Jakub Zwydak, Marcin Wątorek, Jarosław Kwapień, Stanisław Drożdż
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.
Collective dynamics in financial markets can emerge through synchronized movements of large groups of assets. Motivated by analogies with interacting many-body systems, we introduce a spin-lattice representation for analyzing collective states in cryptocurrency markets. In this framework, assets are encoded as binary spin variables according to the sign of their returns, while correlations between assets determine effective interaction strengths. A correlation-based breadth-first search (CBFS) procedure embeds 169 cryptocurrencies into a $13 \times 13$ lattice, enabling the construction of an Ising-like Hamiltonian describing the market configuration, which we call the \emph{Market Crystal}. Macroscopic observables such as magnetization and energy provide a statistical-mechanical characterization of collective market states. The resulting phase-space structure highlights regimes of strong alignment and fragmentation among assets, with an energy--magnetization pattern suggestive of predominantly ferromagnetic interactions. This framework offers a statistical-mechanical viewpoint for studying collective behavior in financial systems.
Deep Nath, Paolo Tasca, Nikhil Vadgama, Marco Alberto Javarone
Quantifying structural stress in transaction networks requires metrics that capture structural organization beyond transaction volume alone. In this work, we introduce the Inefficiency Metric, a deterministic indicator designed to characterize the routing structure of capital flows in decentralized systems. Using Principal Component Analysis and Pearson correlation matrices computed from a six-year Hedera transaction dataset, we identify two dominant and largely independent structural dimensions: the effective diameter, related to the spatial extension of transaction propagation, and the closeness centrality, associated with the efficiency of network-level flow processing. The proposed metric reveals significant topological fluctuations associated with major macroeconomic and ecosystem-level events. Increased inefficiency is observed during periods marked by intermediary fragmentation or rapid smart-contract expansion, whereas lower inefficiency corresponds to phases of network compaction during market stress or institutional concentration. Comparison with a seven-dimensional Isolation Forest approach shows that the metric effectively captures severe multidimensional anomalies while preserving a clear structural interpretation. Overall, these results provide a physics-inspired framework for relating the large-scale organization of decentralized transaction networks to observable economic dynamics.
Stanisław Drożdż, Paweł Jarosz, Jarosław Kwapień, Maria Skupień · 5 authors
Correlations in complex systems are often obscured by nonstationarity, long-range memory, and heavy-tailed fluctuations, which limit the usefulness of traditional covariance-based analyses. To address these challenges, we construct scale- and fluctuation-dependent correlation matrices using the multifractal detrended cross-correlation coefficient ρr that selectively emphasizes fluctuations of different amplitudes. We examine the spectral properties of these detrended correlation matrices and compare them to the spectral properties of the matrices calculated in the same way from synthetic Gaussian and q-Gaussian signals. Our results show that detrending, heavy tails, and the fluctuation-order parameter r jointly produce spectra, which substantially depart from the random case even under the absence of cross-correlations in time series. Applying this framework to one-minute returns of 140 major cryptocurrencies from 2021 to 2024 reveals robust collective modes, including a dominant market factor and several sectoral components whose strength depends on the analyzed scale and fluctuation order. After filtering out the market mode, the empirical eigenvalue bulk aligns closely with the limit of random detrended cross-correlations, enabling clear identification of structurally significant outliers. Overall, the study provides a refined spectral baseline for detrended cross-correlations and offers a promising tool for distinguishing genuine interdependencies from noise in complex, nonstationary, heavy-tailed systems.
Kundan Mukhia, Buddha Nath Sharma, Salam Rabindrajit Luwang, Md. Nurujjaman · 7 authors
We study how the 2024 U.S. presidential election, viewed as a major political risk event, affected cryptocurrency markets by distinguishing human-driven peer-to-peer stablecoin transactions from automated algorithmic activity. Using structural break analysis, we find that human-driven Ethereum Request for Comment 20 (ERC-20) transactions shifted on November 3, two days before the election, while exchange trading volumes reacted only on Election Day. Automated smart-contract activity adjusted much later, with structural breaks appearing in January 2025. We validate these shifts using surrogate-based robustness tests. Complementary energy-spectrum analysis of Bitcoin and Ethereum identifies pronounced post-election turbulence, and a structural vector autoregression confirms a regime shift in stablecoin dynamics. Overall, human-driven stablecoin flows act as early-warning indicators of political stress, preceding both exchange behavior and algorithmic responses.
