arXiv:2605.03574v1 [physics.soc-ph] 5 May 2026
Long-Range Correlation in Code Commit Dynamics as a Novel Indicator of Software Product Stability: A Detrended Fluctuation Analysis Study Fractal structure as a code-inspection-free process quality metric
Goran Mitevski January, 2023 Abstract This work proposes the fractal scaling exponent α, estimated via Detrended Fluctuation Analysis (DFA) on the unaggregated time series of lines of code added per commit event in a software repository, as a novel process-level indicator of software product stability. The proposal rests on the hypothesis that stable software products arise from development processes characterised by long-range temporal correlations in commit behaviour: each code addition is shaped not only by the immediately preceding commits but by patterns extending weeks or months into the past and anticipating work to be done in the future. This hypothesis is tested on two non-overlapping 712-day time series of lines of code added per commit event, drawn from a closed-source software organisation and labeled as stable and unstable by the lead engineer on the basis of crashanalytics data. Applied to these series, DFA yields α = 0.70 (nmin = 16) for the stable period and α = 0.57 for the unstable period, with all estimates substantially above the shuffled-surrogate baseline (α ≈ 0.50 ± 0.01). Results are robust to three parameterisations (nmin ∈ {4, 16, 48}) and validated against 1,000 surrogate time series per condition. Remarkably, the unstable period generated 3.2 times more commit events than the stable period, yet exhibited lower long-range memory, demonstrating that commit volume alone does not predict stability, and that the temporal organisation of development activity is the key variable. This result can be situated in the broader literature on fractality in human creative production, discuss methodological limitations, and outline a research programme for deploying α as a continuous code-health indicator in version-control pipelines. Keywords: detrended fluctuation analysis, fractal scaling, long-range correlation, software stability, commit dynamics, code quality, complexity science, surrogate analysis, distributed software development
1. I NTRODUCTION
Software quality is expensive to measure. The dominant methods of static code analysis, testing, and code review require access to source code or the execution of test suites. Alternatively, production metrics like crash analytics require the prior release of potentially defective software. A metric derivable from the temporal pattern of development activity alone (from the stream of timestamped commits to a repository) would have considerable practical value: it would require no code review, no test execution, no natural language processing, and could be monitored continuously from any version-control log. This work proposes such a metric: the fractal scaling exponent α on the unaggregated time series of lines of code added per commit event in a software repository, estimated via Detrended Fluctuation Analysis (DFA). The core hypothesis is that the temporal organisation of commit behaviour encodes information about the quality of the software being produced. Specifically, that stable, high-quality software is the product of development processes characterised by long-range temporal correlations: each commit is shaped not only by the immediately preceding commits but by patterns extending weeks or months into the past, and in some anticipatory sense by work expected in the future. This interdependence across time scales is precisely what DFA’s scaling exponent α quantifies. The argument has an intuitive foundation. Software architecture is inherently hierarchical and interdependent: a change to a core module today should be consistent with design decisions made three months ago and should anticipate module extensions planned for next month. A developer with deep involvement in a codebase will make commits that reflect this multi-scale awareness, commits that are, in a precise mathematical sense, correlated with the distant past and future of the time series. A developer without such involvement, or a team under deadline pressure patching problems reactively, will make commits that are more locally determined, producing a time series with weaker long-range structure. DFA, introduced by Peng et al. (1994) for the analysis of DNA nucleotide sequences, is designed precisely to detect such long-range structure in the presence of non-stationarities that would confound simpler autocorrelation measures. Its output, the scaling exponent α, is a single number: α ≈ 0.5 indicates white noise (no memory); α ∈ (0.5, 1.0] indicates positive longrange correlations of increasing strength; α > 1.0 indicates non-stationary behaviour. This work proposes that the region α ∈ (0.5, 1.0]—and specifically the distance of α above 0.5—is a continuous indicator of commit-process coherence and, by extension, of likely software product stability.
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1.1 Scope and Contribution This paper makes four contributions. First, it proposes α from DFA on commit time series as a novel, code-inspection-free indicator of software process quality. Second, it provides an empirical test of this proposal against externally validated stability labels from a real organisation. Third, it addresses the methodological challenges of applying DFA to commit data, including non-stationarity, length effects, and window-size sensitivity, with robust parameterisation and surrogate validation. Fourth, it situates the finding within the broader literature on fractality in human creative production and opens a research programme connecting complexity science with software engineering practice.
