Correlations versus noise in the NFT market.

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 inter-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 for capitalization variability exhibit a scale-free character while, for the number of transactions, they are somewhat more decentralized.

Characteristics of price related fluctuations in non-fungible token (NFT) market.

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.

Decomposing cryptocurrency high-frequency price dynamics into recurring and noisy components.

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 s, 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 manifested in three enhanced-activity phases aligning with Asian, European, and U.S. trading sessions. An intriguing pattern of activity surge in 15-min 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.

What Is Mature and What Is Still Emerging in the Cryptocurrency Market?

In relation to the traditional financial markets, the cryptocurrency market is a recent invention and the trading dynamics of all its components are readily recorded and stored. This fact opens up a unique opportunity to follow the multidimensional trajectory of its development since inception up to the present time. Several main characteristics commonly recognized as financial stylized facts of mature markets were quantitatively studied here. In particular, it is shown that the return distributions, volatility clustering effects, and even temporal multifractal correlations for a few highest-capitalization cryptocurrencies largely follow those of the well-established financial markets. The smaller cryptocurrencies are somewhat deficient in this regard, however. They are also not as highly cross-correlated among themselves and with other financial markets as the large cryptocurrencies. Quite generally, the volume V impact on price changes R appears to be much stronger on the cryptocurrency market than in the mature stock markets, and scales as R(V)∼Vα with α≳1.

Genuine multifractality in time series is due to temporal correlations.

Based on the mathematical arguments formulated within the multifractal detrended fluctuation analysis (MFDFA) approach it is shown that, in the uncorrelated time series from the Gaussian basin of attraction, the effects resembling multifractality asymptotically disappear for positive moments when the length of time series increases. A hint is given that this applies to the negative moments as well and extends to the Lévy stable regime of fluctuations. The related effects are also illustrated and confirmed by numerical simulations. This documents that the genuine multifractality in time series may only result from the long-range temporal correlations, and the fatter distribution tails of fluctuations may broaden the width of the singularity spectrum only when such correlations are present. The frequently asked question of what makes multifractality in time series-temporal correlations or broad distribution tails-is thus ill posed. In the absence of correlations only the bifractal or monofractal cases are possible. The former corresponds to the Lévy stable regime of fluctuations while the latter to the ones belonging to the Gaussian basin of attraction in the sense of the central limit theorem.

Cryptocurrencies Are Becoming Part of the World Global Financial Market.

In this study the cross-correlations between the cryptocurrency market represented by the two most liquid and highest-capitalized cryptocurrencies: bitcoin and ethereum, on the one side, and the instruments representing the traditional financial markets: stock indices, Forex, commodities, on the other side, are measured in the period: January 2020-October 2022. Our purpose is to address the question whether the cryptocurrency market still preserves its autonomy with respect to the traditional financial markets or it has already aligned with them in expense of its independence. We are motivated by the fact that some previous related studies gave mixed results. By calculating the q-dependent detrended cross-correlation coefficient based on the high frequency 10 s data in the rolling window, the dependence on various time scales, different fluctuation magnitudes, and different market periods are examined. There is a strong indication that the dynamics of the bitcoin and ethereum price changes since the March 2020 COVID-19 panic is no longer independent. Instead, it is related to the dynamics of the traditional financial markets, which is especially evident now in 2022, when the bitcoin and ethereum coupling to the US tech stocks is observed during the market bear phase. It is also worth emphasizing that the cryptocurrencies have begun to react to the economic data such as the Consumer Price Index readings in a similar way as traditional instruments. Such a spontaneous coupling of the so far independent degrees of freedom can be interpreted as a kind of phase transition that resembles the collective phenomena typical for the complex systems. Our results indicate that the cryptocurrencies cannot be considered as a safe haven for the financial investments.

Analysis of inter-transaction time fluctuations in the cryptocurrency market.

We analyze tick-by-tick data representing major cryptocurrencies traded on some different cryptocurrency trading platforms. We focus on such quantities like the inter-transaction times, the number of transactions in time unit, the traded volume, and volatility. We show that the inter-transaction times show long-range power-law autocorrelations. These lead to multifractality expressed by the right-side asymmetry of the singularity spectra f ( α ) indicating that the periods of increased market activity are characterized by richer multifractality compared to the periods of quiet market. We also show that neither the stretched exponential distribution nor the power-law-tail distribution is able to model universally the cumulative distribution functions of the quantities considered in this work. For each quantity, some data sets can be modeled by the former and some data sets by the latter, while both fail in other cases. An interesting, yet difficult to account for, observation is that parallel data sets from different trading platforms can show disparate statistical properties.

Cryptocurrency Market Consolidation in 2020-2021.

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.

Financial Return Distributions: Past, Present, and COVID-19.