Stanisław Drożdż, Robert Kluszczyński, Jarosław Kwapień, Marcin Wątorek
Multifractality in time series analysis characterizes the presence of multiple scaling exponents, indicating heterogeneous temporal structures and complex dynamical behaviors beyond simple monofractal models. In the context of digital currency markets, multifractal properties arise due to the interplay of long-range temporal correlations and heavy-tailed distributions of returns, reflecting intricate market microstructure and trader interactions. Incorporating multifractal analysis into the modeling of cryptocurrency price dynamics enhances the understanding of market inefficiencies, may improve volatility forecasting and facilitate the detection of critical transitions or regime shifts. Based on the multifractal cross-correlation analysis (MFCCA) whose spacial case is the multifractal detrended fluctuation analysis (MFDFA), as the most commonly used practical tools for quantifying multifractality, in the present contribution a recently proposed method of disentangling sources of multifractality in time series was applied to the most representative instruments from the digital market. They include Bitcoin (BTC), Ethereum (ETH), decentralized exchanges (DEX) and non-fungible tokens (NFT). The results indicate the significant role of heavy tails in generating a broad multifractal spectrum. However, they also clearly demonstrate that the primary source of multifractality are temporal correlations in the series, and without them, multifractality fades out. It appears characteristic that these temporal correlations, to a large extent, do not depend on the thickness of the tails of the fluctuation distribution. These observations, made here in the context of the digital currency market, provide a further strong argument for the validity of the proposed methodology of disentangling sources of multifractality in time series.
Marcin Wątorek, Marija Bezbradica, Martin Crane, Jarosław Kwapień · 5 authors
Based on the cryptocurrency market dynamics, this study presents a general methodology for analyzing evolving correlation structures in complex systems using the $q$-dependent detrended cross-correlation coefficient ρ(q,s). By extending traditional metrics, this approach captures correlations at varying fluctuation amplitudes and time scales. The method employs $q$-dependent minimum spanning trees ($q$MSTs) to visualize evolving network structures. Using minute-by-minute exchange rate data for 140 cryptocurrencies on Binance (Jan 2021-Oct 2024), a rolling window analysis reveals significant shifts in $q$MSTs, notably around April 2022 during the Terra/Luna crash. Initially centralized around Bitcoin (BTC), the network later decentralized, with Ethereum (ETH) and others gaining prominence. Spectral analysis confirms BTC's declining dominance and increased diversification among assets. A key finding is that medium-scale fluctuations exhibit stronger correlations than large-scale ones, with $q$MSTs based on the latter being more decentralized. Properly exploiting such facts may offer the possibility of a more flexible optimal portfolio construction. Distance metrics highlight that major disruptions amplify correlation differences, leading to fully decentralized structures during crashes. These results demonstrate $q$MSTs' effectiveness in uncovering fluctuation-dependent correlations, with potential applications beyond finance, including biology, social and other complex systems.
We employed Multifractal Detrended Fluctuation Analysis (MF-DFA) and Refined Composite Multiscale Sample Entropy (RCMSE) to investigate the complexity of Bitcoin, GBP/USD, gold, and natural gas price log-return time series. This study provides a comparative analysis of these markets and offers insights into their predictability and associated risks. Each tool presents a unique method to quantify time series complexity. The RCMSE and MF-DFA methods demonstrate a higher complexity for the Bitcoin time series than others. It is discussed that the increased complexity of Bitcoin may be attributable to the presence of higher nonlinear correlations within its log-return time series.