2. BACKGROUND AND R ELATED W ORK
2.1 Fractality and Long-Range Correlation in Human Creative Production The concept of statistical self-similarity (fractal structure) has been applied to human language, cognition, and action across several decades. Mandelbrot (1983) observed that word-frequency distributions in natural language follow power laws. Gilden et al. (1995) found that simple reaction times exhibit 1/ f noise—the power spectrum associated with long-range correlations and α ≈ 1 in DFA terms. The broader claim (Van Orden et al., 2003; Riley & Turvey, 2002) is that long-range correlations in performance time series are a signature of flexible, adaptive engagement with a task: a system that is too rigid produces trivially correlated outputs, while one that is too random produces white noise. The productive middle ground is 1/ f noise. In the domain of language and narrative, several studies are directly relevant. Butner et al. (2008) analysed the fractal structure of conversational stories and found that more fractal stories were rated as higher quality by listeners—a striking parallel to the findings of this work that more fractal commit processes produce more stable software. Drożdż et al. (2016) examined long-range correlations in famous literary texts, finding that canonical works exhibit stronger multifractal structure than lesser-known texts. Bhan et al. (2006) found similar structure in Korean literary corpora. These findings collectively suggest that long-range temporal organisation is a hallmark of high-quality human creative production across modalities—and that software development, as a structured creative activity, may be no exception. To the best of the author’s knowledge, no previous study has applied DFA to commit-level code-addition time series as a software quality indicator. 2.2 Detrended Fluctuation Analysis: Origins and Applications DFA was introduced by Peng et al. (1994) to study the long-range correlation properties of DNA nucleotide sequences. Its key advantage over earlier methods (Hurst exponent, spectral analysis) is robustness to non-stationarity: embedded polynomial trends are removed locally 3
within each window, preventing spurious long-range correlation estimates that would arise from global trends. Bashan et al. (2008) provide a systematic comparison of DFA and related methods, confirming its robustness across a wide range of conditions. Goldberger et al. (2000) established the normative framework for interpreting α values in the physiological context, which this work adapts for software. DFA has been applied in physiology (cardiac interbeat intervals, gait, EEG), financial time series, environmental science, and cognitive science (reaction time series, attention dynamics). Its track record across these domains provides a methodological foundation for the present application. 2.3 Creativity, Stability, and the Software Development Process Boden (2007) defines creativity as the production of ideas or artefacts that are new, surprising, and valuable. In the software context, value is reflected in product stability and user satisfaction. Yu et al. (2020) provide an empirical foundation for using stability as a value proxy: stable software products (lower crash rates, faster response times) receive higher user ratings. This establishes the chain: long-range commit correlations → stable software → higher user value → greater product success. 2.4 The Scaling Exponent Interpretation Framework The DFA scaling exponent α admits the following interpretive framework (Goldberger et al., 2000; Bashan et al., 2008): α ≈ 0.5
Uncorrelated white noise (memoryless process)
α < 0.5
Anti-correlated (negatively persistent)
0.5 < α < 1.0
Positively long-range correlated (persistent)
α ≈ 1.0
Pink (1/ f ) noise; signature of complex adaptive systems
α > 1.0
Non-stationary; Brownian-like or unbounded
α ≈ 1.5
Fractional Brownian motion
For commit time series, this work expects α ∈ (0.5, 1.0] for any sustained development process. This work hypothesis is that the distance of α above 0.5 co-varies with product stability.
