Glossary

Adjacency Matrix

Matrix \(\mathbf{W} \in \mathbb{R}^{N \times N}\) where \(W_{ij}\) is the edge weight between vertices \(i\) and \(j\). Zero entries indicate no edge.

Degree Matrix

Diagonal matrix \(\mathbf{D} \in \mathbb{R}^{N \times N}\) where \(D_{ii} = \sum_j W_{ij}\) is the sum of weights connected to vertex \(i\).

Graph Laplacian

The matrix \(\mathbf{L} = \mathbf{D} - \mathbf{W}\) encoding graph structure. Its eigenvalues provide a notion of graph frequency, with physical interpretation depending on the branch weighting scheme.

Poles

Shift values \(q\) in \((\mathbf{L} + q\mathbf{I})^{-1}\). For analytical filters, pole = 1/scale. In VFKernel, stored as Q with shape (n_poles,).

Rational Approximation

Representing a filter function as a ratio of polynomials. Vector Fitting produces pole-residue form \(g(\lambda) \approx d + \sum_k \frac{r_k}{\lambda + q_k}\), enabling efficient linear solves.

Residues

Complex coefficients \(r_k\) in a rational approximation. In VFKernel, stored as R with shape (n_poles, n_dims) where n_dims is the output dimension of the kernel.

Scale

Dilation parameter \(s\) controlling filter bandwidth. Related to poles by \(s = 1/q\) and to wavelength by \(\lambda \approx \sqrt{s}\). Larger scales correspond to lower graph frequencies.

Spectral Domain

The eigenspace of the Graph Laplacian. Signals are decomposed into eigenvector components, analogous to Fourier analysis on regular domains.

Vector Fitting

Algorithm for fitting a rational function to frequency-domain samples. Produces poles and residues stored in VFKernel for use with Convolve and DyConvolve.

Vertex Domain

Signals defined on graph nodes as vectors \(\mathbf{f} \in \mathbb{R}^N\), where the \(i\)-th element is the signal value at vertex \(i\). Contrast with Spectral Domain.

VFKernel

Data structure holding Vector Fitting results:

Q: poles, shape (n_poles,)
R: residues, shape (n_poles, n_dims)
D: direct term, shape (n_dims,)

Wavelength

Spatial extent of oscillation modes, approximately \(\sqrt{s}\) where \(s\) is the scale. Large wavelengths indicate spatially extended (inter-area) modes; small wavelengths indicate localized modes.

Wavenumber

Spatial frequency \(k\). For appropriately weighted graphs (e.g., inverse squared distance), Laplacian eigenvalues correspond to \(k^2\), linking spectral and physical domains.