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.

Self-similarity and quasi-idempotence in neural networks and related dynamical systems.

Self-similarity across length scales is pervasively observed in natural systems. Here, we investigate topological self-similarity in complex networks representing diverse forms of connectivity in the brain and some related dynamical systems, by considering the correlation between edges directly connecting any two nodes in a network and indirect connection between the same via all triangles spanning the rest of the network. We note that this aspect of self-similarity, which is distinct from hierarchically nested connectivity (coarse-grain similarity), is closely related to idempotence of the matrix representing the graph. We introduce two measures, ι(1) and ι(∞), which represent the element-wise correlation coefficients between the initial matrix and the ones obtained after squaring it once or infinitely many times, and term the matrices which yield large values of these parameters "quasi-idempotent". These measures delineate qualitatively different forms of "shallow" and "deep" quasi-idempotence, which are influenced by nodal strength heterogeneity. A high degree of quasi-idempotence was observed for partially synchronized mean-field Kuramoto oscillators with noise, electronic chaotic oscillators, and cultures of dissociated neurons, wherein the expression of quasi-idempotence correlated strongly with network maturity. Quasi-idempotence was also detected for macro-scale brain networks representing axonal connectivity, synchronization of slow activity fluctuations during idleness, and co-activation across experimental tasks, and preliminary data indicated that quasi-idempotence of structural connectivity may decrease with ageing. This initial study highlights that the form of network self-similarity indexed by quasi-idempotence is detectable in diverse dynamical systems, and draws attention to it as a possible basis for measures representing network "collectivity" and pattern formation.

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.

Investigating structural and functional aspects of the brain's criticality in stroke.

This paper addresses the question of the brain's critical dynamics after an injury such as a stroke. It is hypothesized that the healthy brain operates near a phase transition (critical point), which provides optimal conditions for information transmission and responses to inputs. If structural damage could cause the critical point to disappear and thus make self-organized criticality unachievable, it would offer the theoretical explanation for the post-stroke impairment of brain function. In our contribution, however, we demonstrate using network models of the brain, that the dynamics remain critical even after a stroke. In cases where the average size of the second-largest cluster of active nodes, which is one of the commonly used indicators of criticality, shows an anomalous behavior, it results from the loss of integrity of the network, quantifiable within graph theory, and not from genuine non-critical dynamics. We propose a new simple model of an artificial stroke that explains this anomaly. The proposed interpretation of the results is confirmed by an analysis of real connectomes acquired from post-stroke patients and a control group. The results presented refer to neurobiological data; however, the conclusions reached apply to a broad class of complex systems that admit a critical state.

Task-dependent fractal patterns of information processing in working memory.

We applied detrended fluctuation analysis, power spectral density, and eigenanalysis of detrended cross-correlations to investigate fMRI data representing a diurnal variation of working memory in four visual tasks: two verbal and two nonverbal. We show that the degree of fractal scaling is regionally dependent on the engagement in cognitive tasks. A particularly apparent difference was found between memorisation in verbal and nonverbal tasks. Furthermore, the detrended cross-correlations between brain areas were predominantly indicative of differences between resting state and other tasks, between memorisation and retrieval, and between verbal and nonverbal tasks. The fractal and spectral analyses presented in our study are consistent with previous research related to visuospatial and verbal information processing, working memory (encoding and retrieval), and executive functions, but they were found to be more sensitive than Pearson correlations and showed the potential to obtain other subtler results. We conclude that regionally dependent cognitive task engagement can be distinguished based on the fractal characteristics of BOLD signals and their detrended cross-correlation structure.

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.

Wavelet versus detrended fluctuation analysis of multifractal structures.

We perform a comparative study of applicability of the multifractal detrended fluctuation analysis (MFDFA) and the wavelet transform modulus maxima (WTMM) method in proper detecting of monofractal and multifractal character of data. We quantify the performance of both methods by using different sorts of artificial signals generated according to a few well-known exactly soluble mathematical models: monofractal fractional Brownian motion, bifractal Lévy flights, and different sorts of multifractal binomial cascades. Our results show that in the majority of situations in which one does not know a priori the fractal properties of a process, choosing MFDFA should be recommended. In particular, WTMM gives biased outcomes for the fractional Brownian motion with different values of Hurst exponent, indicating spurious multifractality. In some cases WTMM can also give different results if one applies different wavelets. We do not exclude using WTMM in real data analysis, but it occurs that while one may apply MFDFA in a more automatic fashion, WTMM must be applied with care. In the second part of our work, we perform an analogous analysis on empirical data coming from the American and from the German stock market. For this data both methods detect rich multifractality in terms of broad f(alpha), but MFDFA suggests that this multifractality is poorer than in the case of WTMM.

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.

Detrended cross-correlation analysis consistently extended to multifractality.

We propose an algorithm, multifractal cross-correlation analysis (MFCCA), which constitutes a consistent extension of the detrended cross-correlation analysis and is able to properly identify and quantify subtle characteristics of multifractal cross-correlations between two time series. Our motivation for introducing this algorithm is that the already existing methods, like multifractal extension, have at best serious limitations for most of the signals describing complex natural processes and often indicate multifractal cross-correlations when there are none. The principal component of the present extension is proper incorporation of the sign of fluctuations to their generalized moments. Furthermore, we present a broad analysis of the model fractal stochastic processes as well as of the real-world signals and show that MFCCA is a robust and selective tool at the same time and therefore allows for a reliable quantification of the cross-correlative structure of analyzed processes. In particular, it allows one to identify the boundaries of the multifractal scaling and to analyze a relation between the generalized Hurst exponent and the multifractal scaling parameter λ(q). This relation provides information about the character of potential multifractality in cross-correlations and thus enables a deeper insight into dynamics of the analyzed processes than allowed by any other related method available so far. By using examples of time series from the stock market, we show that financial fluctuations typically cross-correlate multifractally only for relatively large fluctuations, whereas small fluctuations remain mutually independent even at maximum of such cross-correlations. Finally, we indicate possible utility of MFCCA to study effects of the time-lagged cross-correlations.