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arxiv: 2605.01689 · v1 · submitted 2026-05-03 · 📡 eess.SY · cs.SY

A Graph Theoretic Approach in Combination With Dynamic Mode Decomposition With Control (DMDc) to Analyze Battery Degradation

Khalid Mahmud Labib , Saad Waheed , Bakhtiar Nafis , Shabbir Ahmed This is my paper

Pith reviewed 2026-05-09 17:20 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords battery degradationDMDcgraph theorylithium-ion batteriesnetwork analysisdynamic mode decompositionaging monitoringdata-driven modeling
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The pith

Graph analysis of DMDc modes shows battery networks becoming fragmented as degradation advances.

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

The paper develops a data-driven method to monitor lithium-ion battery degradation by combining dynamic mode decomposition with control and graph theory. It transforms the DMDc mode matrix into a weighted adjacency matrix that represents the battery's internal states as an evolving network. Graph measures of connectivity and modularity are then tracked across degradation stages. The analysis finds that healthy batteries exhibit highly connected and coherent networks while degraded batteries show weaker, more fragmented structures with greater heterogeneity. A sympathetic reader would care because this supplies interpretable system-level views of aging directly from operational data, sidestepping the fixed-component limits of conventional equivalent circuit models.

Core claim

By converting the DMDc mode matrix into a weighted adjacency matrix, battery dynamics are represented as an evolving network. Graph-theoretic measures applied to this network demonstrate a transition from a highly connected and coherent structure in the healthy state to a progressively weaker and more fragmented structure as degradation advances, accompanied by increasing heterogeneity. This representation captures evolving internal mechanisms from operational data alone.

What carries the argument

The weighted adjacency matrix obtained by transforming the DMDc mode matrix, which encodes interactions among system states so that connectivity and modularity metrics can quantify structural evolution.

If this is right

  • Connectivity metrics decrease as degradation advances through successive stages.
  • The network structure becomes progressively more fragmented and heterogeneous.
  • Modularity acts as a proxy that quantifies the loss of coherent dynamic interactions.
  • The entire procedure operates from operational voltage and current data without explicit physical models.

Where Pith is reading between the lines

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

  • The rate at which graph connectivity declines could be tested as an early predictor of remaining useful life.
  • The same DMDc-to-graph transformation might be applied to other time-varying electrochemical systems such as fuel cells.
  • Correlation of the observed fragmentation with physical measurements like impedance spectra could link specific graph changes to particular degradation modes.

Load-bearing premise

The assumption that connectivity and modularity metrics computed from the DMDc-derived adjacency matrix reflect real electrochemical degradation mechanisms rather than artifacts of the mode decomposition and matrix construction steps.

What would settle it

Apply the same DMDc-to-graph pipeline to voltage-current time series from additional batteries with independently measured capacity loss and find no consistent drop in graph connectivity or rise in fragmentation as capacity declines.

Figures

Figures reproduced from arXiv: 2605.01689 by Bakhtiar Nafis, Khalid Mahmud Labib, Saad Waheed, Shabbir Ahmed.

Figure 1
Figure 1. Figure 1: 4 view at source ↗
Figure 1
Figure 1. Figure 1: Schematic representation of the DMDc enhanced gra view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of network topology in (a) healthy and ( view at source ↗
Figure 3
Figure 3. Figure 3: Spatial distribution of DMD mode magnitude for (a) view at source ↗
Figure 4
Figure 4. Figure 4: Spatial distribution of DMD mode phase for (a) the h view at source ↗
Figure 5
Figure 5. Figure 5: Evolution of connectivity between nodes over differ view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of Modularity over different degradatio view at source ↗
read the original abstract

Accurate monitoring of lithium-ion battery (LIB) degradation is essential, yet it remains challenging due to the complex, nonlinear, and time-varying nature of electrochemical aging processes. Conventional equivalent circuit models (ECMs) provide simplified representations of battery behavior using fixed electrical components, but they cannot capture evolving internal degradation mechanisms and structural changes over time. In this study, a data-driven framework is developed by integrating dynamic mode decomposition with control (DMDc) with graph-theoretic analysis to characterize battery degradation from operational data alone. The mode matrix ($\mathbf{\phi}$) obtained from DMDc is transformed into a weighted adjacency matrix, enabling the representation of battery dynamics as an evolving network of interacting states. Graph-based measures, including connectivity and a modularity (proxy), are then used to quantify structural changes in the system across degradation stages. The results show a clear transition from a highly connected and coherent network in the healthy state to a progressively weaker and more fragmented structure as degradation advances, accompanied by increasing heterogeneity. This work demonstrates that graph-theoretic representations can effectively capture the evolving dynamics of battery degradation and provide interpretable insights into system-level aging behavior.

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

3 major / 2 minor

Summary. The manuscript develops a data-driven framework that applies dynamic mode decomposition with control (DMDc) to operational battery data, transforms the resulting mode matrix φ into a weighted adjacency matrix to represent system states as an evolving network, and computes graph-theoretic measures (connectivity and a modularity proxy) to characterize structural changes across degradation stages in lithium-ion batteries. It reports a transition from highly connected, coherent networks in the healthy state to weaker, more fragmented structures with increasing heterogeneity as degradation advances.

