Compression complexity with ordinal patterns for robust causal inference in irregularly sampled time series.

Distinguishing cause from effect is a scientific challenge resisting solutions from mathematics, statistics, information theory and computer science. Compression-Complexity Causality (CCC) is a recently proposed interventional measure of causality, inspired by Wiener-Granger's idea. It estimates causality based on change in dynamical compression-complexity (or compressibility) of the effect variable, given the cause variable. CCC works with minimal assumptions on given data and is robust to irregular-sampling, missing-data and finite-length effects. However, it only works for one-dimensional time series. We propose an ordinal pattern symbolization scheme to encode multidimensional patterns into one-dimensional symbolic sequences, and thus introduce the Permutation CCC (PCCC). We demonstrate that PCCC retains all advantages of the original CCC and can be applied to data from multidimensional systems with potentially unobserved variables which can be reconstructed using the embedding theorem. PCCC is tested on numerical simulations and applied to paleoclimate data characterized by irregular and uncertain sampling and limited numbers of samples.

Dynamics of social balance on networks: The emergence of multipolar societies.

Within the context of social balance theory, much attention has been paid to the attainment and stability of unipolar or bipolar societies. However, multipolar societies are commonplace in the real world, despite the fact that the mechanism of their emergence is much less explored. Here, we investigate the evolution of a society of interacting agents with friendly (positive) and enmity (negative) relations into a final stable multipolar state. Triads are assigned energy according to the degree of tension they impose on the network. Agents update their connections to decrease the total energy (tension) of the system, on average. Our approach is to consider a variable energy ε∈[0,1] for triads which are entirely made of negative relations. We show that the final state of the system depends on the initial density of the friendly links ρ_{0}. For initial densities greater than an ε-dependent threshold ρ_{0}^{c}(ε), a unipolar (paradise) state is reached. However, for ρ_{0}≤ρ_{0}^{c}(ε), multipolar and bipolar states can emerge. We observe that the number of stable final poles increases with decreasing ε where the first transition from bipolar to multipolar society occurs at ε^{*}≈0.67. We end the paper by providing a mean-field calculation that provides an estimate for the critical (ε dependent) initial positive link density, which is consistent with our simulations.

Reply to "Comment on 'Phase transition in a network model of social balance with Glauber dynamics' ".

Recently, we introduced a stochastic social balance model with Glauber dynamics which takes into account the role of randomness in the individual's behavior [Phys. Rev. E 100, 022303 (2019)2470-004510.1103/PhysRevE.100.022303]. One important finding of our study was a phase transition from a balance state to an imbalance state as the randomness crosses a critical value, which was shown to vanish in the thermodynamic limit. In a similar study [Malarz and Kulakowski, Phys. Rev. E 103, 066301 (2021)10.1103/PhysRevE.103.066301], it was shown that the critical randomness tends to infinity as the system size diverges. This led the authors to question the appropriateness of the results in our Monte Carlo simulations, when compared with the non-normalized form, used in their work. The normalized form of energy in our model is, in fact, a common choice when one deals with systems comprising long-range interactions. Here, we show how their probabilistic definition leads to vanishing possibility of forming negative bonds, thus leading to a frozen ordered (paradise) state for any amount of finite randomness (temperature) for large enough system size. On the other hand, in the same large system size limit, our model is unstable to thermal randomness due to global, long-range effect of changing a bond's sign. We also address the rule of different updating mechanisms (synchronous vs sequential) in the two models. We finally discuss the distinction between the balanced states reached by each model and provide arguments for social relevance of our model.

Causality and Information Transfer Between the Solar Wind and the Magnetosphere-Ionosphere System.

An information-theoretic approach for detecting causality and information transfer is used to identify interactions of solar activity and interplanetary medium conditions with the Earth's magnetosphere-ionosphere systems. A causal information transfer from the solar wind parameters to geomagnetic indices is detected. The vertical component of the interplanetary magnetic field (Bz) influences the auroral electrojet (AE) index with an information transfer delay of 10 min and the geomagnetic disturbances at mid-latitudes measured by the symmetric field in the H component (SYM-H) index with a delay of about 30 min. Using a properly conditioned causality measure, no causal link between AE and SYM-H, or between magnetospheric substorms and magnetic storms can be detected. The observed causal relations can be described as linear time-delayed information transfer.

Nonlinear correlations in multifractals: Visibility graphs of magnitude and sign series.

