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REVIEW 3 major objections 4 minor 90 references

Transaction counts on Bitget for BTC and ETH decouple from volume and returns after May 21, 2025, revealing a noise-like regime in trading activity.

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A complexity-measure analysis of 1-minute crypto trade data reveals a Bitget-specific post-May-2025 surge in small, noise-like BTC and ETH transactions that decouple from volume and returns.

T0 review reviewed 2026-08-02 challenge →

load-bearing objection A solid, honestly-hedged case study of a Bitget-specific transaction-count anomaly; the finding is real but the single data source keeps it from being conclusive. the 3 major comments →

arxiv 2607.13916 v1 pith:BRHDNOPS submitted 2026-07-15 q-fin.TR cs.CEecon.EMphysics.data-anstat.AP

Detecting unusual trading patterns on cryptocurrency exchanges by means of complexity measures

Jakub Zwydak , Marcin W\k{a}torek , Jaros{\l}aw Kwapie\'n , Stanis{\l}aw Dro\.zd\.z This is my paper
classification q-fin.TR cs.CEecon.EMphysics.data-anstat.AP
keywords wash tradingcryptocurrency exchangescomplexity measuresmultifractal analysisdetrended cross-correlationapproximate entropytransaction countsBitget anomaly
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

This paper tries to establish that complexity-based statistics computed from high-frequency trade-level data can detect exchange-specific trading anomalies that standard price-based diagnostics miss. Its central finding is a pronounced anomaly on Bitget for BTC and ETH after mid-May 2025: the number of transactions per minute rises sharply while traded volume and return fluctuations do not rise proportionally. The paper shows that this post-May regime is statistically distinct from Bitget's earlier behavior and from other exchanges, characterized by many tiny trades, a nearly Gaussian transaction-count distribution, weak autocorrelations, reduced multifractal organization, higher short-pattern irregularity, and weak cross-correlations involving transaction counts. The authors interpret these features as consistent with a noise-like component that may indicate artificially increased transaction counts, while explicitly stating this is not direct proof of wash trading. If correct, the paper demonstrates that transaction-count complexity measures can serve as diagnostic tools for market quality and liquidity reliability on centralized exchanges.

Core claim

The paper's core claim is that, after May 21, 2025, the transaction-count process for BTC and ETH on Bitget becomes statistically different from its own earlier behavior and from the corresponding processes on Binance, Kraken, and KuCoin, while price dynamics remain largely synchronized across exchanges. The elevated transaction counts are driven by low-volume trades that do not translate into proportional volume or price impact. This is supported by a within-minute decomposition: active seconds per minute rise from about 12 to 55.5 for BTC, and the average records per active second rise from 1.9 to 3.3. Removing the smallest trades makes no-trade intervals visible again. The anomaly is not

What carries the argument

The framework combines multifractal detrended fluctuation analysis (MFDFA), multifractal detrended cross-correlation analysis (MFCCA), the q-dependent detrended cross-correlation coefficient ρ(q,s), approximate entropy (ApEn) and sample entropy (SampEn) computed in rolling windows, autocorrelation functions, tail distributions, and a formal change-point detection procedure. The central object carrying the argument is the detrended cross-correlation coefficient ρ(q=2,s) between pairs of |R|, V, and N, together with the entropy measures on N. These tools identify a time-localized structural break around May 21, 2025, and quantify how the transaction-count process loses its usual coupling to vo

Load-bearing premise

The analysis defines 'usual' trading behavior empirically from Binance, Kraken, KuCoin, and Bitget's pre-May data; if Bitget changed its reporting, fee structure, or incentives around May 21, the detected anomaly could be a benign platform change rather than artificially generated transactions.

What would settle it

If public records show that Bitget altered its timestamp precision, minimum trade size, fee schedule, or market-making rebates on or around May 21, 2025, or if account-level data reveal that the low-volume trades originate from a small set of known market makers, the anomaly interpretation would be weakened. Conversely, if the same framework flags no similar pattern on other exchanges over the same window, that would support an exchange-specific artificial-activity explanation.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Transaction-count series are a highly informative diagnostic for exchange-specific anomalies that remain hidden in price-based measures.
  • A persistent departure from empirically defined regular trading patterns -- narrower distributions, weakened autocorrelations, reduced multifractal organization, and weaker cross-correlations -- can flag unusual trading activity even without account-level data.
  • The Bitget anomaly is specific to BTC and ETH, not exchange-wide, since XRP on the same exchange does not exhibit the same regime shift.
  • The post-break regime is consistent with a noise-like component in trading activity, likely driven by extremely small trades that add records without adding proportionally to volume or price impact.
  • Complexity-based indicators can complement standard liquidity and price diagnostics for assessing the reliability of reported market activity on centralized cryptocurrency exchanges.

Where Pith is reading between the lines

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

  • If the anomaly stems from a benign platform change -- such as altered timestamp granularity, fee tiers, or market-making incentives -- then the same framework applied to order-book depth, trade direction, or account-level data could separate natural fragmentation from artificial inflation.
  • The simultaneity of the BTC and ETH regime changes, combined with weak cross-asset transaction-count correlations, suggests independent or separately coordinated generation processes rather than a single synchronized market-wide event; testing the timing of order submissions could clarify this.
  • The rolling-window entropy and cross-correlation approach could be applied prospectively to other exchanges and assets as a near-real-time surveillance tool for wash-trading-like patterns.
  • A testable extension is to compare Bitget's post-May behavior with specific exchange announcements or fee changes: if the regime shift aligns with a known policy change, the benign-mechanism explanation becomes more likely than artificial activity.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes a diagnostic framework for detecting unusual trading patterns on centralized cryptocurrency exchanges using complexity and statistical-structure measures. It analyzes 1-minute log-returns, trading volume, and transaction counts for BTC, ETH, and XRP on Binance, Bitget, Kraken, and KuCoin over April 1–June 30, 2025, using CCDFs, autocorrelations, MFDFA, MFCCA, detrended cross-correlation coefficients, ApEn/SampEn, and change-point detection. The central claim is that after May 21, 2025, the BTC and ETH transaction-count processes on Bitget changed structurally: transaction counts increased sharply, became near-Gaussian and weakly autocorrelated, lost multifractal organization, and became weakly cross-correlated with volume and returns, while XRP on Bitget and all other exchanges remained unaffected. The authors interpret this as a noise-like component possibly consistent with artificial activity, but explicitly caution that direct proof of wash trading is absent.