We analyze the price return distributions of currency exchange rates, cryptocurrencies, and contracts for differences (CFDs) representing stock indices, stock shares, and commodities. Based on recent data from the years 2017-2020, we model tails of the return distributions at different time scales by using power-law, stretched exponential, and q-Gaussian functions. We focus on the fitted function parameters and how they change over the years by comparing our results with those from earlier studies and find that, on the time horizons of up to a few minutes, the so-called "inverse-cubic power-law" still constitutes an appropriate global reference. However, we no longer observe the hypothesized universal constant acceleration of the market time flow that was manifested before in an ever faster convergence of empirical return distributions towards the normal distribution. Our results do not exclude such a scenario but, rather, suggest that some other short-term processes related to a current market situation alter market dynamics and may mask this scenario. Real market dynamics is associated with a continuous alternation of different regimes with different statistical properties. An example is the COVID-19 pandemic outburst, which had an enormous yet short-time impact on financial markets. We also point out that two factors-speed of the market time flow and the asset cross-correlation magnitude-while related (the larger the speed, the larger the cross-correlations on a given time scale), act in opposite directions with regard to the return distribution tails, which can affect the expected distribution convergence to the normal distribution.

Complexity in Economic and Social Systems.

During recent years we have witnessed a systematic progress in the understanding of complex systems, both in the case of particular systems that are classified into this group and, in general, as regards the phenomenon of complexity [...].

Complexity in Economic and Social Systems: Cryptocurrency Market at around COVID-19.

Social systems are characterized by an enormous network of connections and factors that can influence the structure and dynamics of these systems. Among them the whole economical sphere of human activity seems to be the most interrelated and complex. All financial markets, including the youngest one, the cryptocurrency market, belong to this sphere. The complexity of the cryptocurrency market can be studied from different perspectives. First, the dynamics of the cryptocurrency exchange rates to other cryptocurrencies and fiat currencies can be studied and quantified by means of multifractal formalism. Second, coupling and decoupling of the cryptocurrencies and the conventional assets can be investigated with the advanced cross-correlation analyses based on fractal analysis. Third, an internal structure of the cryptocurrency market can also be a subject of analysis that exploits, for example, a network representation of the market. In this work, we approach the subject from all three perspectives based on data from a recent time interval between January 2019 and June 2020. This period includes the peculiar time of the Covid-19 pandemic; therefore, we pay particular attention to this event and investigate how strong its impact on the structure and dynamics of the market was. Besides, the studied data covers a few other significant events like double bull and bear phases in 2019. We show that, throughout the considered interval, the exchange rate returns were multifractal with intermittent signatures of bifractality that can be associated with the most volatile periods of the market dynamics like a bull market onset in April 2019 and the Covid-19 outburst in March 2020. The topology of a minimal spanning tree representation of the market also used to alter during these events from a distributed type without any dominant node to a highly centralized type with a dominating hub of USDT. However, the MST topology during the pandemic differs in some details from other volatile periods.

Competition of noise and collectivity in global cryptocurrency trading: Route to a self-contained market.

Cross correlations in fluctuations of the daily exchange rates within the basket of the 100 highest-capitalization cryptocurrencies over the period October 1, 2015-March 31, 2019 are studied. The corresponding dynamics predominantly involve one leading eigenvalue of the correlation matrix, while the others largely coincide with those of Wishart random matrices. However, the magnitude of the principal eigenvalue, and thus the degree of collectivity, strongly depends on which cryptocurrency is used as a base. It is largest when the base is the most peripheral cryptocurrency; when more significant ones are taken into consideration, its magnitude systematically decreases, nevertheless preserving a sizable gap with respect to the random bulk, which in turn indicates that the organization of correlations becomes more heterogeneous. This finding provides a criterion for recognizing which currencies or cryptocurrencies play a dominant role in the global cryptomarket. The present study shows that over the period under consideration, the Bitcoin (BTC) predominates, hallmarking exchange rate dynamics at least as influential as the U.S. dollar (USD). Even more, the BTC started dominating around the year 2017, while other cryptocurrencies, such as the Ethereum and even Ripple, assumed similar trends. At the same time, the USD, an original value determinant for the cryptocurrency market, became increasingly disconnected, and its related characteristics eventually started approaching those of a fictitious currency. These results are strong indicators of incipient independence of the global cryptocurrency market, delineating a self-contained trade resembling the Forex.

Warped phase coherence: An empirical synchronization measure combining phase and amplitude information.

