Spectral Analysis

Spectral analysis tools for financial networks.

This module provides functions for spectral analysis of financial networks, including eigendecomposition, spectral gap identification, and diffusion modes.

scr_financial.network.spectral.compute_laplacian(adjacency_matrix, normalized=True)[source]

Compute the graph Laplacian from an adjacency matrix.

Parameters:

  • adjacency_matrix (ndarray) – Adjacency matrix of the graph.

  • normalized (bool) – Whether to compute the normalized Laplacian. Defaults to True.

Returns:

Graph Laplacian matrix as an ndarray.

Return type:

ndarray

scr_financial.network.spectral.eigendecomposition(laplacian)[source]

Perform eigendecomposition of the Laplacian matrix.

Parameters:

laplacian (ndarray) – Graph Laplacian matrix.

Returns:

Tuple of (eigenvalues, eigenvectors).

Return type:

Tuple[ndarray, ndarray]

scr_financial.network.spectral.find_spectral_gap(eigenvalues, min_index=1, max_index=None)[source]

Identify the spectral gap in the eigenvalue spectrum.

Parameters:

  • eigenvalues (ndarray) – Array of eigenvalues.

  • min_index (int) – Minimum index to consider. Defaults to 1.

  • max_index (int | None) – Maximum index to consider. Defaults to None (half of spectrum).

Returns:

Tuple of (gap index, gap size).

Return type:

Tuple[int, float]

scr_financial.network.spectral.compute_diffusion_modes(laplacian, k)[source]

Compute the first k diffusion modes of the network.

Parameters:

  • laplacian (ndarray) – Graph Laplacian matrix.

  • k (int) – Number of modes to compute.

Returns:

Tuple of (eigenvalues, eigenvectors) for the first k modes.

Return type:

Tuple[ndarray, ndarray]

scr_financial.network.spectral.analyze_spectral_properties(eigenvalues, eigenvectors)[source]

Analyze spectral properties of the network.

Parameters:

  • eigenvalues (ndarray) – Array of eigenvalues.

  • eigenvectors (ndarray) – Matrix of eigenvectors.

Returns:

Dictionary containing spectral properties.

Raises:

ValueError – If eigenvalues has fewer than 2 elements.

Return type:

dict

scr_financial.network.spectral.compute_spectral_embedding(laplacian, dim=2)[source]

Compute spectral embedding of the network.

Parameters:

  • laplacian (ndarray) – Graph Laplacian matrix.

  • dim (int) – Embedding dimension. Defaults to 2.

Returns:

Spectral embedding matrix of shape (n_nodes, dim).

Return type:

ndarray

scr_financial.network.spectral.compute_diffusion_distance(laplacian, t=1.0, k=None)[source]

Compute diffusion distance matrix at time t.

Uses a fully vectorised NumPy/SciPy implementation. Memory complexity is O(n²·k) due to the pairwise squared-distance computation.

Parameters:

  • laplacian (ndarray) – Graph Laplacian matrix.

  • t (float) – Diffusion time. Defaults to 1.0.

  • k (int | None) – Number of eigenmodes to use. Defaults to None (use all).

Returns:

Symmetric diffusion distance matrix of shape (n_nodes, n_nodes) with zeros on the diagonal.

Return type:

ndarray