Graph Signals and Permutation Entropy (PEG)

• Graph Signals (PEG) are functions defined on graph vertices that capture both signal values and the underlying graph topology.
• They rely on methods like the Graph Fourier Transform and structured dictionaries to analyze smooth, piecewise-constant, and complex signal patterns.
• PEG extends permutation entropy to graph signals, enabling detection of local irregularities and nuanced dynamics in various applications.

A graph signal is a function $f:V \to \mathbb{R}$ defined on the vertex set $V$ of a graph $G=(V,E)$, and is naturally represented as an $n$-dimensional vector $f = (f(v_1),\ldots,f(v_n))^T$ for a graph with $n=|V|$ vertices. Such signals are central objects in graph signal processing (GSP), where the interplay between signal values and underlying graph structure supports sophisticated analysis, operator design, and learning tasks. Graph signals arise in numerous domains, including sensor networks, brain connectomics, transport systems, and irregular domains more generally. This article surveys the mathematical foundations and advanced research on graph signals, focusing on their structure, associated geometric and spectral analysis tools, their classes and representations, the role of entropy and irregularity, as well as extensions to distributed, multivariate, and uncertain (distributional) settings.

1. Definitions, Representations, and Classes of Graph Signals

A graph signal $f$ can be considered in various contexts depending on its regularity and locality with respect to $G$. The primary structural classes include:

• Smooth graph signals: These admit small variations between nodes connected by edges. Multiple smoothness formulations exist:
  • Pairwise Lipschitz: $|x_i-x_j| \leq C\,d(v_i,v_j)$ for all $i,j$, for some graph metric $d$.
  • Laplacian smoothness: $$x^T L x = \sum_{(i,j)\in E} W_{ij}(x_i-x_j)^2 \leq C$$, where $L$ is the combinatorial Laplacian.
• Piecewise-constant graph signals: Signals constant within certain connected subgraphs, with few discontinuities (cuts) across the graph. Characterized by sparsity in the edge-difference vector $\Delta x$.
• Piecewise-smooth graph signals: Signals that are smooth within subgraphs, potentially having polynomial or bandlimited structure restricted to local clusters.

These definitions induce structured dictionaries for approximation, such as Laplacian or adjacency eigenvector bases (for smooth signals), local-set indicators or wavelet-type bases (for piecewise-constant), and local-polynomial or local-bandlimited dictionaries for piecewise-smooth signals (Chen et al., 2015).

2. Spectral Analysis and the Graph Fourier Transform

Central to GSP is the concept of the Graph Fourier Transform (GFT), which generalizes classical Fourier analysis by diagonalizing a graph shift operator, typically the adjacency or Laplacian matrix: $$A = U \Lambda U^T,\quad f \mapsto \widehat{f}_i = \langle u_i, f \rangle$$ Here, $U=[u_1,\ldots,u_n]$ is the matrix of eigenvectors, $\Lambda$ the (usually ordered) eigenvalues. Spectral content is then defined with respect to these eigenvectors, and bandlimitedness (support of $\widehat{f}$ on low-frequency modes) becomes a surrogate for signal smoothness.

Decisions about operator choice (adjacency, normalized Laplacian, etc.) and their implications for transform domain properties are fundamental. Notably, the GFT description alone may fail to capture all geometric or hidden structure present in graph signals, motivating geometric generalizations (Ji et al., 2022).

The design of graph filters—operators of the form $H(A) = \sum_k h_k A^k$—enables processing such as denoising and bandpass filtering, with efficient implementation via Chebyshev polynomials (Stankovic et al., 2019). Multiscale versions include graph wavelets and Laplacian pyramids (Tanaka, 2017).

3. Geometric and Non-Spectral Perspectives

Beyond GFT-based approaches, graph signals may encode geometry inaccessible to any choice of shift operator. The graph-signal coupling framework introduces a parametrized family of graphs $G_F: M_F \to \mathcal{G}_n$, where $M_F = \mathbb{R}_{\geq 0}^{k+1}$, and the family reflects both shortest-path distances of $G$ and weighted increments of a signal set $F = \{f_1,...,f_k\}$: $$\Delta_F(u,v;x) = \sqrt{ d_G(u,v)^2 + \sum_{i=1}^k x_i (f_i(u) - f_i(v))^2 } \leq x_0$$ This leads to a new geometry on the space of signals and graphs, independent of classical GFT, and yields an alternative smoothness metric computed via optimal partitions of node pairs (Ji et al., 2022). Illustrative applications include recovery of hidden manifold structure in "perpendicular" signals (e.g., helix embeddings) and improved PCA energy concentration when incorporating signal-augmented graphs.