Significance. If the empirical finding holds, the paper demonstrates that complexity-based measures can reveal an exchange- and asset-specific anomaly that is invisible in standard price-based indicators. The study's main strengths are the converging evidence across complementary measures, the XRP control, the within-minute active-second decomposition, and the robustness checks for ApEn/SampEn with varying embedding dimensions and tolerances. The principal weakness is the unresolved ambiguity between a genuine trading anomaly and a data-reporting artifact, which the authors acknowledge in Section 9. The contribution is a useful empirical case study and diagnostic demonstration, conditional on data fidelity.

major comments (3)
  1. [Sec. 3 and Sec. 9] The load-bearing assumption is that the post-May 21 Bitget BTC/ETH transaction-record series measures the same economic construct as the pre-break series. Section 9 explicitly concedes that the analysis 'could not unambiguously distinguish between these possible mechanisms.' The XRP control and the active-second decomposition (Sec. 8, Eqs. 18-20) mitigate exchange-wide or timestamp-granularity artifacts, but they do not rule out an asset-specific reporting change (e.g., a new fee tier, market-making incentive, or feed duplication for BTC/ETH only). Because the title and abstract claim detection of 'unusual trading patterns,' the manuscript must either validate the anomaly with an independent tick-level data source or reframe the claim as an anomaly in the transaction-record series and explain how the framework separates feed artifacts from trading behavior.
  2. [Sec. 8] The two-period comparison is built on the change point returned by findchangepts applied to the full sample (maximum one change point, minimum distance 1440). All Bitget1-versus-Bitget2 contrasts in Figs. 29-35 are therefore in-sample and may overstate the regime difference. The rolling-window analyses provide supporting evidence, but the formal pre/post comparison would be substantially stronger with an out-of-sample validation or a post-selection inference procedure (e.g., a permutation or block-bootstrap test that accounts for the change-point search). Please report the statistical uncertainty of the change point and the size of the between-period differences.
  3. [Sec. 8, Eq. (20)] The claim that 'even under the conservative aggregation ... the post-break activity remains more than four times higher' assumes that the only possible reporting artifact is the duplication of records within the same one-second timestamp. If a reporting change introduced records with distinct second timestamps (e.g., synthetic trades or additional record types), the active-second decomposition is not conservative. The manuscript should state this assumption explicitly and, if feasible, validate the timestamp behavior against order-book or other independent data.
minor comments (4)
  1. [Throughout] The exchange name is spelled 'KuCoin' in the text but 'Kucoin' in several figure labels and captions (e.g., Figs. 4, 7, 10). Please standardize.
  2. [Author affiliations] The correspondence line lists two email addresses and the asterisk is placed next to Stanisław Drozdz; please clarify who is the corresponding author.
  3. [Sec. 2.1, Eq. (7)] The denominator of ρ(q,s) writes FqXX(s)FqYY(s) without the 1/q exponent; this is consistent with the cited literature but may confuse readers. A parenthetical reminder that the denominator is evaluated at the same q would help.
  4. [Data Availability] The Data Availability Statement lists only public APIs; providing analysis scripts or a reproducibility repository would strengthen the paper's usefulness, especially because the anomaly claim rests on a specific data-processing pipeline.

Circularity Check

0 steps flagged

No significant circularity: the paper is an empirical comparative analysis whose anomaly claim is a direct statistical comparison, not a fitted parameter renamed as a prediction.

full rationale

The paper does not derive a prediction from a fitted parameter. The central claim—that BTC and ETH transaction counts on Bitget after May 21, 2025, are statistically unusual—rests on direct comparisons between raw 1-min series of N, V, and |R| across exchanges and across pre/post subsamples. The benchmark for 'usual' behavior is explicitly operationalized as the empirical regularities observed on Binance, Kraken, KuCoin, and on Bitget before the break; the anomaly is then detected as a departure from those regularities. This is a control-comparison design, not a self-referential derivation. The change point is estimated from the log(1+N) series, and the sample is split at that point, but the downstream characterizations (trade-size histograms, V–N scatter separation, autocorrelations, entropy, detrended cross-correlations) are measured independently and are not logically entailed by the mere existence of a mean shift in N. The paper also explicitly concedes that it cannot distinguish between artifactual reporting changes and genuine unusual trading, which further limits any claim that the mechanism is derived from the framework. The many self-citations are to established or previously published methodology and stylized-fact references; none of them is used as a uniqueness theorem or as the sole justification for the empirical anomaly. Thus no circular step can be quoted or exhibited.

Axiom & Free-Parameter Ledger

6 free parameters · 5 axioms · 0 invented entities

No new entities are postulated. The analysis uses established statistical methods and applies them to public data; the central claim depends on chosen method parameters and the empirical definition of 'usual' behavior, but not on forcing parameters or invented mechanisms.

free parameters (6)
  • ApEn/SampEn embedding dimension m and tolerance r = m=2, r=0.2σ
    Chosen by hand for the main calculations; robustness tested for m=3–5 and r=0.05–0.3 (Sec. 2.2, Figs. 25–27).
  • Rolling window length and step = 10,080 min (7 days) window, 1,440 min (1 day) step
    Chosen as a compromise between temporal resolution and statistical stability (Sec. 7.1).
  • MFDFA polynomial detrending order = l=2
    Standard choice for financial time series (Sec. 2.1).
  • Time scale s for rolling-window ρ = s=10
    Used in rolling-window cross-correlations; surrogate baseline is near zero at this scale (Sec. 7).
  • Trade-size thresholds for filtering = 0.001 BTC, 0.01 ETH
    Chosen to reveal periods without trading after removing small trades (Sec. 8, Fig. 31).
  • Change-point minimum distance = 1440 observations (1 day)
    Imposed to identify persistent mean shifts rather than intraday fluctuations (Sec. 8).
axioms (5)
  • standard math MFDFA, MFCCA, DCCA, ApEn, and SampEn definitions and their properties are taken as correct.
    The paper relies on these established algorithms without deriving them (Sec. 2).
  • domain assumption The empirical regularities on Binance, Kraken, and KuCoin, and in Bitget's early period, define the 'usual' baseline.
    Stated in Sec. 1 as an empirical benchmark; the anomaly is defined as persistent departure from this baseline.
  • domain assumption Public exchange APIs provide accurate and complete trade-level data.
    The analysis uses tick data from exchange archives (Sec. 3); no independent verification is provided.
  • domain assumption A single mean-shift change point with a 1-day minimum distance is adequate to characterize the regime break.
    MATLAB findchangepts with max 1 point is used to split the sample (Sec. 8).
  • domain assumption One-second timestamp resolution limits the identification of order-splitting; identical timestamps are not treated as child fills.
    Sec. 8: 'identical timestamps cannot be interpreted as direct evidence of child-fill splitting'—this constrains interpretation.

reviewed 2026-08-02 · how reviews work

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Cite this review

Pith. "Pith review of Detecting unusual trading patterns on cryptocurrency exchanges by means of complexity measures." pith.science (2026). https://pith.science/paper/BRHDNOPS

@misc{pith2026260713916,
  author       = {Pith},
  title        = {Pith review of: Detecting unusual trading patterns on cryptocurrency exchanges by means of complexity measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BRHDNOPS}},
  note         = {Machine review of arXiv:2607.13916}
}
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read the original abstract

Artificial transaction generation remains an important source of potential market manipulation on cryptocurrency exchanges, as it may distort reported liquidity and reduce market transparency. This study proposes a diagnostic framework for detecting unusual trading patterns based on complexity and statistical-structure measures derived from high-frequency trade-level data. The analysis considers log-returns, trading volume, and transaction counts, using tail distributions, autocorrelation functions, multifractal characteristics, approximate entropy, and detrended cross-correlations. The methodology is applied to BTC, ETH, and XRP traded on Binance, Bitget, KuCoin, and Kraken over the period from April 1 to June 30, 2025. The results reveal a pronounced anomaly on Bitget for BTC and ETH after mid-May 2025. The number of transactions increases sharply, but there is no proportional increase in traded volume or return fluctuations. This regime is characterised by numerous low-volume trades, weaker autocorrelations, reduced multifractal organisation, higher short-pattern irregularity, and weaker cross-correlations involving the transaction-count series. These features are consistent with a noise-like component in trading activity and may indicate artificially increased transaction counts, although they do not provide direct proof of wash trading. The findings show that complexity-based indicators can be useful for detecting exchange-specific trading anomalies that remain hidden in price-based measures.

