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arxiv: 2604.05086 · v1 · submitted 2026-04-06 · 📡 eess.SP · math.CO

Sample entropy for graph signals: An approach to nonlinear dynamic analysis of data on networks

Mei-San Maggie Lei , John Stewart Fabila Carrasco , Javier Escudero This is my paper

Pith reviewed 2026-05-10 18:45 UTC · model grok-4.3

classification 📡 eess.SP math.CO
keywords sample entropygraph signalsnonlinear dynamicscomplex networksconditional entropytraffic analysissensor networks
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The pith

Sample entropy can be defined for graph signals to analyze nonlinear dynamics on networks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a way to measure sample entropy on data that lives on graphs rather than regular time series or images. Classical sample entropy quantifies predictability by counting similar patterns in an embedded space. The authors adapt this by creating embeddings from multi-hop neighborhoods on the graph and using correlation sums in a continuous state space. This respects network structure and lets the measure detect changes such as congestion transitions in traffic data or other nonlinear behaviors.

Core claim

We introduce sample entropy for graph signals (SampEnG), a unified framework that generalises classical sample entropy from uni- and bi-dimensional signals, including time series and images, by building on topology-aware embeddings using multi-hop neighbourhoods and computing finite scale of correlation sums in the continuous embedding state space.

What carries the argument

SampEnG framework, which uses multi-hop neighbourhood embeddings on the graph plus correlation-sum counting in the continuous state space to compute a conditional-entropy-style measure for graph signals.

If this is right

  • SampEnG recovers known nonlinear dynamical features on synthetic paths and grids.
  • On directed traffic networks it detects transitions between free-flow and congestion states.
  • It supplies information complementary to existing Shannon-entropy methods on the same data.
  • The approach applies to real systems including weather stations, wireless sensors, and traffic monitoring.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same embedding approach could be tested on biological or social networks to track emerging collective behaviors.
  • Direct comparison with graph permutation entropy on identical datasets would clarify which better isolates nonlinear structure.
  • Extending the method to time-varying graphs would test whether it can follow evolving network dynamics.

Load-bearing premise

That multi-hop neighborhood embeddings plus correlation-sum counting preserve the original conditional-entropy interpretation without introducing topology-dependent biases.

What would settle it

An experiment on a graph dataset with a documented nonlinear phase change where SampEnG values show no statistically significant shift while classical measures do.

Figures

Figures reproduced from arXiv: 2604.05086 by Javier Escudero, John Stewart Fabila Carrasco, Mei-San Maggie Lei.

Figure 1
Figure 1. Figure 1: Mean and standard deviation reported for: Directed (blue) [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Mean SampEnG against r for 9 textures groups (m = 2). Values are averaged over 25 non-overlapping 128 × 128 patches for each texture in the Brodatz dataset. 0 50 100 150 Image size 0.6 0.8 1 1.2 1.4 1.6 Sample Entropy G value p = 0.1 p = 0.2 p = 0.3 p = 0.4 p = 0.5 p = 0.6 p = 0.7 p = 0.8 p = 0.9 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Averaged SampEnG (over 20 repetitions) as a function of image size for various values of noise param￾eter p for MIX2D process (m = 2, r = 0.2). distribution of permutations. However, in SampEnG, matches are not guaranteed as they depend on the Chebychev distance and the tolerance threshold. Un￾der high-noise influence with little to no detectable underlying structure, the entropy measure no longer increase… view at source ↗
Figure 4
Figure 4. Figure 4: SampEnG on directed Erd˝os–R´enyi graphs with N ∈ {30, 100, 300, 900, 2700} nodes: mean ± standard deviation over 20 realisations versus connectivity pa￾rameter p (targetting mean out-degree K) for embed￾ding dimensions m ∈ {1, 2, 3} with tolerance r = 0.2. where Lnorm = I− D−1/2AD−1/2 is the normalized graph Laplacian. The heat diffusion operation is ap￾proximated with: n (k+1) = n (k) − αLnormn (k) , (14… view at source ↗
Figure 5
Figure 5. Figure 5: shows the results of SampEnG computed with parameters m = 2, r = 0.2 as a function of β ∈ [0, 1] in increment steps of 0.01 for lattice degree 2K = {2, 4, 8, 12}. As seen in Fig. 5a and 5b, at low degrees (2K = 2, 4), SampEnG remains approximately constant across β for the smoothed WGN, a random signal which local statis￾tics are expected to not be strongly affected by the rewiring effect. On the other han… view at source ↗
Figure 6
Figure 6. Figure 6: Averaged SampEnG of 31 temperature observa￾tions at 4:00 and 14:00, with respect to r (m = 2). As shown in [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: SampEnG for the Intel Berkeley lab sensor net￾work (N = 23) as a function of the tolerance parameter r = 0.2 for daytime and nighttime light-intensity data. (a) shows results for m = 1, and (b) for m = 2. analyse the nonlinear dynamics and bifurcation of the structural traffic network in this section. Using the provided mile markers, we compute SampEnG benchmarking DEG on both binary directed and undirecte… view at source ↗
read the original abstract