Significance. If the reported graph metrics can be shown to track underlying electrochemical mechanisms rather than DMDc approximation artifacts, the approach could offer an interpretable, model-free complement to equivalent-circuit models for battery health monitoring. The integration of DMDc dynamics with network analysis is a novel direction, but the current lack of quantitative validation, baseline comparisons, and robustness checks limits its assessed significance.

major comments (3)
  1. [Abstract] Abstract: the central claim of a 'clear transition' from highly connected to fragmented networks is asserted without any quantitative results, error bars, statistical tests, baseline comparisons, or details on adjacency-matrix construction and modularity computation, leaving the claim unsupported by visible evidence.
  2. [Method] Method section: no derivation or explicit equation is supplied for the transformation rule that converts the DMDc mode matrix φ into the weighted adjacency matrix; without this, it is impossible to determine whether connectivity and modularity changes reflect electrochemical degradation or are induced by the linear DMDc approximation on progressively noisier trajectories.
  3. [Results] Results section: the manuscript provides no comparison of the graph metrics against independent physical degradation indicators (e.g., capacity fade, SEI growth proxies, or impedance spectra) and no robustness tests to DMDc hyperparameters or measurement-noise levels that increase with degradation, which are required to substantiate that the metrics track real aging mechanisms.
minor comments (2)
  1. [Method] Notation for the mode matrix is introduced as bold phi without an explicit definition or reference to the DMDc literature equation from which it is obtained.
  2. [Method] The term 'modularity (proxy)' is used without specifying which modularity algorithm or resolution parameter is employed.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the constructive comments. We address each major comment below and commit to revisions that enhance the clarity, methodological transparency, and validation of our findings.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim of a 'clear transition' from highly connected to fragmented networks is asserted without any quantitative results, error bars, statistical tests, baseline comparisons, or details on adjacency-matrix construction and modularity computation, leaving the claim unsupported by visible evidence.

    Authors: We agree that the abstract should provide more quantitative support for the central claim to make it self-contained. In the revision, we will incorporate specific quantitative results from our analysis, such as average connectivity values and modularity indices at healthy and degraded states, along with indications of variability. Details on the adjacency matrix construction and modularity computation will also be briefly noted. The full supporting evidence, including figures, remains in the Results section. revision: yes

  2. Referee: [Method] Method section: no derivation or explicit equation is supplied for the transformation rule that converts the DMDc mode matrix φ into the weighted adjacency matrix; without this, it is impossible to determine whether connectivity and modularity changes reflect electrochemical degradation or are induced by the linear DMDc approximation on progressively noisier trajectories.

    Authors: The referee is correct that an explicit equation and derivation for the transformation from the DMDc mode matrix φ to the weighted adjacency matrix were omitted. We will add this in the revised Method section, providing the precise rule used to define edge weights based on mode similarities and a brief justification of how this representation captures system dynamics independent of the DMDc linear approximation. revision: yes

  3. Referee: [Results] Results section: the manuscript provides no comparison of the graph metrics against independent physical degradation indicators (e.g., capacity fade, SEI growth proxies, or impedance spectra) and no robustness tests to DMDc hyperparameters or measurement-noise levels that increase with degradation, which are required to substantiate that the metrics track real aging mechanisms.

    Authors: We recognize the value of such validations. We will revise the Results section to include comparisons of the graph metrics with capacity fade, which is available in our dataset, and perform correlation analyses. Robustness tests to DMDc hyperparameters and added noise will also be conducted and reported. However, SEI growth proxies and impedance spectra are not available in the operational data used, limiting direct comparison there. revision: partial

standing simulated objections not resolved
  • Direct comparison to SEI growth proxies and impedance spectra, as these measurements were not collected in the operational dataset used for this study.

Circularity Check

0 steps flagged

No circularity in the DMDc-to-graph transformation chain

full rationale

The derivation applies standard DMDc to operational voltage/current trajectories to extract the mode matrix φ, converts φ to a weighted adjacency matrix via an explicit (if heuristic) mapping, and computes conventional graph metrics (connectivity, modularity) on the resulting network. None of these steps is defined in terms of the target degradation indicators, fitted to the same quantities being reported, or justified solely by self-citation; the observed transition from coherent to fragmented graphs is an empirical outcome of the pipeline rather than a quantity forced by construction or by prior author results. The framework therefore remains self-contained against external data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger reflects assumptions stated or implied in the summary. No explicit free parameters or invented entities are described.

axioms (1)
  • domain assumption DMDc modes extracted from operational battery data capture the relevant dynamics of degradation
    Required for the mode-to-graph transformation to be meaningful.

pith-pipeline@v0.9.0 · 5515 in / 1129 out tokens · 32328 ms · 2026-05-09T17:20:41.136434+00:00 · methodology

discussion (0)

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