Erveton P. Pinto, Marcelo A. Pires, Rone N. da Silva, Sı́lvio M. Duarte Queirós
We report the first application of a tailored Complexity-Entropy Plane designed for binary sequences and structures. We do so by considering the daily up/down price fluctuations of the largest cryptocurrencies in terms of capitalization (stable-coins excluded) that are worth $circa \,\, 90 \%$ of the total crypto market capitalization. With that, we focus on the basic elements of price motion that compare with the random walk backbone features associated with mathematical properties of the Efficient Market Hypothesis. From the location of each crypto on the Binary Complexity-Plane (BiCEP) we define an inefficiency score, $\mathcal I$, and rank them accordingly. The results based on the BiCEP analysis, which we substantiate with statistical testing, indicate that only Shiba Inu (SHIB) is significantly inefficient, whereas the largest stake of crypto trading is reckoned to operate in close-to-efficient conditions. Generically, our $\mathcal I$-based ranking hints the design and consensus architecture of a crypto is at least as relevant to efficiency as the features that are usually taken into account in the appraisal of the efficiency of financial instruments, namely canonical fiat money. Lastly, this set of results supports the validity of the binary complexity analysis.
Some blockchain networks employ a distributed consensus algorithm featuring Byzantine fault tolerance. Notably, certain public chains, such as Cosmos and Tezos, which operate on a proof-of-stake mechanism, have adopted this algorithm. While it is commonly assumed that these blockchains maintain a nearly constant block creation time, empirical analysis reveals fluctuations in this interval; this phenomenon has received limited attention. In this paper, we propose a mathematical model to account for the processes of block propagation and validation within Byzantine fault-tolerant consensus blockchains, aiming to theoretically analyze the probability distribution of block time. First, we propose stochastic processes governing the broadcasting communications among validator nodes. Consequently, we theoretically demonstrate that the probability distribution of broadcast time among validator nodes adheres to the Gumbel distribution. This finding indicates that the distribution of block time typically arises from convolving multiple Gumbel distributions. Additionally, we derive an approximate formula for the block time distribution suitable for data analysis purposes. By fitting this approximation to real-world block time data, we demonstrate the consistent estimation of block time distribution parameters,
Marcin Wątorek, Paweł Szydło, Jarosław Kwapień, Stanisław Drożdż
The non-fungible token (NFT) market emerges as a recent trading innovation leveraging blockchain technology, mirroring the dynamics of the cryptocurrency market. The current study is based on the capitalization changes and transaction volumes across a large number of token collections on the Ethereum platform. In order to deepen the understanding of the market dynamics, the collection-collection dependencies are examined by using the multivariate formalism of detrended correlation coefficient and correlation matrix. It appears that correlation strength is lower here than that observed in previously studied markets. Consequently, the eigenvalue spectra of the correlation matrix more closely follow the Marchenko-Pastur distribution, still, some departures indicating the existence of correlations remain. The comparison of results obtained from the correlation matrix built from the Pearson coefficients and, independently, from the detrended cross-correlation coefficients suggests that the global correlations in the NFT market arise from higher frequency fluctuations. Corresponding minimal spanning trees (MSTs) for capitalization variability exhibit a scale-free character while, for the number of transactions, they are somewhat more decentralized.
Paweł Szydło, Marcin Wątorek, Jarosław Kwapień, Stanisław Drożdż
A non-fungible token (NFT) market is a new trading invention based on the blockchain technology, which parallels the cryptocurrency market. In the present work, we study capitalization, floor price, the number of transactions, the inter-transaction times, and the transaction volume value of a few selected popular token collections. The results show that the fluctuations of all these quantities are characterized by heavy-tailed probability distribution functions, in most cases well described by the stretched exponentials, with a trace of power-law scaling at times, long-range memory, persistence, and in several cases even the fractal organization of fluctuations, mostly restricted to the larger fluctuations, however. We conclude that the NFT market-even though young and governed by somewhat different mechanisms of trading-shares several statistical properties with the regular financial markets. However, some differences are visible in the specific quantitative indicators.