3. DATA
3.1 Source and Labeling Commit-level time series were extracted from the complete GitHub activity log of a small, globally distributed closed-source software (CSS) organisation with more than five years of continuous operation. Two non-overlapping periods of 712 days each were identified and labeled by the lead software engineer as stable and unstable. The labeling criterion was the frequency of 4
software crashes (instances in which the software became unresponsive) as reported by Google Analytics and equivalent services. This is an operationally clear criterion grounded in user-facing behaviour rather than developer judgment of code quality. 3.2 Time Series Construction The time series were constructed directly from the raw commit log: each commit event constitutes one observation, ordered chronologically by its UTC-aware timestamp. The value at each observation is the number of lines of code added in that commit. Key descriptive statistics: the stable period contained 976 commit events across 712 days (mean: 1.37 commits/day); the unstable period contained 3,129 commit events across 712 days (mean: 4.39 commits/day)— approximately 3.2 times more commit activity. This asymmetry is a central feature of the comparison. 3.3 Crossover Analysis Following the methodological framework of Peng et al. (1995), both time series were inspected for crossovers, which are pronounced changes in scaling behavior across different timescales. In DFA, a crossover in the log F(n) vs. log n plot typically indicates a transition between distinct scaling regimes, potentially signaling a shift in the underlying system dynamics (Ogata et al., 2006) or the influence of non-stationary trends that could confound the interpretation of a single scaling exponent (Hu et al., 2001). Visual and linear regression analysis of the fluctuation functions revealed that while minor local fluctuations are present (typical of empirical human-activity time series) the scaling behavior remains sufficiently consistent across the examined window ranges. The absence of systematic breaks or slope shifts suggests that the code-commit dynamics are governed by a dominant monofractal process rather than a multifractal one. Therefore, the application of a single scaling exponent α is a parsimonious and appropriate descriptor for characterizing both periods (see Appendix I for the detailed visual diagnostic).
4. M ETHODS
4.1 Detrended Fluctuation Analysis Let x(i), i = 1, . . . , N be the commit-level time series. DFA proceeds in three stages. 4.1.1 Stage 1: Integration The series is mean-centred and integrated: k y(k) = ∑ x(i) − ⟨x⟩ i=1
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(4.1)
where ⟨x⟩ is the series mean. 4.1.2 Stage 2: Local Detrending The integrated series y is divided into non-overlapping windows of length n. Within each window, a least-squares polynomial of degree d = 1 is fitted to estimate the local trend yn (k). The root-mean-square fluctuation is: v u N u1 2 F(n) = t ∑ y(k) − yn (k) N k=1
(4.2)
Local detrending suppresses non-stationarities that would otherwise inflate α estimates—the key innovation of DFA relative to earlier fluctuation methods. 4.1.3 Stage 3: Scaling Analysis Stage 2 is repeated over a range of window sizes n. If F(n) scales as a power law, F(n) ∝ nα
(4.3)
then α is estimated as the slope of the log F(n) vs. log n regression. Linearity of the log-log plot is a necessary condition for fractal scaling and was verified visually and quantitatively for both series. 4.2 Parameterisation Strategy To assess robustness, three parameterisations were tested: Parameterisation 1 (recommended): nmin = 16, nmax = N/4. For the stable series (N = 976 events), nmax = 244, yielding 13 distinct window sizes. For the unstable series (N = 3,129), nmax = 782, yielding 18 window sizes. The nmin = 16 choice corresponds to approximately two weeks of stable-period activity. Parameterisation 2: nmin = 48, nmax = N/4. A larger minimum window reduces sensitivity to short-timescale fluctuations. Parameterisation 3 (exploratory): nmin = 4, nmax = N/2. An aggressive parameterisation serving as a stress test. 4.3 Surrogate Validation To confirm that observed α values reflect genuine long-range correlations rather than statistical artefacts, 1,000 surrogate series were generated for each condition by randomly shuffling the commit-level values without replacement (Theiler et al., 1992; Moulder et al., 2018). Shuffling destroys temporal order—and thus all serial correlation—while preserving the marginal distribution. DFA was applied to each surrogate series. The resulting distribution provides an empirical
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null distribution. An observed α is considered to reflect genuine long-range memory if and only if it lies outside the 95% confidence interval of the surrogate distribution. Surrogate series consistently yield α ≈ 0.50 ± 0.01. 4.4 Evaluation Criteria Long-range memory is inferred when: (1) α > 0.5 in the original series; (2) α < 1.0; (3) α lies outside the surrogate 95% CI; and (4) the surrogate mean is ≈ 0.5. Criterion (3) is decisive. Comparison between periods is made directly on α values; a larger α in the stable period, consistent across all parameterisations, supports the hypothesis.