Correlations in a multifractal series have been investigated extensively. Almost all approaches try to find scaling features of a given time series. However, the scaling analysis has always been encountered with some difficulties. Of particular importance is finding a proper scaling region and removing the impact of the probability distribution function of the series on the correlation extraction methods. In this article, we apply the horizontal visibility graph algorithm to map a stochastic time series into networks. By investigating the magnitude and sign of a multifractal time series, we show that one can detect linear as well as nonlinear correlations, even for situations that have been considered as uncorrelated noises by typical approaches such as the multifractal detrended fluctuation analysis. Furthermore, we introduce a topological parameter that can well measure the strength of nonlinear correlations. This parameter is independent of the probability distribution function and calculated without the need to find any scaling region. Our findings may provide new insights about the multifractal analysis of a time series in a variety of complex systems.

Phase transition in a network model of social balance with Glauber dynamics.

We study the evolution of a social network with friendly or enmity connections into a balanced state by introducing a dynamical model with an intrinsic randomness, similar to Glauber dynamics in statistical mechanics. We include the possibility of the tension promotion as well as the tension reduction in our model. Such a more realistic situation enables the system to escape from local minima in its energy landscape and thus to exit out of frozen imbalanced states, which are unwanted outcomes observed in previous models. On the other hand, in finite networks the dynamics takes the system into a balanced phase, if the randomness is lower than a critical value. For large networks, we also find a sharp phase transition at the initial positive link density of ρ_{0}^{*}=1/2, where the system transitions from a bipolar state into a paradise. This modifies the gradual phase transition at a nontrivial value of ρ_{0}^{*}≃0.65, observed in recent studies.

Interoccurrence time statistics in fully-developed turbulence.

Emergent extreme events are a key characteristic of complex dynamical systems. The main tool for detailed and deep understanding of their stochastic dynamics is the statistics of time intervals of extreme events. Analyzing extensive experimental data, we demonstrate that for the velocity time series of fully-developed turbulent flows, generated by (i) a regular grid; (ii) a cylinder; (iii) a free jet of helium, and (iv) a free jet of air with the Taylor Reynolds numbers Reλ from 166 to 893, the interoccurrence time distributions P(τ) above a positive threshold Q in the inertial range is described by a universal q- exponential function, P(τ) = ß(2 - q)[1 - ß(1 - q)τ](1/(1-q)), which may be due to the superstatistical nature of the occurrence of extreme events. Our analysis provides a universal description of extreme events in turbulent flows.

Complex network approach to fractional time series.

In order to extract correlation information inherited in stochastic time series, the visibility graph algorithm has been recently proposed, by which a time series can be mapped onto a complex network. We demonstrate that the visibility algorithm is not an appropriate one to study the correlation aspects of a time series. We then employ the horizontal visibility algorithm, as a much simpler one, to map fractional processes onto complex networks. The degree distributions are shown to have parabolic exponential forms with Hurst dependent fitting parameter. Further, we take into account other topological properties such as maximum eigenvalue of the adjacency matrix and the degree assortativity, and show that such topological quantities can also be used to predict the Hurst exponent, with an exception for anti-persistent fractional Gaussian noises. To solve this problem, we take into account the Spearman correlation coefficient between nodes' degrees and their corresponding data values in the original time series.

Anomalous fluctuations of vertical velocity of Earth and their possible implications for earthquakes.

High-quality measurements of seismic activities around the world provide a wealth of data and information that are relevant to understanding of when earthquakes may occur. If viewed as complex stochastic time series, such data may be analyzed by methods that provide deeper insights into their nature, hence leading to better understanding of the data and their possible implications for earthquakes. In this paper, we provide further evidence for our recent proposal [P. Mansour, Phys. Rev. Lett. 102, 014101 (2009)10.1103/PhysRevLett.102.014101] for the existence of a transition in the shape of the probability density function (PDF) of the successive detrended increments of the stochastic fluctuations of Earth's vertical velocity V_{z} , collected by broadband stations before moderate and large earthquakes. To demonstrate the transition, we carried out extensive analysis of the data for V_{z} for 12 earthquakes in several regions around the world, including the recent catasrophic one in Haiti. The analysis supports the hypothesis that before and near the time of an earthquake, the shape of the PDF undergoes significant and discernable changes, which can be characterized quantitatively. The typical time over which the PDF undergoes the transition is about 5-10 h prior to a moderate or large earthquake.