Figures

Figures reproduced from arXiv: 2607.13916 by Jakub Zwydak, Jaros{\l}aw Kwapie\'n, Marcin W\k{a}torek, Stanis{\l}aw Dro\.zd\.z.

Figure 1
Figure 1. Figure 1: Evolution of the cumulative log-returns Rˆ(t), trading volume V∆t=1min(t), and the number of trans￾actions N∆t=1min(t) for BTC on Binance (top left), Bitget (top right), Kraken (bottom left), and KuCoin (bottom right). The period of increased transaction activity is marked by a red dashed ellipse. The temporal evolution of cumulative log-returns, Rˆ(ti) = ∑ i k=1 R(tk ), V∆t=1min(t), and N∆t=1min(t) is pre… view at source ↗
Figure 2
Figure 2. Figure 2: The same as in [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The same as in [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Complementary cumulative distribution functions of standardised absolute log-returns |R| for BTC, ETH, and XRP on Binance, Bitget, Kraken, and KuCoin, together with the Gaussian and power-law (γ = 3) reference distributions. 4.2. Autocorrelation The specific behaviour of the transaction-count series on Bitget is further confirmed by the autocorrelation analysis. The autocorrelation function (ACF) is define… view at source ↗
Figure 5
Figure 5. Figure 5: Complementary cumulative distribution functions of standardised trading volume V for BTC, ETH, and XRP on Binance, Bitget, Kraken, and KuCoin, together with reference distributions: the power-law distribution with γ = 2 and the stretched exponential distribution with β = 0.4. liquidity conditions than absolute log-returns. However, the exchange-specific differences in volume autocorrelations are less prono… view at source ↗
Figure 6
Figure 6. Figure 6: Complementary cumulative distribution functions of the standardised number of transactions N for BTC, ETH, and XRP on Binance, Bitget, Kraken, and KuCoin, together with the reference distributions: Gaussian, power law with γ = 2, and stretched exponential with β = 0.4. More pronounced exchange-specific differences are observed for trading volume. The correspond￾ing fluctuation functions FVV(s) are shown in… view at source ↗
Figure 7
Figure 7. Figure 7: Autocorrelation functions of the standardised number of transactions N for BTC, ETH, and XRP on Binance, Bitget, Kraken, and KuCoin. suggests a systematic disturbance in the temporal organisation of the transaction-count process. This is consistent with the regime-change behaviour of N∆t=1min(t) observed earlier for Bitget. Interestingly, the same effect is not observed for XRP on Bitget. In this case, the… view at source ↗
Figure 8
Figure 8. Figure 8: Autocorrelation functions of standardised absolute log-returns |R| for BTC, ETH, and XRP on Binance, Bitget, Kraken, and KuCoin. both larger turnover and stronger return fluctuations [81]. A positive relationship between trading volume and the number of transactions is therefore also expected. Figs. 14, 15, and 16 show the relationships between N∆t=1min(t), V∆t=1min(t), and R∆t=1min(t) for BTC, ETH, and XR… view at source ↗
Figure 9
Figure 9. Figure 9: Autocorrelation functions of standardised trading volume V for BTC, ETH, and XRP on Binance, Bitget, Kraken, and KuCoin. towards either positive or negative returns. However, the dispersion of returns clearly increases with N∆t=1min(t). For small values of N∆t=1min(t), returns are tightly concentrated around zero, whereas for larger values of N∆t=1min(t), the range of observed returns expands. Thus, what t… view at source ↗
Figure 10
Figure 10. Figure 10: Univariate fluctuation functions FRR(s) for BTC, ETH, and XRP log-returns R on Binance, Bitget, Kraken, and KuCoin. Dashed red lines indicate the scale range selected for determining the multifractal spectra. increases with both trading volume and transaction intensity. KuCoin follows the same general structure, but the relationships are visibly noisier and more dispersed, which may reflect lower liquidit… view at source ↗
Figure 11
Figure 11. Figure 11: Univariate fluctuation functions FVV(s) for BTC, ETH, and XRP trading volume V on Binance, Bitget, Kraken, and KuCoin. Dashed red lines indicate the scale range selected for determining the multifractal spectra. V, and the number of transactions N. The results presented in [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Univariate fluctuation functions FNN(s) for BTC, ETH, and XRP number of transactions N on Binance, Bitget, Kraken, and KuCoin. Dashed red lines indicate the scale range selected for determining the multifractal spectra. Red dashed ellipses mark the distorted scaling region observed for BTC and ETH on Bitget. dependence to very strong long-scale correlations, frequently approaching ρ(q = 2,s) ≈ 0.9. ETH sh… view at source ↗
Figure 13
Figure 13. Figure 13: Multifractal spectra for BTC, ETH, and XRP log-returns R (top panels), trading volume V (middle panels), and number of transactions N (bottom panels) on Binance, Bitget, Kraken, and KuCoin [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Relationship between trading volume V∆t=1min(t) and the number of transactions N∆t=1min(t) on Binance, Bitget, Kraken, and KuCoin for BTC, ETH, and XRP. Each point corresponds to a 1-min interval. the interpretation that, for BTC and ETH on Bitget, the transaction dynamics differ from those observed on the remaining exchanges [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Relationship between log-returns R∆t=1min(t) and the number of transactions N∆t=1min(t) on Binance, Bitget, Kraken, and KuCoin for BTC, ETH, and XRP. Each point corresponds to a 1-min interval [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Relationship between log-returns R∆t=1min(t) and trading volume V∆t=1min(t) on Binance, Bitget, Kraken, and KuCoin for BTC, ETH, and XRP. Each point corresponds to a 1-min interval. 7. Rolling window analysis In the previous section, the differences in the cross-correlations between |R|, V, and N were observed. For BTC and ETH, the correlations involving the number of transactions were clearly weaker on B… view at source ↗
Figure 17
Figure 17. Figure 17: Detrended cross-correlation coefficient ρ(q = 2,s) calculated between absolute log-returns |R| and trading volume V (top panels), absolute log-returns |R| and the number of transactions N (middle panels), and trading volume V and the number of transactions N (bottom panels) for BTC, ETH, and XRP traded on each considered exchange. The dashed curve denotes the scale-dependent standard deviation of ρ(q = 2,… view at source ↗
Figure 18