4. Nonlinear Measures: Permutation Entropy and Pattern Complexity

Nonlinear descriptors such as Permutation Entropy for Graph signals (PEG) generalize classical time-series complexity measures to graphs:

• For a graph signal $X$, for each node $i$, PEG forms a vector $$y_i = (x_i, \langle x \rangle_{N_1(i)}, ..., \langle x \rangle_{N_{m-1}(i)})$$, where $N_k(i)$ denotes the $k$-hop neighborhood.
• Each $y_i$ is mapped to an ordinal pattern (permutation), frequencies of which define the entropy: $H_{PEG}(m,L) = -\sum_{\pi} P_\pi \log P_\pi$ PEG inherits invariance properties under monotonic transforms and is sensitive to local irregularity, modulated by graph topology. Denser or more regular graphs tend to yield lower entropy for random signals, since local averaging suppresses random fluctuations (Fabila-Carrasco et al., 2021).

Vertex-level permutation patterns further extend PEG by recording the pattern at each node, mapping the support of local dynamical regimes across the graph. Pattern contrasts such as turning rate $\alpha$ and up-down balance $\beta$ capture features undetectable by global permutation entropy, with applications to brain networks and detection of disease-related functional changes (Fabila-Carrasco et al., 2023).

5. Sampling, Subsampling, and Distributed Graph Signal Processing

Sampling theory for graph signals studies conditions and strategies for recovering signals from a subset of node measurements. For bandlimited signals, recovery is possible from any set of nodes corresponding to a full-rank submatrix of the retained GFT basis.

Recent work establishes that for signals generated via low-rank linear models $y(t) = Bc(t)$ with $B = \sum_k \gamma_k L^k$, optimal subsampling can be performed by selecting nodes so that the sampled rows of a low-rank basis $T$ are invertible, with explicit bounds on recovery error tied to the singular values of $B$. Node selection can be optimized greedily according to correlation structure, and near-perfect recovery is possible at rates governed by the prescribed tolerance (Ravi et al., 2024).

Distributed algorithms, including abstract message passing and one-pass function-valued message protocols, allow signal processing to be performed efficiently while preserving privacy and reducing communication overhead. Solvability is characterized in terms of convexity, dimension balances, and Morse function properties (Ji et al., 2022).

6. Extensions: Vector-Valued, Multivariate, and Distributional Graph Signals

• Vector-valued signals and stratified spectra: When signals at each vertex are vectors rather than scalars, classical GFT cannot be naively applied. Generalized spectral transforms and stratified adjacency levels (based on hop distance) allow extraction of interpretable spectra. Tasks such as diagnosing over-smoothing in learned node embeddings, or profiling multi-scale spectral content, rely on these methods (Meng et al., 2022).
• Cross-spectral analysis: For multivariate graph signal processes, joint stationarity and cross-spectral density estimation generalize the concept of power spectral density. Methods include graph cross-periodograms and windowed GFTs. Magnitude-squared coherence enables quantification of frequency-localized relationships between processes, with robust estimators handling outliers (Kim et al., 2024).
• Distributional graph signals: To model stochasticity and uncertainty in both signal and graph, the classical vector space of signals is replaced by the 2-Wasserstein space of probability measures $\mathcal{P}(\mathbb{R}^n)$. Signals and graphs are both treated as random, enabling pushforward definitions of filtering, Fourier transforms, and sampling. Continuity properties and empirical studies demonstrate improved robustness and flexibility in tasks such as forecasting, anomaly detection, and learning with uncertainty (Ji et al., 2023).

7. Applications, Case Studies, and Future Directions

Applications of graph signals span environmental sensor networks (multisensor temperature estimation), epidemic modeling, social networks, functional brain analysis (via DTI–fMRI fusion), and image processing domains. Geometry-aware filtering and entropy-based analysis have demonstrated empirical gains in denoising (e.g., MNIST digit recovery with sharpness preservation) and in the population-level analysis of cortical networks in disease progression.

Open problems include the characterization and construction of geometry-aware bandlimited subspaces; generalization to higher-order relational domains (simplicial complexes, hypergraphs); integration of distributional and multivariate methodologies; and the design of scalable, adaptive pipelines that combine signal processing and dynamic graph learning. Computational aspects such as sub-quadratic algorithms remain topics of ongoing research (Ji et al., 2022).