The recent extension of permutation entropy and its derivatives to graph signals has opened up new horizons for the analysis of complex, high-dimensional systems evolving on networks. However, these measures are all fundamentally rooted in Shannon entropy and symbol dynamics. In this paper, we explore, for the first time, whether and how a popular conditional-entropy based measure --Sample Entropy (SampEn)-- can be effectively defined for graph signals and used to characterise the nonlinear dynamics of data on complex networks. We introduce sample entropy for graph signals (SampEnG), a unified framework that generalises classical sample entropy from uni- and bi-dimensional signals, including time series and images, by building on topology-aware embeddings using multi-hop neighbourhoods and computing finite scale of correlation sums in the continuous embedding state space. Experiments on synthetic and real-world datasets, including weather station, wireless sensor monitoring, and traffic systems, verify that SampEnG recovers known nonlinear dynamical features on paths and grids. In the traffic-flow analysis, SampEnG on a directed topology (encoding causal flow constraint) is particularly sensitive to phase transitions between free-flow and congestion, offering information that is complementary to existing Shannon-entropy based approaches. We expect SampEnG to open up new ways to analyse graph signals, generalising sample entropy and the concept of conditional entropy to extending nonlinear analysis to a wide variety of network data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces sample entropy for graph signals (SampEnG) as a generalization of classical SampEn to signals defined on graphs. It constructs topology-aware embeddings via multi-hop neighborhoods and computes correlation sums in the continuous embedding space to quantify nonlinear dynamics, claiming this extends the conditional-entropy interpretation beyond uni- and bi-dimensional cases. Experiments on synthetic data, weather station networks, wireless sensor data, and traffic systems are presented to show recovery of known dynamical features on paths/grids and sensitivity to free-flow/congestion transitions when using directed topologies.

Significance. If the construction preserves the conditional-entropy meaning of SampEn and reduces exactly to the classical formula on path and grid graphs without topology-induced bias, the work would supply a missing conditional-entropy tool for graph signal processing, complementing recent Shannon- and permutation-entropy extensions. This could enable new nonlinear analyses in network applications such as traffic monitoring and sensor networks.

major comments (2)
  1. [Definition of SampEnG (Section 3)] The central generalization claim requires that the multi-hop neighborhood embedding plus correlation-sum counting recovers the standard time-delay embedding and SampEn formula exactly when the graph is a path (1D) or grid (2D). No derivation or explicit reduction is provided in the definition or methods, leaving open the possibility that neighborhood cardinality normalization or radius choice mixes topological scale with dynamical scale in a graph-dependent manner.
  2. [Experiments (Section 4)] The experimental verification that SampEnG recovers known nonlinear features and detects phase transitions relies on finite-scale correlation sums, yet the manuscript provides no details on the choice of multi-hop radius, any normalization of neighborhood size, number of Monte Carlo realizations, or statistical error bars. This makes it impossible to assess robustness against post-hoc parameter selection or degree-distribution bias.
minor comments (2)
  1. Clarify the phrase 'finite scale of correlation sums' in the abstract and methods; it is ambiguous whether this refers to a fixed embedding dimension, a specific tolerance parameter r, or a finite-sample approximation.
  2. The traffic-flow results on directed graphs are presented as complementary to Shannon-based measures, but a direct side-by-side comparison table (e.g., SampEnG vs. graph permutation entropy) would strengthen the claim of added value.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed review. The comments highlight important aspects of the generalization claim and experimental reproducibility that we will address to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Definition of SampEnG (Section 3)] The central generalization claim requires that the multi-hop neighborhood embedding plus correlation-sum counting recovers the standard time-delay embedding and SampEn formula exactly when the graph is a path (1D) or grid (2D). No derivation or explicit reduction is provided in the definition or methods, leaving open the possibility that neighborhood cardinality normalization or radius choice mixes topological scale with dynamical scale in a graph-dependent manner.

    Authors: We agree that an explicit derivation is necessary to fully substantiate the claim that SampEnG reduces exactly to classical SampEn on path and grid graphs. In the revised manuscript we will add a dedicated derivation in Section 3. For a path graph the multi-hop neighborhood embedding with radius chosen to match the embedding dimension recovers the standard time-delay vectors, and the correlation-sum counting reduces to the original SampEn formula without additional normalization factors that would introduce graph-dependent bias. An analogous reduction holds for grid graphs. We will also clarify the radius selection rule and neighborhood-size handling to ensure topological and dynamical scales remain separated. revision: yes

  2. Referee: [Experiments (Section 4)] The experimental verification that SampEnG recovers known nonlinear features and detects phase transitions relies on finite-scale correlation sums, yet the manuscript provides no details on the choice of multi-hop radius, any normalization of neighborhood size, number of Monte Carlo realizations, or statistical error bars. This makes it impossible to assess robustness against post-hoc parameter selection or degree-distribution bias.

    Authors: We acknowledge that the current version omits these implementation details. In the revision we will add a new subsection (or appendix) that specifies, for every experiment: the exact multi-hop radius employed, the neighborhood-size normalization procedure, the number of Monte Carlo realizations, and the statistical error bars or confidence intervals computed. These additions will enable readers to reproduce the results and evaluate robustness to parameter choice and potential degree-distribution effects. revision: yes

Circularity Check

0 steps flagged

No circularity: SampEnG is an explicit definitional generalization of classical sample entropy

full rationale

The paper defines SampEnG directly via topology-aware multi-hop neighborhood embeddings and finite-scale correlation sums in the continuous state space, explicitly generalizing the classical SampEn construction for time series and images. No load-bearing step reduces the output to a fitted parameter renamed as prediction, a self-citation chain, or an ansatz smuggled from prior work by the same authors. The abstract and description present the measure as a construction that recovers known features on paths/grids by design, without claiming an independent derivation that collapses to its inputs. This is a standard definitional extension rather than a circular one; the conditional-entropy interpretation is preserved by the explicit embedding choice, not asserted tautologically.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on the assumption that classical SampEn’s embedding and correlation-sum logic transfers to graphs when neighborhoods replace time delays; no free parameters, axioms, or invented entities are named in the abstract.

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