5. R ESULTS
5.1 Stable Period Table 1 reports α estimates for the stable period across all parameterisations. Under the recommended parameterisation (nmin = 16), α = 0.70. Under nmin = 48, α = 0.66. Under the exploratory nmin = 4, α = 0.67. All three estimates lie within the long-range memory range (0.5, 1.0) and all lie well outside the corresponding 95% CI of the surrogate distribution. The consistency across parameterisations (range: 0.66–0.70) indicates that the long-range memory in the stable period is robustly present across timescales from days to months. Values in the range 0.66–0.70 correspond to moderate persistent correlations—below the 1/ f boundary (α = 1.0) but substantially above white noise. 5.2 Unstable Period Table 1 reports α estimates for the unstable period across all parameterisations. For the unstable period, α = 0.57 (nmin = 16), α = 0.62 (nmin = 48), and α = 0.56 (nmin = 4). All three estimates exceed the surrogate upper bound, indicating long-range memory above chance. However, all three are lower than the corresponding stable-period estimates by a consistent margin of 0.04– 0.14 units. The unstable period values cluster closer to 0.5, indicating weaker (though still present) long-range memory. 5.3 Commit frequency volume vs. Long-Range Memory The most striking finding is the dissociation between commit frequency volume and α. The unstable period generated 3,129 commits versus 976 in the stable period (3.2 times more activity) yet consistently produced lower α. This directly refutes the hypothesis that more development activity produces more stable software. The temporal organisation of that activity, not its volume, is the discriminating variable.
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Table 1: DFA scaling exponents with surrogate validation across three parameterisations. Surrogate values are mean ± 95% CI across 1,000 shuffled realisations. † Non-recommended exploratory parameterisation. Parameterisation
Stable α
Stable surrogates
Unstable α
Unstable surrogates
∆α
nmin = 16, nmax = N/4 nmin = 48, nmax = N/4 nmin = 4, nmax = N/2 †
0.70 0.66 0.67
[0.49, 0.50] [0.49, 0.50] [0.50, 0.51]
0.57 0.62 0.56
[0.49, 0.50] [0.49, 0.50] [0.50, 0.50]
+0.13 +0.04 +0.11
This pattern is consistent with the ‘thrashing’ phenomenon well known in software project management: teams under deadline pressure commit frequently but without deep integration, producing many small reactive fixes that collectively degrade rather than improve software quality. The DFA exponent may be capturing exactly this difference: architecturally integrated development (high α) versus reactive, patch-driven development (low α). 5.4 Comparison to Adjacent Literature The stable period α values (0.66–0.70) are comparable to those found in high-quality literary texts (Drożdż et al., 2016), conversational stories rated as high quality (Butner et al., 2008), and physiological signals from healthy, adaptive systems (Goldberger et al., 2000). The unstable period values (0.56–0.62) are closer to values found in more repetitive or less structurally complex materials. This convergence across domains strengthens the interpretation of α as a general indicator of structured, adaptive creative production.
6. D ISCUSSION
6.1 Interpretation of the Core Finding The central finding supports the proposed framework: stable software development is characterised by commit patterns in which current decisions are shaped by a memory stretching weeks and months into the past. This coherence manifests as statistical self-similarity in the commit time series. In practical terms, the rhythm of development is not reset at arbitrary boundaries: a productive week tends to follow a productive month, and a slow day tends to follow a slow week, with these tendencies persisting across the full range of timescales examined. The fact that both periods exceed the surrogate baseline (both α > 0.5 + surrogate bound) indicates that commit dynamics are not random in either condition—some degree of multi-scale structure is present in all sustained development processes. What distinguishes stable from unstable development is the strength of this structure.