Figure 18. Figure 18: Detrended cross-correlation coefficient ρ(q = 2,s = 10) calculated in a rolling window of length 7 days, with a step of 24 hours, for BTC on Binance, Bitget, Kraken, and KuCoin. The top, middle, and bottom panels show the pairs |R| − V, |R| − N, and V − N, respectively. The time coordinate of each point denotes the end of the corresponding rolling window. A closely related pattern is observed for ETH in … view at source ↗
Figure 19
Figure 19. Figure 19: The same as in [PITH_FULL_IMAGE:figures/full_fig_p023_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: The same as in [PITH_FULL_IMAGE:figures/full_fig_p024_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Detrended cross-correlation coefficient ρ(q = 2,s = 10) calculated in a rolling window of length 7 days, with a step of 24 hours, for BTC (left), ETH (middle) and XRP (right) R (top), V (middle), N (bottom) between Binance, Bitget, Kraken, and KuCoin exchanges. The time coordinate of each point denotes the end of the corresponding rolling window. heterogeneous across exchanges. Binance and KuCoin remain a… view at source ↗
Figure 22
Figure 22. Figure 22: Approximate entropy with m = 2 and τ = 1, calculated in a rolling window of length 7 days with a step of 24 hours for BTC log-returns R (top), trading volume V (middle), and the number of transactions N (bottom) on Binance, Bitget, Kraken, and KuCoin. The time coordinate of each point denotes the end of the corresponding rolling window. For XRP, the behaviour is clearly different, as shown in [PITH_FULL_… view at source ↗
Figure 23
Figure 23. Figure 23: The same as in [PITH_FULL_IMAGE:figures/full_fig_p027_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: The same as in [PITH_FULL_IMAGE:figures/full_fig_p028_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Approximate entropy calculated in a rolling window of length 7 days with a step of 24 hours for BTC and ETH log-returns R (top), trading volume V (middle), and the number of transactions N (bottom) on Bitget for different embedding dimensions m = 2, 3, 4, 5 and fixed τ = 1. The time coordinate of each point denotes the end of the corresponding rolling window. keeping m = 2 fixed. Second, the embedding dim… view at source ↗
Figure 26
Figure 26. Figure 26 [PITH_FULL_IMAGE:figures/full_fig_p029_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Sample entropy calculated in a rolling window of length 7 days with a step of 24 hours for BTC and ETH log-returns R (top), trading volume V (middle), and the number of transactions N (bottom) on Bitget for different embedding dimensions m = 2, 3, 4, 5 and fixed r = 0.2. The time coordinate of each point denotes the end of the corresponding rolling window. The sensitivity to the embedding dimension is als… view at source ↗
Figure 28
Figure 28. Figure 28: Decomposition of the one-minute transaction count on Bitget. Left: number of active seconds per one-minute interval, At . Right: average number of transaction records per active second, Mt = Nt/At . Results are shown for BTC, ETH, and XRP [PITH_FULL_IMAGE:figures/full_fig_p031_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Relations between the number of transactions N∆t=1min and trading volume V∆t=1min (top panels), and between log-returns R∆t=1min and N∆t=1min (bottom panels), on Bitget in the two periods of interest for BTC (left) and ETH (right). does not translate into higher traded volume, especially for ETH. This indicates that the second period is dominated by numerous low-volume transactions, which substantially in… view at source ↗
Figure 30
Figure 30. Figure 30: Trade-size histograms for BTC, ETH, and XRP on Bitget in the two periods of interest. The vertical axis shows the number of trades n(V) in each trade-size bin [PITH_FULL_IMAGE:figures/full_fig_p032_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: Time series from Bitget after removing transactions with volumes below 0.001 BTC (left) and 0.01 ETH (right). The average volume per transaction is 0.0078 BTC in Bitget1 and 0.0024 BTC in Bitget2, and 0.6636 ETH in Bitget1 and 0.0268 ETH in Bitget2. The periods with increased transaction activity from Figs. 1 and 2 are marked by a red dashed ellipse. are not visible in the original full transaction-count … view at source ↗
Figure 32
Figure 32. Figure 32: Complementary cumulative distribution functions for the number of transactions N for BTC, ETH, and XRP on Bitget in the two periods of interest, together with the Gaussian, power-law with γ = 2, and stretched exponential with β = 0.4 reference distributions. A similar conclusion follows from ACFs of the number of transactions shown in [PITH_FULL_IMAGE:figures/full_fig_p033_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: Autocorrelation functions of the number of transactions N for BTC, ETH, and XRP on Bitget in the two periods of interest. 1 102 FNN(s) Bitget1 10 100 1000 10 100 1000 scale s [1min] 10 100 1000 1 102 FNN(s) Bitget2 BTC ETH XRP 0.2 0.4 0.6 0.8 1 1.2 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 f(α) N 0.2 0.4 0.6 0.8 1 1.2 α 0.2 0.4 0.6 0.8 1 1.2 Bitget1 Bitget2 BTC ETH XRP [PITH_FULL_IMAGE:figures/full_fig_p034_33.png] view at source ↗
Figure 34
Figure 34. Figure 34: Fluctuation functions FNN(s) (left panel) and the corresponding multifractal spectra (right panel) for the number of transactions N for BTC, ETH, and XRP on Bitget in two periods. Dashed red lines indicate the scale range selected for determining the multifractal spectra. The previous observations indicate that BTC and ETH trading on Bitget in the second period is characterised by a large number of weakly… view at source ↗
Figure 35
Figure 35. Figure 35: Detrended cross-correlation coefficient ρ(q = 2,s) calculated between absolute log-returns |R| and the number of transactions N (top panels) and between trading volume V and the number of transactions N (bottom panels), for BTC, ETH, and XRP on Bitget in the two periods of interest. of whether the trading characteristics of BTC and ETH became mutually synchronised during the anomalous period. This issue w… view at source ↗
Figure 36
Figure 36. Figure 36: Detrended cross-correlation coefficient ρ(q = 2,s) calculated between cryptocurrency pairs for log￾returns R (top panels), trading volume V (middle panels), and the number of transactions N (bottom panels). The columns correspond to BTC–ETH, BTC–XRP, and ETH–XRP. Results are shown for Binance, Kraken, KuCoin, and Bitget split into Bitget1 and Bitget2. The dashed curves denote the scale-dependent ±1 standa… view at source ↗

