approx \frac{f(x+h) - 2f(x) + f(x-h)}{h^2}$$
In matrix form for $n$ grid points with periodic boundary:
$$L_{grid} = \frac{1}{h^2} \begin{pmatrix} 2 & -1 & 0 & \cdots & -1 \ -1 & 2 & -1 & \cdots & 0 \ \vdots & & \ddots & & \vdots \ -1 & 0 & \cdots & -1 & 2 \end{pmatrix}$$
This is exactly the graph Laplacian for a cycle graph (up to the $1/h^2$ scaling)!
Why This Connection Matters:
The continuous Laplacian has well-studied spectral properties:
On a circle (1D torus): Eigenfunctions are sines and cosines of increasing frequency. Lower eigenfunctions vary slowly (smooth); higher eigenfunctions vary rapidly (oscillatory).
On a rectangle with Dirichlet boundary: Eigenfunctions are products of sines, with eigenvalues depending on spatial frequencies.
On general manifolds: Eigenfunctions encode geometric structure. The first nontrivial eigenfunction often separates the manifold into two "halves."
The graph Laplacian inherits these properties in discrete form:
Manifold Learning Connection:
When data lies on a manifold sampled with noise, the graph Laplacian (constructed with appropriate bandwidth) converges to the Laplace-Beltrami operator on the manifold as sample size grows. This justifies using spectral methods for manifold learning and explains why they discover intrinsic geometry.
Think of heat diffusion on a graph: each node has a temperature, and heat flows along edges proportional to temperature difference weighted by edge weight. The heat equation on graphs is df/dt = -Lf, where f(t) is the temperature vector at time t. The eigenvectors of L are the 'modes' of this diffusion—the first eigenvector (constant) is the equilibrium, and higher eigenvectors decay proportional to their eigenvalue. This diffusion view underlies diffusion maps and connects to random walks.
We now connect the Laplacian's spectral properties directly to the clustering problem. This section explains why eigenvectors of $L$ provide cluster information.
The Rayleigh Quotient Perspective:
For the unnormalized Laplacian, the Rayleigh quotient of a vector $f$ is:
$$R(f) = \frac{f^T L f}{f^T f} = \frac{\sum_{i,j} W_{ij}(f_i - f_j)^2 / 2}{\sum_i f_i^2}$$
Minimizing $R(f)$ finds vectors that vary minimally across graph edges. By the Rayleigh-Ritz theorem:
Why Eigenvectors Reveal Clusters:
If the graph has $k$ well-separated clusters (clusters connected internally but weakly between each other), then:
The Ideal Case: Disconnected Clusters
Consider a graph with exactly $k$ disconnected components. The Laplacian is block-diagonal:
$$L = \begin{pmatrix} L_1 & 0 & \cdots & 0 \ 0 & L_2 & \cdots & 0 \ \vdots & & \ddots & \vdots \ 0 & 0 & \cdots & L_k \end{pmatrix}$$
Each block $L_i$ is the Laplacian of component $i$, each having its own zero eigenvalue with constant eigenvector on that component.
The first $k$ eigenvectors of $L$ are indicator vectors: $$v_j^{(i)} = \begin{cases} 1 & \text{if node } j \text{ is in component } i \ 0 & \text{otherwise} \end{cases}$$
(scaled appropriately for orthonormality)
Clustering in the space of these $k$ eigenvectors is trivial—each cluster occupies a distinct corner of the $k$-dimensional cube.
The Near-Ideal Case: Weakly Connected Clusters
In practice, clusters are connected by weak edges. Perturbation theory tells us:
This is why spectral clustering uses K-Means on eigenvectors: even with perturbation, cluster structure is preserved in the embedding.
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import numpy as npfrom scipy.linalg import eighfrom scipy.spatial.distance import cdistimport matplotlib.pyplot as plt def analyze_eigenvector_clustering(): """ Demonstrate how Laplacian eigenvectors encode cluster structure. We construct a graph with clear cluster structure and analyze how the eigenvectors reflect this structure. """ np.random.seed(42) # Create 3 well-separated Gaussian clusters n_per_cluster = 50 cluster1 = np.random.randn(n_per_cluster, 2) * 0.5 + [0, 0] cluster2 = np.random.randn(n_per_cluster, 2) * 0.5 + [5, 0] cluster3 = np.random.randn(n_per_cluster, 2) * 0.5 + [2.5, 4] X = np.vstack([cluster1, cluster2, cluster3]) true_labels = np.array([0]*n_per_cluster + [1]*n_per_cluster + [2]*n_per_cluster) n = X.shape[0] # Construct Gaussian similarity graph distances = cdist(X, X) sigma = 1.0 # Controls connectivity W = np.exp(-distances**2 / (2 * sigma**2)) np.fill_diagonal(W, 0) # Compute unnormalized Laplacian d = np.sum(W, axis=1) D = np.diag(d) L = D - W # Eigendecomposition eigenvalues, eigenvectors = eigh(L) print("Eigenvector Cluster Analysis") print("=" * 55) print(f"\nNumber of points: {n} ({n_per_cluster} per cluster)") print(f"\nFirst 6 eigenvalues: {eigenvalues[:6].round(4)}") print("\nNote: ~3 near-zero eigenvalues suggest 3 clusters") print(f"Spectral gap (λ₄ - λ₃): {eigenvalues[3] - eigenvalues[2]:.4f}") # Analyze eigenvector values by cluster print("\nEigenvector 2 (Fiedler) statistics by cluster:") v2 = eigenvectors[:, 1] for c in range(3): mask = true_labels == c print(f" Cluster {c}: mean={v2[mask].mean():.3f}, std={v2[mask].std():.3f}") print("\nEigenvector 3 statistics by cluster:") v3 = eigenvectors[:, 2] for c in range(3): mask = true_labels == c print(f" Cluster {c}: mean={v3[mask].mean():.3f}, std={v3[mask].std():.3f}") # Spectral embedding k = 3 embedding = eigenvectors[:, 1:k+1] # Skip first (constant) eigenvector # Row normalize (for normalized spectral clustering) row_norms = np.linalg.norm(embedding, axis=1, keepdims=True) row_norms[row_norms == 0] = 1 embedding_normalized = embedding / row_norms print("\nSpectral Embedding Analysis:") print("After embedding in eigenvector space, clusters are linearly separable:") # Simple analysis: within-cluster vs between-cluster distances within_dists = [] between_dists = [] for i in range(n): for j in range(i+1, n): dist = np.linalg.norm(embedding_normalized[i] - embedding_normalized[j]) if true_labels[i] == true_labels[j]: within_dists.append(dist) else: between_dists.append(dist) print(f" Within-cluster mean distance: {np.mean(within_dists):.3f}") print(f" Between-cluster mean distance: {np.mean(between_dists):.3f}") print(f" Separation ratio: {np.mean(between_dists)/np.mean(within_dists):.2f}x") return X, eigenvalues, eigenvectors, true_labels def visualize_spectral_gap(): """ Visualize how the spectral gap indicates cluster quality. We vary the separation between clusters and observe the eigenvalue spectrum. """ print("\n" + "=" * 55) print("Spectral Gap Analysis: Varying Cluster Separation") print("=" * 55) separations = [1, 2, 4, 8] # Distance between cluster centers for sep in separations: np.random.seed(42) n = 60 # Two clusters with varying separation c1 = np.random.randn(n//2, 2) * 0.5 c2 = np.random.randn(n//2, 2) * 0.5 + [sep, 0] X = np.vstack([c1, c2]) # Build graph distances = cdist(X, X) sigma = 1.0 W = np.exp(-distances**2 / (2 * sigma**2)) np.fill_diagonal(W, 0) d = np.sum(W, axis=1) L = np.diag(d) - W eigenvalues = np.linalg.eigvalsh(L) gap = eigenvalues[2] - eigenvalues[1] print(f"\nSeparation = {sep}:") print(f" λ₁={eigenvalues[0]:.4f}, λ₂={eigenvalues[1]:.4f}, λ₃={eigenvalues[2]:.4f}") print(f" Spectral gap (λ₃-λ₂) = {gap:.4f}") print(f" Interpretation: {'Clear clusters' if gap > 0.5 else 'Weak separation'}") if __name__ == "__main__": analyze_eigenvector_clustering() visualize_spectral_gap()A large gap between eigenvalue λₖ and λₖ₊₁ suggests the data has k natural clusters. This "eigengap heuristic" is widely used to choose the number of clusters: plot eigenvalues and look for the largest gap. However, this is a heuristic, not a guarantee—real data often has gradual eigenvalue decay without clear gaps.
Working with graph Laplacians at scale requires understanding their computational properties.
Matrix Properties:
Eigendecomposition Complexity:
Memory Considerations:
Practical Limit: Direct spectral clustering is typically feasible for $n \lesssim 10^4-10^5$ with sparse graphs. Beyond this, approximations (Nyström, landmark methods) are necessary.
| Operation | Dense Graph | Sparse Graph (KNN with k neighbors) |
|---|---|---|
| Graph construction | O(n²d) for n points in d dimensions | O(n log n · d) with KD-tree |
| Laplacian storage | O(n²) | O(nk) |
| Full eigendecomp | O(n³) | Not applicable |
| Top-k eigendecomp | O(n²k) | O(nk² · iterations) |
| K-Means on embedding | O(n · k² · iters) | Same |
Computing the Laplacian requires care: (1) Ensure degree matrix is computed accurately—small errors accumulate. (2) For normalized Laplacians, handle zero-degree nodes (isolated nodes) explicitly. (3) Use established libraries (scipy.sparse.linalg, scikit-learn) rather than implementing from scratch. (4) The smallest eigenvalue should be exactly 0—numerical error may give small positive or negative values; threshold appropriately.
We have developed a deep understanding of the graph Laplacian matrix—the mathematical heart of spectral clustering. Let's consolidate the key insights:
What's Next:
With the Laplacian understood, we next examine the eigenstructure in detail: how eigenvalues and eigenvectors together reveal graph and cluster properties. We'll prove the key theorems connecting spectra to cuts and develop intuition for the spectral embedding that makes clustering work.
You now understand the graph Laplacian matrix—its construction, properties, variants, and connection to continuous mathematics. This matrix is the computational engine of spectral clustering: its eigenvectors encode the cluster structure we seek. Next, we explore how to interpret and use this eigenstructure for clustering.