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6.2 Frequency Volume vs. Quality: A Fundamental Asymmetry The 3.2× greater commit frequency volume of the unstable period refutes the trivial hypothesis that more activity produces better software. A process characterised by reactive, patch-driven commits will produce high volume but weak long-range structure. Each commit is determined primarily by the immediately preceding problem, not by the multi-week and multi-month patterns of the development trajectory. This interpretation connects to the established concept of technical debt in software engineering: short-term expedient decisions that solve an immediate problem without considering long-term consequences. A low α may be an early indicator of technical debt accumulation, identifiable from commit timestamps alone, before its consequences become visible in crash statistics or user complaints. 6.3 Methodological Considerations Several methodological issues deserve explicit discussion. First, the choice of minimum window size in DFA is consequential: the nmin = 48 result for the unstable period (α = 0.62) is higher than other parameterisations, suggesting that at longer timescales, the unstable period shows more structure than at shorter timescales. This could indicate that while short-term commit patterns are reactive, there is some longer-term periodicity at the multi-month scale. Second, the series differ in length (in events, not days). This work adjusted minimum window sizes proportionally (nmin = 48 ≈ 3 × 16) to account for this difference, and found that the stable-period α remains higher even under this adjustment. Third, DFA as applied here assumes monofractal scaling. This assumption was verified through crossover analysis. If future data exhibit crossovers, multifractal DFA (MFDFA) would be required, yielding a spectrum of exponents rather than a single α. Fourth, the DFA analysis is based solely on lines of code added per commit; deletions were not included. To the best of the author’s knowledge, no prior study has examined the fractal structure of code-deletion dynamics, and additions alone capture the primary signal of constructive development activity required to test the core hypothesis advanced here. Fifth, the stability label was assigned by a single domain expert, grounded in objective crash analytics. Future work should use multiple independent raters and finer-grained stability measures to enable continuous rather than binary analysis. 6.4 Practical Deployment: α as a Continuous Code-Health Metric The practical appeal of α lies in its computational simplicity and its inputs. Computing α requires only a time series of commit timestamps, data that any organisation using version control generates as a by-product of normal development activity. The DFA algorithm is O(N log N) and takes seconds for time series of the lengths considered here. It requires no code
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access, no build system, no test suite, and no language-specific tooling. A practical monitoring system could compute α on a rolling 30-, 60-, or 90-day window of commit data, updating daily. A decline in α, particularly a sustained decline across multiple window sizes, would serve as an early-warning signal that the development process is losing its multi-scale coherence. For integration into CI/CD pipelines, α computation could be triggered on each push to the main branch, with the result logged alongside conventional quality metrics. 6.5 Connections to Creativity Research and Team Science This work is positioned within Batey (2012)’s taxonomic framework as a team-level, processfacet, objective-measurement approach. It measures the temporal organisation of the development process, the fractal texture of collective creative production, rather than the creativity of individual developers or the novelty of specific features. In this sense, α is a measure of creative coherence rather than creative output. A high α does not guarantee that good software will result, nor does a low α preclude good outcomes. But on average, across the timescales of sustained software development projects, this work proposes that higher α is associated with more stable outcomes. 6.6 Future Directions Multifractal DFA (MFDFA) and Wavelet Transform Modulus Maxima (WTMM) analysis would reveal whether the full spectrum of scaling exponents, not only the dominant α, varies between conditions. Cross-team complexity matching, examining whether teams on the same project have commit time series with different α values, could index coordination difficulties, analogous to complexity matching in motor coordination research (Marmelat & Delignières, 2012). An important avenue of research is the difference in long-term memory and complexity between teams that produce software solely through human effort and teams that rely heavily on generative LLM-based AI for code generation, and whether higher long-range correlation in lines of code added per commit remains a valid indicator of software stability when the commit process is driven predominantly by LLM-based generation. Combining DFA on commits with the Lyapunov exponent analysis of discussion dynamics, opens the possibility of a two-dimensional characterisation of the creative state of a software development team: the fractal coherence of the code process and the chaotic vitality of the surrounding discussion. Together, these metrics can provide a richer picture than either alone.
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7. C ONCLUSION
This work proposes the DFA scaling exponent α of the unaggregated time series of lines of code added per commit event as a novel, code-inspection-free indicator of software product stability, and provides an empirical test of this proposal using data from a real distributed software organisation. The stable software period yields α = 0.66–0.70 across parameterisations; the unstable period yields α = 0.56–0.62; both are validated against 1,000 surrogate series per condition. The stable period’s higher α persists despite having only 31% as many commit events as the unstable period, suggesting that the temporal organisation of development activity may be a more informative signal than commit frequency volume alone. These findings add software commit dynamics to the class of human creative productions whose fractal structure co-varies with quality, alongside literary texts, conversational stories, and physiological signals of adaptive system function. DFA on commit time series is computationally trivial, platform-agnostic, and producible from any version-control log. This work proposes it as a complementary metric to existing code-quality tools, deployable continuously within any development organisation, and as a foundation for a new research programme connecting complexity science with software engineering.