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Works this paper leans on

90 extracted references · 31 canonical work pages

  1. [2]

    Do cryptocurrency exchanges fake trading volumes? An empirical analysis of wash trading based on data mining.Physica A2022,586, 126405

    Chen, J.; Lin, D.; Wu, J. Do cryptocurrency exchanges fake trading volumes? An empirical analysis of wash trading based on data mining.Physica A2022,586, 126405. https://doi.org/10.1016/j.physa.2021.126405

    arXiv 2021

  2. [3]

    Trading Volume Manipulation and Competition Among Centralized Crypto Exchanges.Management Science2025,71, 8604–8622

    Amiram, D.; Lyandres, E.; Rabetti, D. Trading Volume Manipulation and Competition Among Centralized Crypto Exchanges.Management Science2025,71, 8604–8622. https://doi.org/10.1287/mnsc.2021.02903

    arXiv 2021

  3. [4]

    Wash trading at cryptocurrency exchanges.Finance Research Letters2021, 43, 101982

    Pennec, G.L.; Fiedler, I.; Ante, L. Wash trading at cryptocurrency exchanges.Finance Research Letters2021, 43, 101982. https://doi.org/10.1016/j.frl.2021.101982

    arXiv 2021

  4. [5]

    Detecting and Quantifying Wash Trading on Decentralized Cryptocurrency Exchanges

    Victor, F.; Weintraud, A.M. Detecting and Quantifying Wash Trading on Decentralized Cryptocurrency Exchanges. In Proceedings of the The Web Conference 2021, 2021, pp. 23–32. https://doi.org/10.1145/3442 381.3449824

    work page doi:10.1145/3442 2021

  5. [6]

    Crypto Wash Trading.Management Science2023,69, 6427–6454

    Cong, L.W.; Li, X.; Tang, K.; Yang, Y. Crypto Wash Trading.Management Science2023,69, 6427–6454

  6. [7]

    Wash Trading Detection Techniques for Centralised Cryptocurrency Exchange Services

    Di Francesco Maesa, D.; Ricci, L.; Santarella, L.; Yimame, Y. Wash Trading Detection Techniques for Centralised Cryptocurrency Exchange Services. In Proceedings of the 2024 5th International Artificial Intelligence and Blockchain Conference, 2024, AIBC ’24. https://doi.org/10.1145/3702359.3702363

    arXiv 2024

  7. [8]

    Determinants of wash trading in major cryptoexchanges

    Sila, J.; Kocenda, E.; Kristoufek, L.; Kukacka, J. Determinants of wash trading in major cryptoexchanges. International Review of Financial Analysis2026,117, 105283. https://doi.org/https://doi.org/10.1016/j.irfa.20 26.105283

    work page doi:10.1016/j.irfa.20

  8. [9]

    Approaching Multifractal Complexity in Decentralized Cryptocurrency Trading.Fractal and Fractional2024,8

    W ˛ atorek, M.; Królczyk, M.; Kwapie ´ n, J.; Stanisz, T.; Dro˙zd˙z, S. Approaching Multifractal Complexity in Decentralized Cryptocurrency Trading.Fractal and Fractional2024,8. https://doi.org/10.3390/fractalfract8 110652

    work page doi:10.3390/fractalfract8

  9. [10]

    Characteristics of price related fluctuations in non-fungible token (NFT) market.Chaos2024,34, 013108

    Szydło, P .; W ˛ atorek, M.; Kwapie ´ n, J.; Dro˙zd ˙z, S. Characteristics of price related fluctuations in non-fungible token (NFT) market.Chaos2024,34, 013108. https://doi.org/10.1063/5.0185306

    work page doi:10.1063/5.0185306

  10. [11]

    Correlations versus noise in the NFT market.Chaos2024, 34, 073112

    W ˛ atorek, M.; Szydło, P .; Kwapie ´ n, J.; Dro˙zd˙z, S. Correlations versus noise in the NFT market.Chaos2024, 34, 073112. https://doi.org/10.1063/5.0214399. 39 of 42

    work page doi:10.1063/5.0214399

  11. [12]

    Understanding Flash-Loan-based Wash Trading

    Gan, R.; Wang, L.; Ruan, X.; Lin, X. Understanding Flash-Loan-based Wash Trading. In Proceedings of the 4th ACM Conference on Advances in Financial Technologies, New York, NY, USA, 2023; AFT ’22, p. 74–88. https://doi.org/10.1145/3558535.3559793

    arXiv 2023

  12. [13]

    Patterns and centralisation in Ethereum-based token transaction networks.Frontiers in Physics2024,12, 1305167

    Collibus, F.M.D.; Campajola, C.; Caldarelli, G.; Tessone, C.J. Patterns and centralisation in Ethereum-based token transaction networks.Frontiers in Physics2024,12, 1305167. https://doi.org/10.3389/fphy.2024.13051 67

    arXiv 2024

  13. [14]

    The Dark Side of NFTs: A Large-Scale Empirical Study of Wash Trading

    Chen, S.; Chen, J.; Yu, J.; Luo, X.; Wang, Y. The Dark Side of NFTs: A Large-Scale Empirical Study of Wash Trading. In Proceedings of the 15th Asia-Pacific Symposium on Internetware, New York, NY, USA, 2024; Internetware ’24, p. 447–456. https://doi.org/10.1145/3671016.3674808

    arXiv 2024

  14. [15]

    Exposing Stealthy Wash Trading on Automated Market Maker Exchanges

    Gan, R.; Wang, L.; Xue, L.; Lin, X. Exposing Stealthy Wash Trading on Automated Market Maker Exchanges. ACM Transactions on Internet Technology2024,24, 1–30

  15. [16]

    Unveiling Wash Trading in Popular NFT Markets

    Niu, Y.; Li, X.; Peng, H.; Li, W. Unveiling Wash Trading in Popular NFT Markets. In Proceedings of the Companion Proceedings of the ACM Web Conference 2024, New York, NY, USA, 2024; WWW ’24, p. 730–733. https://doi.org/10.1145/3589335.3651580

    arXiv 2024

  16. [17]

    Beyond the surface: advanced wash-trading detection in decentralized NFT markets.Financial Innovation2025,11, 86

    Toši´ c, A.; Viˇ ciˇ c, J.; Hrovatin, N. Beyond the surface: advanced wash-trading detection in decentralized NFT markets.Financial Innovation2025,11, 86. https://doi.org/10.1186/s40854-025-00766-z

    work page doi:10.1186/s40854-025-00766-z

  17. [18]

    Physical approach to complex systems.Physics Reports2012,515, 115–226

    Kwapie ´ n, J.; Dro˙zd˙z, S. Physical approach to complex systems.Physics Reports2012,515, 115–226. https: //doi.org/10.1016/j.physrep.2012.01.007

    work page doi:10.1016/j.physrep.2012.01.007 2012

  18. [19]

    Scaling behaviour in the dynamics of an economic index.Nature1995, 376, 46–49

    Mantegna, R.N.; Stanley, H.E. Scaling behaviour in the dynamics of an economic index.Nature1995, 376, 46–49. https://doi.org/10.1038/376046a0

    doi:10.1038/376046a0

  19. [20]

    Inverse cubic law for the distribution of stock price variations.The European Physical Journal B1998,3, 139–140

    Gopikrishnan, P .; Meyer, M.; Amaral, L.N.; Stanley, H.E. Inverse cubic law for the distribution of stock price variations.The European Physical Journal B1998,3, 139–140. https://doi.org/10.1007/s100510050292

    doi:10.1007/s100510050292

  20. [21]

    Scaling of the distribution of fluctuations of financial market indices.Physical Review E1999,60, 5305–5316

    Gopikrishnan, P .; Plerou, V .; Nunes Amaral, L.A.; Meyer, M.; Stanley, H.E. Scaling of the distribution of fluctuations of financial market indices.Physical Review E1999,60, 5305–5316

  21. [22]

    Empirical properties of asset returns: stylized facts and statistical issues.Quantitative Finance2001, 1, 223–236

    Cont, R. Empirical properties of asset returns: stylized facts and statistical issues.Quantitative Finance2001, 1, 223–236. https://doi.org/10.1080/713665670

    doi:10.1080/713665670

  22. [23]