The entrainment between weakly coupled nonlinear oscillators, as well as between complex signals such as those representing physiological activity, is frequently assessed in terms of whether a stable relationship is detectable between the instantaneous phases extracted from the measured or simulated time-series via the analytic signal. Here, we demonstrate that adding a possibly complex constant value to this normally null-mean signal has a non-trivial warping effect. Among other consequences, this introduces a level of sensitivity to the amplitude fluctuations and average relative phase. By means of simulations of Rössler systems and experiments on single-transistor oscillator networks, it is shown that the resulting coherence measure may have an empirical value in improving the inference of the structural couplings from the dynamics. When tentatively applied to the electroencephalogram recorded while performing imaginary and real movements, this straightforward modification of the phase locking value substantially improved the classification accuracy. Hence, its possible practical relevance in brain-computer and brain-machine interfaces deserves consideration.

High-dimensional dynamics in a single-transistor oscillator containing Feynman-Sierpinski resonators: Effect of fractal depth and irregularity.

Fractal structures pervade nature and are receiving increasing engineering attention towards the realization of broadband resonators and antennas. We show that fractal resonators can support the emergence of high-dimensional chaotic dynamics even in the context of an elementary, single-transistor oscillator circuit. Sierpinski gaskets of variable depth are constructed using discrete capacitors and inductors, whose values are scaled according to a simple sequence. It is found that in regular fractals of this kind, each iteration effectively adds a conjugate pole/zero pair, yielding gradually more complex and broader frequency responses, which can also be implemented as much smaller Foster equivalent networks. The resonators are instanced in the circuit as one-port devices, replacing the inductors found in the initial version of the oscillator. By means of a highly simplified numerical model, it is shown that increasing the fractal depth elevates the dimension of the chaotic dynamics, leading to high-order hyperchaos. This result is overall confirmed by SPICE simulations and experiments, which however also reveal that the non-ideal behavior of physical components hinders obtaining high-dimensional dynamics. The issue could be practically mitigated by building the Foster equivalent networks rather than the verbatim fractals. Furthermore, it is shown that considerably more complex resonances, and consequently richer dynamics, can be obtained by rendering the fractal resonators irregular through reshuffling the inductors, or even by inserting a limited number of focal imperfections. The present results draw attention to the potential usefulness of fractal resonators for generating high-dimensional chaotic dynamics, and underline the importance of irregularities and component non-idealities.

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

Based on 1-min price changes recorded since year 2012, the fluctuation properties of the rapidly emerging Bitcoin market are assessed over chosen sub-periods, in terms of return distributions, volatility autocorrelation, Hurst exponents, and multiscaling effects. The findings are compared to the stylized facts of mature world markets. While early trading was affected by system-specific irregularities, it is found that over the months preceding April 2018 all these statistical indicators approach the features hallmarking maturity. This can be taken as an indication that the Bitcoin market, and possibly other cryptocurrencies, carry concrete potential of imminently becoming a regular market, alternative to the foreign exchange. Since high-frequency price data are available since the beginning of trading, the Bitcoin offers a unique window into the statistical characteristics of a market maturation trajectory.

Apparent remote synchronization of amplitudes: A demodulation and interference effect.

A form of "remote synchronization" was recently described, wherein amplitude fluctuations across a ring of non-identical, non-linear electronic oscillators become entrained into spatially-structured patterns. According to linear models and mutual information, synchronization and causality dip at a certain distance, then recover before eventually fading. Here, the underlying mechanism is finally elucidated through novel experiments and simulations. The system non-linearity is found to have a dual role: it supports chaotic dynamics, and it enables the energy exchange between the lower and higher sidebands of a predominant frequency. This frequency acts as carrier signal in an arrangement resembling standard amplitude modulation, wherein the lower sideband and the demodulated baseband signals spectrally overlap. Due to a spatially-dependent phase relationship, at a certain distance near-complete destructive interference occurs between them, causing the observed dip. Methods suitable for detecting non-trivial entrainment, such as transfer entropy and the auxiliary system approach, nevertheless, reveal that synchronization and causality actually decrease with distance monotonically. Remoteness is, therefore, arguably only apparent, as also reflected in the propagation of external perturbations. These results demonstrate a complex mechanism of dynamical interdependence, and exemplify how it can lead to incorrectly inferring synchronization and causality.

Atypical transistor-based chaotic oscillators: Design, realization, and diversity.

In this paper, we show that novel autonomous chaotic oscillators based on one or two bipolar junction transistors and a limited number of passive components can be obtained via random search with suitable heuristics. Chaos is a pervasive occurrence in these circuits, particularly after manual adjustment of a variable resistor placed in series with the supply voltage source. Following this approach, 49 unique circuits generating chaotic signals when physically realized were designed, representing the largest collection of circuits of this kind to date. These circuits are atypical as they do not trivially map onto known topologies or variations thereof. They feature diverse spectra and predominantly anti-persistent monofractal dynamics. Notably, we recurrently found a circuit comprising one resistor, one transistor, two inductors, and one capacitor, which generates a range of attractors depending on the parameter values. We also found a circuit yielding an irregular quantized spike-train resembling some aspects of neural discharge and another one generating a double-scroll attractor, which represent the smallest known transistor-based embodiments of these behaviors. Through three representative examples, we additionally show that diffusive coupling of heterogeneous oscillators of this kind may give rise to complex entrainment, such as lag synchronization with directed information transfer and generalized synchronization. The replicability and reproducibility of the experimental findings are good.