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DATA AVAILABILITY
The commit-level time series analysed in this study are made available as two comma-separatedvalues (CSV) files deposited on Zenodo under a Creative Commons Attribution 4.0 International (CC BY 4.0) licence (Mitevski & Efremova, 2026): • unstable_period_code_additions.csv — 3,129 commit events recorded during the 712-day unstable period (mean: 4.39 commits/day). Each row contains a UTC-aware ISO 8601 timestamp (time_point) and the number of lines of code added in that commit event (additions). • stable_period_code_additions.csv — 976 commit events recorded during the 712day stable period (mean: 1.37 commits/day). Columns are identical to those of the unstable-period file. The source repository and organisational identity of the software company remain confidential to protect commercially sensitive information; the commit-level time series provided are sufficient to reproduce all statistical analyses reported in this study. The dataset is available at https: //doi.org/10.5281/zenodo.19986248.
D ECLARATION OF GENERATIVE AI AND AI- ASSISTED TECHNOLOGIES
This work was created by the author in January 2023. In May 2026, LLM based generative AI was used to improve the readability and language of the original work. The author takes full responsibility for the content of this publication.
ACKNOWLEDGEMENT
First and foremost, I would like to thank my wife, Marija Efremova, whose incredible talent and work were the inspiration and driving force behind this work. I would also like to acknowledge Dr. Travis Wiltshire for his exceptional support, knowledge, and guidance in Complex Systems, as well as for mentoring my thesis work, of which this work was a part. I would like to acknowledge my mother Ana Mitevska, a talented mathematician and a Germanist, Velin Mitevski, whose work in expert and decision support systems in the 1970s and 1990s sparked my interest in this problem space. Last but not least, I would like to acknowledge Mitko Efremov, whose work and involvement in centralised automation design, and engineering sustainable mass water supply systems were an inspiration behind some of the insights in this work.
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A PPENDICES
I. C ROSSOVER C HECK
This appendix details the diagnostic procedure used to verify the absence of significant crossovers in the DFA log-log plots for the CSS ‘stable’ and ‘unstable’ commit-level time series. The Significance of Crossovers In the context of DFA, a crossover is defined as a change in the scaling exponent α at a specific characteristic scale n. According to Peng et al. (1995), such phenomena suggest that the system’s underlying dynamics change as a function of the observation window. Furthermore, Hu et al. (2001) demonstrated that crossovers can emerge as artifacts of non-stationarity (e.g., linear or power-law trends) in the data. While empirical data rarely yields a perfectly linear scaling law, identifying a sufficiently consistent linear region is a prerequisite for the monofractal interpretation. If pronounced crossovers were present, a single α would represent an oversimplified average of multiple scaling regimes, necessitating multifractal analysis. Detection Procedure Crossover detection involves evaluating the linearity of the fluctuation function F(n) on a log-log scale. A robust scaling law is indicated by data points that closely follow a single straight line across the window sizes n. The composite results for our datasets are shown in Figure 1, while a representative example of a definitive crossover (for comparison) is shown in Figure 2.
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Figure 1: DFA log-log plots for stable and unstable periods. The strong overall linearity suggests that the monofractal interpretation is sufficient for comparative analysis.
Results and Validation Inspection of the DFA plots (Figure 1) reveals no systematic shifts in the scaling exponent. The scaling behavior is sufficiently consistent for monofractal interpretation, unlike systems where a delayed or blunted crossover signals a fundamental shift in regulatory dynamics (Varela et al., 2016).
Figure 2: Illustrative example of a definitive crossover, identifying the intersection of two distinct scaling limbs. This diagnostic check confirms that our code-commit data lacks the structural "kink" characteristic of multi-regime systems (adapted from Figure 2 in Varela et al., 2016).
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