Directed Laplacian Jordan GFT

• Directed Laplacian (Jordan-based) GFT is a framework that defines graph frequencies via the Jordan decomposition of the directed Laplacian, enabling spectral analysis on directed graphs.
• It orders frequencies using total variation, with low frequencies indicating smoother signals and high frequencies signifying stronger local variation.
• The method accommodates both diagonalizable and non-diagonalizable matrices and supports the design of linear shift-invariant graph filters through polynomial representations.

The Directed Laplacian (Jordan-based) Graph Fourier Transform (GFT) is a formalism for the spectral analysis of signals defined on directed graphs, where the Fourier basis is derived from the Jordan decomposition of the graph's directed Laplacian matrix. In this framework, the eigenvectors (and generalized eigenvectors) of the directed Laplacian serve as the graph harmonics, and their corresponding eigenvalues are the graph frequencies. This approach extends the Laplacian-based GFT for undirected graphs to general directed graphs, offering a natural frequency order based on total variation and aligning with the digital signal processing on graphs (DSP$_\mathrm{G}$) methodology (Singh et al., 2016).

1. Directed Graphs, Laplacian Construction, and Signal Representation

Let $\mathcal{G} = (\mathcal{V}, W)$ be a directed graph with node set $\mathcal{V} = \{v_0, v_1, \dots, v_{N-1}\}$ and (possibly weighted or complex) adjacency matrix $W \in \mathbb{R}^{N \times N}$, where element $w_{ij}$ encodes the weight from node $v_j$ to $v_i$. A graph signal is a function $x: \mathcal{V}\rightarrow\mathbb{C}$, collected as $x \in \mathbb{C}^N$, with $x(i)$ being the value at node $v_i$.

The in-degree of node $i$ is $d_i^{\rm in} = \sum_{j=0}^{N-1} w_{ij}$. Define $$D_{\rm in} = \operatorname{diag}(d^{\rm in}_0, ..., d^{\rm in}_{N-1})$$. The directed Laplacian is given by

$L_d = D_{\rm in} - W.$

For adjacency matrices with nonnegative weights, $L_d$ has all eigenvalues in the right half-plane ($\operatorname{Re}(\lambda)\geq 0$), and every row of $L_d$ sums to zero, so $0$ is always an eigenvalue (Singh et al., 2016).

2. Jordan Decomposition and Graph Harmonics

Since $L_d$ is generally non-symmetric and possibly non-diagonalizable, the spectral decomposition is expressed by the Jordan decomposition: $L_d = V J V^{-1},$ where $V$ contains the (generalized) eigenvectors, and $J$ is block-diagonal with Jordan blocks $J_\ell$ for each eigenvalue $\lambda_\ell$. Each Jordan block is of the form: $$J_\ell = \begin{pmatrix} \lambda_\ell & 1 & 0 & \cdots & 0 \ 0 & \lambda_\ell & 1 & \cdots & 0 \ \vdots & & \ddots & \ddots & \vdots \ 0 & & & \lambda_\ell & 1 \ 0 & & & 0 & \lambda_\ell \end{pmatrix}$$ with size $k_\ell \times k_\ell$. The corresponding Jordan chain $\{v_\ell^{(0)}, ..., v_\ell^{(k_\ell-1)}\}$ satisfies: $$(L_d - \lambda_\ell I) v_\ell^{(0)} = 0, \qquad (L_d - \lambda_\ell I) v_\ell^{(1)} = v_\ell^{(0)}, \dots$$ The columns of $V$ are adopted as the graph harmonics (Fourier basis), making this approach applicable to both diagonalizable and non-diagonalizable $L_d$.

3. Shift Operator and Signal Shift

A shift operator is defined analogously to the cycle graph: $S = I - L_d.$ The action of the shift operator on a graph signal $x$ is given by $\tilde{x} = S x = (I - L_d)x$. This operation replaces each node's value by its own value minus the (weighted) Laplacian-difference to its inbound neighbors. The shift operator plays a central role in defining notions of frequency and shift-invariance in the graph setting (Singh et al., 2016).

4. Graph Fourier Transform: Construction and Inverse

The GFT maps a graph signal $x \in \mathbb{C}^N$ to its spectral representation $\hat{x} = V^{-1} x$, where $V$ is as above. The inverse is $x = V \hat{x}$. The eigenvalues $\lambda_i$ of $L_d$, appearing on the diagonal(s) of $J$, serve as the frequencies. In the special case where $L_d$ is diagonalizable, the GFT reduces to the familiar Laplacian eigendecomposition-based transform. For non-diagonalizable $L_d$, the full Jordan structure is employed, and $V$ consists of proper and generalized eigenvectors (Singh et al., 2016).

5. Frequency Ordering via Total Variation

Total variation (TV) provides a principled way to order frequencies: $$TV(x) = \sum_{i=0}^{N-1} |x(i) - \tilde{x}(i)| = \|x - Sx\|_1 = \|L_d x\|_1.$$ For a proper eigenvector $v$ satisfying $L_d v = \lambda v$, $TV(v) = |\lambda| \|v\|_1$. If all eigenvectors are normalized to the same $\ell_1$-norm, this implies that $TV(v) \propto |\lambda|$. Thus, eigenmodes with small $|\lambda|$ are "low frequencies" (little variation across the directed graph), and large $|\lambda|$ correspond to "high frequencies" (strong oscillation or non-smoothness) (Singh et al., 2016).

6. Linear Shift-Invariant Filters and Polynomial Representation

A linear operator $H$ is shift-invariant if $S(Hx) = H(Sx)$, or equivalently, $L_d H = H L_d$. Under mild conditions (where each eigenvalue of $L_d$ has geometric multiplicity one), any shift-invariant operator is a polynomial in $L_d$: $H = h(L_d) = \sum_{m=0}^{M-1} h_m L_d^m,$ with appropriate coefficients $\{h_0, ..., h_{M-1}\}$. The shift $S$ itself is the two-tap filter $H(L_d) = I - L_d$. Eigenvectors and generalized eigenvectors of $L_d$ are simultaneously (generalized) eigenvectors for any such $H$, guaranteeing a consistent spectral structure under filtering (Singh et al., 2016).

7. Interpretation, Special Cases, and Unification

Low-frequency graph harmonics are characterized by smoothness with respect to the directed graph, as quantified by small total variation, while high-frequency harmonics oscillate with strong local variation. In undirected graphs with nonnegative weights, $L_d$ is symmetric positive semidefinite, $V$ is orthonormal, and the GFT reduces to the classical Laplacian eigendecomposition. For constant signals $x = c\,[1, ..., 1]^T$, all energy resides at $\lambda=0$ (the zero frequency), satisfying $L_d x = 0$. For general directed graphs, eigenvalues have non-negative real part and radial ordering in the complex plane. The Jordan-based GFT offers a unifying perspective, connecting the Laplacian-based approach for undirected graphs and the DSP$_\mathrm{G}$ weight-matrix formalism for directed graphs, while restoring the intuitive link between smoothness and frequency for signals on graphs (Singh et al., 2016).