Marcin Wątorek, Maria Skupień, Jarosław Kwapień, Stanisław Drożdż
This paper investigates the temporal patterns of activity in the cryptocurrency market with a focus on Bitcoin, Ethereum, Dogecoin, and WINkLink from January 2020 to December 2022. Market activity measures - logarithmic returns, volume, and transaction number, sampled every 10 seconds, were divided into intraday and intraweek periods and then further decomposed into recurring and noise components via correlation matrix formalism. The key findings include the distinctive market behavior from traditional stock markets due to the nonexistence of trade opening and closing. This was manifest in three enhanced-activity phases aligning with Asian, European, and U.S. trading sessions. An intriguing pattern of activity surge in 15-minute intervals, particularly at full hours, was also noticed, implying the potential role of algorithmic trading. Most notably, recurring bursts of activity in bitcoin and ether were identified to coincide with the release times of significant U.S. macroeconomic reports such as Nonfarm payrolls, Consumer Price Index data, and Federal Reserve statements. The most correlated daily patterns of activity occurred in 2022, possibly reflecting the documented correlations with U.S. stock indices in the same period. Factors that are external to the inner market dynamics are found to be responsible for the repeatable components of the market dynamics, while the internal factors appear to be substantially random, which manifests itself in a good agreement between the empirical eigenvalue distributions in their bulk and the random matrix theory predictions expressed by the Marchenko-Pastur distribution. The findings reported support the growing integration of cryptocurrencies into the global financial markets.
Marcin Wątorek, Jarosław Kwapień, Stanisław Drożdż
Unlike price fluctuations, the temporal structure of cryptocurrency trading has seldom been a subject of systematic study. In order to fill this gap, we analyse detrended correlations of the price returns, the average number of trades in time unit, and the traded volume based on high-frequency data representing two major cryptocurrencies: bitcoin and ether. We apply the multifractal detrended cross-correlation analysis, which is considered the most reliable method for identifying nonlinear correlations in time series. We find that all the quantities considered in our study show an unambiguous multifractal structure from both the univariate (auto-correlation) and bivariate (cross-correlation) perspectives. We looked at the bitcoin--ether cross-correlations in simultaneously recorded signals, as well as in time-lagged signals, in which a time series for one of the cryptocurrencies is shifted with respect to the other. Such a shift suppresses the cross-correlations partially for short time scales, but does not remove them completely. We did not observe any qualitative asymmetry in the results for the two choices of a leading asset. The cross-correlations for the simultaneous and lagged time series became the same in magnitude for the sufficiently long scales.
Jarosław Kwapień, Marcin Wątorek, Stanisław Drożdż
Time series of price returns for 80 of the most liquid cryptocurrencies listed on Binance are investigated for the presence of detrended cross-correlations. A spectral analysis of the detrended correlation matrix and a topological analysis of the minimal spanning trees calculated based on this matrix are applied for different positions of a moving window. The cryptocurrencies become more strongly cross-correlated among themselves than they used to be before. The average cross-correlations increase with time on a specific time scale in a way that resembles the Epps effect amplification when going from past to present. The minimal spanning trees also change their topology and, for the short time scales, they become more centralized with increasing maximum node degrees, while for the long time scales they become more distributed, but also more correlated at the same time. Apart from the inter-market dependencies, the detrended cross-correlations between the cryptocurrency market and some traditional markets, like the stock markets, commodity markets, and Forex, are also analyzed. The cryptocurrency market shows higher levels of cross-correlations with the other markets during the same turbulent periods, in which it is strongly cross-correlated itself.