    Price impact

    Bouchaud, J.P . Price impact. InEncyclopedia of Quantitative Finance; Cambridge University Press, 2010; pp. 1–6

    2010

  23. [24]

    International Financial Markets Through 150 Years: Evaluating Stylized Facts.The Journal of Finance and Data Science2026, p

    S’afari, S.A.; Janisch, M.; Lehéricy, T. International Financial Markets Through 150 Years: Evaluating Stylized Facts.The Journal of Finance and Data Science2026, p. 100196. https://doi.org/10.1016/j.jfds.2026.100196

    arXiv 2026

  24. [25]

    Scaling properties of extreme price fluctuations in Bitcoin markets.Physica A2018,510, 400–406

    Beguši´ c, S.; Kostanjˇ car, Z.; Stanley, H.E.; Podobnik, B. Scaling properties of extreme price fluctuations in Bitcoin markets.Physica A2018,510, 400–406. https://doi.org/10.1016/j.physa.2018.06.131

    doi:10.1016/j.physa.2018.06.131 2018

  25. [26]

    Bitcoin market route to maturity? Evidence from return fluctuations, temporal correlations and multiscaling effects.Chaos2018,28, 071101

    Dro˙zd˙z, S.; G˛ ebarowski, R.; Minati, L.; O´ swi˛ ecimka, P .; W ˛ atorek, M. Bitcoin market route to maturity? Evidence from return fluctuations, temporal correlations and multiscaling effects.Chaos2018,28, 071101. https://doi.org/10.1063/1.5036517

    work page doi:10.1063/1.5036517

  26. [27]

    Signatures of the crypto-currency market decoupling from the Forex.Future Internet2019,11

    Dro˙zd ˙z, S.; Minati, L.; O´ swi˛ ecimka, P .; Stanuszek, M.; W ˛ atorek, M. Signatures of the crypto-currency market decoupling from the Forex.Future Internet2019,11. https://doi.org/10.3390/fi11070154

    work page doi:10.3390/fi11070154

  27. [28]

    Complexity in economic and social systems: Cryptocurrency market at around COVID-19.Entropy2020,22, 1043

    Dro˙zd ˙z, S.; Kwapie ´ n, J.; O´ swi˛ ecimka, P .; Stanisz, T.; W ˛ atorek, M. Complexity in economic and social systems: Cryptocurrency market at around COVID-19.Entropy2020,22, 1043. https://doi.org/10.3390/E22091043

    work page doi:10.3390/e22091043

  28. [29]

    Changes to the extreme and erratic behaviour of cryptocurrencies during COVID-19.Physica A2021,565, 125581

    James, N.; Menzies, M.; Chan, J. Changes to the extreme and erratic behaviour of cryptocurrencies during COVID-19.Physica A2021,565, 125581. https://doi.org/10.1016/j.physa.2020.125581

    arXiv 2020

  29. [30]

    Stylized facts of cryptocurrency markets: Robust definitions and inference approaches.Emerging Markets Review2026,72, 101440

    Abdullaev, N.; Ibragimov, R. Stylized facts of cryptocurrency markets: Robust definitions and inference approaches.Emerging Markets Review2026,72, 101440. https://doi.org/10.1016/j.ememar.2026.101440

    arXiv 2026

  30. [31]

    Statistical properties and multifractality of Bitcoin.Physica A2018,506, 507–519

    Takaishi, T. Statistical properties and multifractality of Bitcoin.Physica A2018,506, 507–519. https: //doi.org/10.1016/j.physa.2018.04.046

    work page doi:10.1016/j.physa.2018.04.046 2018

  31. [32]

    Multifractal analysis of Bitcoin market.Physica A 2018,512, 954–967

    da Silva Filho, A.C.; Maganini, N.D.; de Almeida, E.F. Multifractal analysis of Bitcoin market.Physica A 2018,512, 954–967

    2018

  32. [33]

    Mensi, W.; Lee, Y.J.; Al-Yahyaee, K.H.; Sensoy, A.; Yoon, S.M. Intraday downward/upward multifractality and long memory in Bitcoin and Ethereum markets: An asymmetric multifractal detrended fluctuation analysis.Finance Research Letters2019,31, 19–25

  33. [34]

    The high frequency multifractal properties of Bitcoin.Physica A2019,520, 62–71

    Stavroyiannis, S.; Babalos, V .; Bekiros, S.; Lahmiri, S.; Uddin, G.S. The high frequency multifractal properties of Bitcoin.Physica A2019,520, 62–71. https://doi.org/10.1016/j.physa.2018.12.037

    work page doi:10.1016/j.physa.2018.12.037 2018

  34. [35]

    Multifractal behavior of price and volume changes in the cryptocurrency market.Physica A2019,520, 54–61

    Stosic, D.; Stosic, D.; Ludermir, T.B.; Stosic, T. Multifractal behavior of price and volume changes in the cryptocurrency market.Physica A2019,520, 54–61. https://doi.org/10.1016/j.physa.2018.12.038. 40 of 42

    work page doi:10.1016/j.physa.2018.12.038 2018

  35. [36]

    Market efficiency, liquidity, and multifractality of Bitcoin: A dynamic study.Asia- Pacific Financial Markets2020,27, 145–154

    Takaishi, T.; Adachi, T. Market efficiency, liquidity, and multifractality of Bitcoin: A dynamic study.Asia- Pacific Financial Markets2020,27, 145–154

  36. [37]

    Long-range dependence, multi-fractality and volume-return causality of ether market.Chaos2020,30, 011101

    Han, Q.; Wu, J.; Zheng, Z. Long-range dependence, multi-fractality and volume-return causality of ether market.Chaos2020,30, 011101

  37. [38]

    Analysis of inter-transaction time fluctuations in the cryptocurrency market.Chaos2022,32, 083142

    Kwapie ´ n, J.; W ˛ atorek, M.; Bezbradica, M.; Crane, M.; Tan Mai, T.; Dro˙zd˙z, S. Analysis of inter-transaction time fluctuations in the cryptocurrency market.Chaos2022,32, 083142. https://doi.org/10.1063/5.0104707

    work page doi:10.1063/5.0104707

  38. [39]

    Cryptocurrency market efficiency in short- and long-term horizons during COVID- 19: An asymmetric multifractal analysis approach.Finance Research Letters2022,46, 102319

    Kakinaka, S.; Umeno, K. Cryptocurrency market efficiency in short- and long-term horizons during COVID- 19: An asymmetric multifractal analysis approach.Finance Research Letters2022,46, 102319

  39. [40]

    Inner Multifractal Dynamics in the Jumps of Cryptocurrency and Forex Markets.Fractal and Fractional2024,8

    Ali, H.; Aftab, M.; Aslam, F.; Ferreira, P . Inner Multifractal Dynamics in the Jumps of Cryptocurrency and Forex Markets.Fractal and Fractional2024,8. https://doi.org/10.3390/fractalfract8100571

    work page doi:10.3390/fractalfract8100571

  40. [41]

    Choi, I. Intrinsic multifractality and adaptive efficiency in Bitcoin: Disentangling sources of scaling complexity across market regimes.Physica A: Statistical Mechanics and its Applications2026,697, 131735. https://doi.org/10.1016/j.physa.2026.131735

    arXiv 2026

  41. [42]