Minimum spanning tree filtering of correlations for varying time scales and size of fluctuations.

Based on a recently proposed q-dependent detrended cross-correlation coefficient, ρ_{q} [J. Kwapien, P. Oswiecimka, and S. Drozdz, Phys. Rev. E 92, 052815 (2015)PLEEE81539-375510.1103/PhysRevE.92.052815], we generalize the concept of the minimum spanning tree (MST) by introducing a family of q-dependent minimum spanning trees (qMSTs) that are selective to cross-correlations between different fluctuation amplitudes and different time scales of multivariate data. They inherit this ability directly from the coefficients ρ_{q}, which are processed here to construct a distance matrix being the input to the MST-constructing Kruskal's algorithm. The conventional MST with detrending corresponds in this context to q=2. In order to illustrate their performance, we apply the qMSTs to sample empirical data from the American stock market and discuss the results. We show that the qMST graphs can complement ρ_{q} in disentangling "hidden" correlations that cannot be observed in the MST graphs based on ρ_{DCCA}, and therefore, they can be useful in many areas where the multivariate cross-correlations are of interest. As an example, we apply this method to empirical data from the stock market and show that by constructing the qMSTs for a spectrum of q values we obtain more information about the correlation structure of the data than by using q=2 only. More specifically, we show that two sets of signals that differ from each other statistically can give comparable trees for q=2, while only by using the trees for q≠2 do we become able to distinguish between these sets. We also show that a family of qMSTs for a range of q expresses the diversity of correlations in a manner resembling the multifractal analysis, where one computes a spectrum of the generalized fractal dimensions, the generalized Hurst exponents, or the multifractal singularity spectra: the more diverse the correlations are, the more variable the tree topology is for different q's. As regards the correlation structure of the stock market, our analysis exhibits that the stocks belonging to the same or similar industrial sectors are correlated via the fluctuations of moderate amplitudes, while the largest fluctuations often happen to synchronize in those stocks that do not necessarily belong to the same industry.

Multifractal cross-correlation effects in two-variable time series of complex network vertex observables.

We investigate the scaling of the cross-correlations calculated for two-variable time series containing vertex properties in the context of complex networks. Time series of such observables are obtained by means of stationary, unbiased random walks. We consider three vertex properties that provide, respectively, short-, medium-, and long-range information regarding the topological role of vertices in a given network. In order to reveal the relation between these quantities, we applied the multifractal cross-correlation analysis technique, which provides information about the nonlinear effects in coupling of time series. We show that the considered network models are characterized by unique multifractal properties of the cross-correlation. In particular, it is possible to distinguish between Erdös-Rényi, Barabási-Albert, and Watts-Strogatz networks on the basis of fractal cross-correlation. Moreover, the analysis of protein contact networks reveals characteristics shared with both scale-free and small-world models.

Detrended fluctuation analysis made flexible to detect range of cross-correlated fluctuations.

The detrended cross-correlation coefficient ρ(DCCA) has recently been proposed to quantify the strength of cross-correlations on different temporal scales in bivariate, nonstationary time series. It is based on the detrended cross-correlation and detrended fluctuation analyses (DCCA and DFA, respectively) and can be viewed as an analog of the Pearson coefficient in the case of the fluctuation analysis. The coefficient ρ(DCCA) works well in many practical situations but by construction its applicability is limited to detection of whether two signals are generally cross-correlated, without the possibility to obtain information on the amplitude of fluctuations that are responsible for those cross-correlations. In order to introduce some related flexibility, here we propose an extension of ρ(DCCA) that exploits the multifractal versions of DFA and DCCA: multifractal detrended fluctuation analysis and multifractal detrended cross-correlation analysis, respectively. The resulting new coefficient ρ(q) not only is able to quantify the strength of correlations but also allows one to identify the range of detrended fluctuation amplitudes that are correlated in two signals under study. We show how the coefficient ρ(q) works in practical situations by applying it to stochastic time series representing processes with long memory: autoregressive and multiplicative ones. Such processes are often used to model signals recorded from complex systems and complex physical phenomena like turbulence, so we are convinced that this new measure can successfully be applied in time-series analysis. In particular, we present an example of such application to highly complex empirical data from financial markets. The present formulation can straightforwardly be extended to multivariate data in terms of the q-dependent counterpart of the correlation matrices and then to the network representation.