Nicolò Vallarano, Matteo Bruno, Emiliano Marchese, Giuseppe Di Trapani · 8 authors
Exponential Random Graph Models (ERGMs) have gained increasing popularity over the years. Rooted into statistical physics, the ERGMs framework has been successfully employed for reconstructing networks, detecting statistically significant patterns in graphs, counting networked configurations with given properties. From a technical point of view, the ERGMs workflow is defined by two subsequent optimization steps: the first one concerns the maximization of Shannon entropy and leads to identify the functional form of the ensemble probability distribution that is maximally non-committal with respect to the missing information; the second one concerns the maximization of the likelihood function induced by this probability distribution and leads to its numerical determination. This second step translates into the resolution of a system of $O(N)$ non-linear, coupled equations (with $N$ being the total number of nodes of the network under analysis), a problem that is affected by three main issues, i.e. accuracy, speed and scalability. The present paper aims at addressing these problems by comparing the performance of three algorithms (i.e. Newton's method, a quasi-Newton method and a recently-proposed fixed-point recipe) in solving several ERGMs, defined by binary and weighted constraints in both a directed and an undirected fashion. While Newton's method performs best for relatively little networks, the fixed-point recipe is to be preferred when large configurations are considered, as it ensures convergence to the solution within seconds for networks with hundreds of thousands of nodes (e.g. the Internet, Bitcoin). We attach to the paper a Python code implementing the three aforementioned algorithms on all the ERGMs considered in the present work.
In this article, we discuss the random graph, Barabási-Albert (BA) model, and lattice networks from a unified view point, with the parameter $ω$ with values $1,0,-1$ characterizing these networks, respectively. The parameter is related to the preferential attachment of nodes in the networks and has different weights for the incoming and outgoing links. In addition, we discuss the correspondence between quantum statistics and the networks. Positive and negative $ω$ correspond to Bose and Fermi-like statistics, respectively, and we obtain the distribution that connects the two. When $ω$ is positive, it is related to the threshold of Bose-Einstein condensation (BEC). As $ω$ decreases, the area of the BEC phase is narrowed, and disappears in the limit $ω=0$. When $ω$ is negative, nodes have limits in the number of attachments for newly added nodes (outgoing links), which corresponds to Fermi statistics. We also observe the Fermi degeneracy of the network. When $ω=-1$, a standard Fermion-like network is observed. Fermion networks are realized in the cryptocurrency network "Tangle."
Information transfer between time series is calculated using the asymmetric information-theoretic measure known as transfer entropy. Geweke’s autoregressive formulation of Granger causality is used to compute linear transfer entropy, and Schreiber’s general, non-parametric, information-theoretic formulation is used to quantify nonlinear transfer entropy. We first validate these measures against synthetic data. Then we apply these measures to detect statistical causality between social sentiment changes and cryptocurrency returns. We validate results by performing permutation tests by shuffling the time series, and calculate the Z -score. We also investigate different approaches for partitioning in non-parametric density estimation which can improve the significance. Using these techniques on sentiment and price data over a 48-month period to August 2018, for four major cryptocurrencies, namely bitcoin (BTC), ripple (XRP), litecoin (LTC) and ethereum (ETH), we detect significant information transfer, on hourly timescales, with greater net information transfer from sentiment to price for XRP and LTC, and instead from price to sentiment for BTC and ETH. We report the scale of nonlinear statistical causality to be an order of magnitude larger than the linear case.
This letter investigates the dynamic relationship between market efficiency, liquidity, and multifractality of Bitcoin. We find that before 2013 liquidity is low and the Hurst exponent is less than 0.5, indicating that the Bitcoin time series is anti-persistent. After 2013, as liquidity increased, the Hurst exponent rose to approximately 0.5, improving market efficiency. For several periods, however, the Hurst exponent was found to be significantly less than 0.5, making the time series anti-persistent during those periods. We also investigate the multifractal degree of the Bitcoin time series using the generalized Hurst exponent and find that the multifractal degree is related to market efficiency in a non-linear manner.
The Bitcoin has emerged as a fascinating phenomenon in the Financial markets. Without any central authority issuing the currency, the Bitcoin has been associated with controversy ever since its popularity, accompanied by increased public interest, reached high levels. Here, we contribute to the discussion by examining the potential drivers of Bitcoin prices, ranging from fundamental sources to speculative and technical ones, and we further study the potential influence of the Chinese market. The evolution of relationships is examined in both time and frequency domains utilizing the continuous wavelets framework, so that we not only comment on the development of the interconnections in time but also distinguish between short-term and long-term connections. We find that the Bitcoin forms a unique asset possessing properties of both a standard financial asset and a speculative one.