    Multifractal cross-correlations of bitcoin and ether trading characteristics in the post-COVID-19 time.Future Internet2022,14

    W ˛ atorek, M.; Kwapie ´ n, J.; Dro˙zd ˙z, S. Multifractal cross-correlations of bitcoin and ether trading characteristics in the post-COVID-19 time.Future Internet2022,14

  42. [43]

    What is mature and what is still emerging in the cryptocurrency market?Entropy2023,25

    Dro˙zd˙z, S.; Kwapie ´ n, J.; W ˛ atorek, M. What is mature and what is still emerging in the cryptocurrency market?Entropy2023,25. https://doi.org/10.3390/e25050772

    work page doi:10.3390/e25050772

  43. [44]

    Detrended cross-correlation analysis: A new method for analyzing two nonstation- ary time series.Physical Review Letters2008,100, 084102

    Podobnik, B.; Stanley, H.E. Detrended cross-correlation analysis: A new method for analyzing two nonstation- ary time series.Physical Review Letters2008,100, 084102. https://doi.org/10.1103/PhysRevLett.100.084102

    doi:10.1103/physrevlett.100.084102

  44. [45]

    DCCA cross-correlation coefficient: Quantifying level of cross-correlation.Physica A2011, 390, 614–618

    Zebende, G. DCCA cross-correlation coefficient: Quantifying level of cross-correlation.Physica A2011, 390, 614–618. https://doi.org/10.1016/j.physa.2010.10.022

    work page doi:10.1016/j.physa.2010.10.022 2010

  45. [46]

    Detrended fluctuation analysis made flexible to detect range of cross- correlated fluctuations.Physical Review E2015,92, 052815

    Kwapie ´ n, J.; O´ swi˛ ecimka, P .; Dro˙zd ˙z, S. Detrended fluctuation analysis made flexible to detect range of cross- correlated fluctuations.Physical Review E2015,92, 052815. https://doi.org/10.1103/PhysRevE.92.052815

    work page doi:10.1103/physreve.92.052815

  46. [47]

    Approximate entropy as a measure of system complexity.Proceedings of the National Academy of Sciences1991,88, 2297–2301

    Pincus, S.M. Approximate entropy as a measure of system complexity.Proceedings of the National Academy of Sciences1991,88, 2297–2301. https://doi.org/10.1073/pnas.88.6.2297

    doi:10.1073/pnas.88.6.2297

  47. [48]

    Trading and arbitrage in cryptocurrency markets.Journal of Financial Economics2020, 135, 293–319

    Makarov, I.; Schoar, A. Trading and arbitrage in cryptocurrency markets.Journal of Financial Economics2020, 135, 293–319

  48. [49]

    The Role of Binance in Bitcoin Volatility Transmission.Applied Mathematical Finance2022,29, 1–32

    Carol Alexander, D.F.H.; Kaeck, A. The Role of Binance in Bitcoin Volatility Transmission.Applied Mathematical Finance2022,29, 1–32

  49. [50]

    Rise of the machines? Intraday high-frequency trading patterns of cryptocurrencies.The European Journal of Finance2021,27, 8–30

    Petukhina, A.A.; Reule, R.C.G.; Härdle, W.K. Rise of the machines? Intraday high-frequency trading patterns of cryptocurrencies.The European Journal of Finance2021,27, 8–30. https://doi.org/10.1080/1351847X.2020.1 789684

    work page doi:10.1080/1351847x.2020.1 2020

  50. [51]

    Bitcoin spot and futures market microstructure.Journal of Futures Markets2021, 41, 194–225

    Aleti, S.; Mizrach, B. Bitcoin spot and futures market microstructure.Journal of Futures Markets2021, 41, 194–225

  51. [52]

    Decomposing cryptocurrency high-frequency price dynamics into recurring and noisy components.Chaos2023,33, 083146

    W ˛ atorek, M.; Skupie ´ n, M.; Kwapie ´ n, J.; Dro˙zd˙z, S. Decomposing cryptocurrency high-frequency price dynamics into recurring and noisy components.Chaos2023,33, 083146. https://doi.org/10.1063/5.0165635

    work page doi:10.1063/5.0165635

  52. [53]

    Time-of-day periodicities of trading volume and volatility in Bitcoin exchange: Does the stock market matter?Finance Research Letters2020,36, 101303

    Wang, J.N.; Liu, H.C.; Hsu, Y.T. Time-of-day periodicities of trading volume and volatility in Bitcoin exchange: Does the stock market matter?Finance Research Letters2020,36, 101303. https://doi.org/10.1016/ j.frl.2019.101303

    arXiv 2019

  53. [54]

    Periodicity in Cryptocurrency Volatility and Liquidity.Journal of Financial Econometrics2024,22, 224–251

    Hansen, P .R.; Kim, C.; Kimbrough, W. Periodicity in Cryptocurrency Volatility and Liquidity.Journal of Financial Econometrics2024,22, 224–251. https://doi.org/10.1093/jjfinec/nbac034

    work page doi:10.1093/jjfinec/nbac034

  54. [55]

    How investible is Bitcoin? Analyzing the liquidity and transaction costs of Bitcoin markets.Economics Letters2018,171, 140–143

    Dyhrberg, A.H.; Foley, S.; Svec, J. How investible is Bitcoin? Analyzing the liquidity and transaction costs of Bitcoin markets.Economics Letters2018,171, 140–143. https://doi.org/10.1016/j.econlet.2018.07.032

    work page doi:10.1016/j.econlet.2018.07.032 2018

  55. [56]

    Multifractal detrended fluctuation analysis of nonstationary time series.Physica A2002,316, 87–114

    Kantelhardt, J.W.; Zschiegner, S.A.; Koscielny-Bunde, E.; Havlin, S.; Bunde, A.; Stanley, H.E. Multifractal detrended fluctuation analysis of nonstationary time series.Physica A2002,316, 87–114. https://doi.org/10 .1016/s0378-4371(02)01383-3

  56. [57]

    Mosaic organization of DNA nucleotides.Physical Review E1994,49, 1685–1689

    Peng, C.K.; Buldyrev, S.V .; Havlin, S.; Simons, M.; Stanley, H.E.; Goldberger, A.L. Mosaic organization of DNA nucleotides.Physical Review E1994,49, 1685–1689. https://doi.org/10.1103/PhysRevE.49.1685

    doi:10.1103/physreve.49.1685

  57. [58]

    Multifractal detrended cross-correlation analysis for two nonstationary signals.Physical Review E2008,77, 066211

    Zhou, W.X. Multifractal detrended cross-correlation analysis for two nonstationary signals.Physical Review E2008,77, 066211. https://doi.org/10.1103/PhysRevE.77.066211

    work page doi:10.1103/physreve.77.066211

  58. [59]

    Detrended cross-correlation analysis consistently extended to multifractality.Physical Review E2014,89, 023305

    O´ swi˛ ecimka, P .; Dro˙zd˙z, S.; Forczek, M.; Jadach, S.; Kwapie ´ n, J. Detrended cross-correlation analysis consistently extended to multifractality.Physical Review E2014,89, 023305. https://doi.org/10.1103/ PhysRevE.89.023305

  59. [60]

    Multifractal analysis of financial markets: A review.Reports on Progress in Physics2019,82, 125901

    Jiang, Z.Q.; Xie, W.J.; Zhou, W.X.; Sornette, D. Multifractal analysis of financial markets: A review.Reports on Progress in Physics2019,82, 125901. https://doi.org/10.1088/1361-6633/ab42fb. 41 of 42

    work page doi:10.1088/1361-6633/ab42fb

  60. [61]

    Genuine multifractality in time series is due to temporal correlations.Physical Review E2023,107, 034139

    Kwapie ´ n, J.; Blasiak, P .; Dro˙zd ˙z, S.; O´ swi˛ ecimka, P . Genuine multifractality in time series is due to temporal correlations.Physical Review E2023,107, 034139. https://doi.org/10.1103/PhysRevE.107.034139

    work page doi:10.1103/physreve.107.034139

  61. [62]

    Multifractality and its sources in the digital currency market.Future Internet2025,17

    Dro˙zd ˙z, S.; Kluszczy ´ nski, R.; Kwapie ´ n, J.; W ˛ atorek, M. Multifractality and its sources in the digital currency market.Future Internet2025,17. https://doi.org/10.3390/fi17100470

    work page doi:10.3390/fi17100470

  62. [63]

    Effect of Detrending on Multifractal Characteristics

    O´ swi˛ ecimka, P .; Dro˙zd˙z, S.; Kwapie ´ n, J.; Górski, A.Z. Effect of Detrending on Multifractal Characteristics. Acta Physica Polonica A2013,123, 597–603. https://doi.org/10.12693/APhysPolA.123.597

    work page doi:10.12693/aphyspola.123.597

  63. [64]

    Fractal measures and their singularities: The characterization of strange sets.Physical Review A1986,33, 1141–1151

    Halsey, T.C.; Jensen, M.H.; Kadanoff, L.P .; Procaccia, I.; Shraiman, B.I. Fractal measures and their singularities: The characterization of strange sets.Physical Review A1986,33, 1141–1151

  64. [65]

    Detecting and interpreting distortions in hierarchical organization of complex time series.Physical Review E2015,91, 030902

    Dro˙zd˙z, S.; O´ swi˛ ecimka, P . Detecting and interpreting distortions in hierarchical organization of complex time series.Physical Review E2015,91, 030902. https://doi.org/10.1103/PhysRevE.91.030902

    doi:10.1103/physreve.91.030902

  65. [66]

    Physiological time-series analysis using approximate entropy and sample entropy.American Journal of Physiology-Heart and Circulatory Physiology2000,278, H2039–H2049

    Richman, J.S.; Moorman, J.R. Physiological time-series analysis using approximate entropy and sample entropy.American Journal of Physiology-Heart and Circulatory Physiology2000,278, H2039–H2049. https: //doi.org/10.1152/ajpheart.2000.278.6.H2039

    doi:10.1152/ajpheart.2000.278.6.h2039 2000

  66. [67]

    https://www.binance.com/

    Binance. https://www.binance.com/

  67. [68]

    https://www.bitget.com/

    Bitget. https://www.bitget.com/

  68. [69]

    https://www.kucoin.com/

    KuCoin. https://www.kucoin.com/

  69. [70]

    https://www.kraken.com/

    Kraken. https://www.kraken.com/

  70. [71]

    https://www.coingecko.com/en/exchanges

    Top Crypto Exchanges Ranked by Trust Score. https://www.coingecko.com/en/exchanges

  71. [72]

    fat tails

    Laherrère, J.; Sornette, D. Stretched exponential distributions in nature and economy: "fat tails" with characteristic scales.The European Physical Journal B1998,2, 525–539

  72. [73]

    The Variation of Certain Speculative Prices.The Journal of Business1963,36, 394–419

    Mandelbrot, B. The Variation of Certain Speculative Prices.The Journal of Business1963,36, 394–419

  73. [74]

    Financial return distributions: Past, present, and covid-19.Entropy 2021,23, 884

    W ˛ atorek, M.; Kwapie ´ n, J.; Dro˙zd˙z, S. Financial return distributions: Past, present, and covid-19.Entropy 2021,23, 884. https://doi.org/10.3390/e23070884

    work page doi:10.3390/e23070884 2021

  74. [75]

    A long memory property of stock market returns and a new model

    Ding, Z.; Granger, C.W.; Engle, R.F. A long memory property of stock market returns and a new model. Journal of Empirical Finance1993,1, 83–106. https://doi.org/10.1016/0927-5398(93)90006-D

    doi:10.1016/0927-5398(93)90006-d

  75. [76]

    Dynamics, behaviours, and anomaly persistence in cryptocurrencies and equities surrounding COVID-19.Physica A2021,570, 125831

    James, N. Dynamics, behaviours, and anomaly persistence in cryptocurrencies and equities surrounding COVID-19.Physica A2021,570, 125831. https://doi.org/10.1016/j.physa.2021.125831

    arXiv 2021

  76. [77]

    Collective correlations, dynamics, and behavioural inconsistencies of the cryptocur- rency market over time.Nonlinear Dynamics2022,107, 4001–4017

    James, N.; Menzies, M. Collective correlations, dynamics, and behavioural inconsistencies of the cryptocur- rency market over time.Nonlinear Dynamics2022,107, 4001–4017. https://doi.org/10.1007/s11071-021-071 66-9

    work page doi:10.1007/s11071-021-071

  77. [78]

    Fractal properties, information theory, and market efficiency.Chaos, Solitons & Fractals 2024,180, 114543

    Brouty, X.; Garcin, M. Fractal properties, information theory, and market efficiency.Chaos, Solitons & Fractals 2024,180, 114543. https://doi.org/10.1016/j.chaos.2024.114543

    arXiv 2024

  78. [79]

    Long-range correlations in cryptocurrency markets: A multi-scale DFA approach.Physica A2025,661, 130417

    Bui, H.Q.; Schinckus, C.; Al-Jaifi, H. Long-range correlations in cryptocurrency markets: A multi-scale DFA approach.Physica A2025,661, 130417. https://doi.org/10.1016/j.physa.2025.130417

    arXiv 2025

  79. [80]

    Disentangling Sources of Multifractality in Time Series.Mathematics2025,13

    Kluszczy ´ nski, R.; Dro˙zd˙z, S.; Kwapie ´ n, J.; Stanisz, T.; W ˛ atorek, M. Disentangling Sources of Multifractality in Time Series.Mathematics2025,13. https://doi.org/10.3390/math13020205

    work page doi:10.3390/math13020205

  80. [81]

    What really causes large price changes?Quantitative Finance2004,4, 383–397

    Farmer, J.D.; Gillemot, L.; Lillo, F.; Mike, S.; Sen, A. What really causes large price changes?Quantitative Finance2004,4, 383–397. https://doi.org/10.1080/14697680400008627

    work page doi:10.1080/14697680400008627

Showing first 80 references.

This paper was first reviewed by deepseek-v4-flash on August 2, 2026.