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The description of the dynamics of complex systems, in particular the capture of the interaction structure and causal relationships between elements of the system, is one of the central questions of interdisciplinary research. While the characterization of pairwise causal interactions is a relatively ripe field with established theoretical concepts and the current focus is on technical issues of their efficient estimation, it turns out that the standard concepts such as Granger causality or transfer entropy may not faithfully reflect possible synergies or interactions of higher orders, phenomena highly relevant for many real-world complex systems. In this paper, we propose a generalization and refinement of the information-theoretic approach to causal inference, enabling the description of truly multivariate, rather than multiple pairwise, causal interactions, and moving thus from causal networks to causal hypernetworks. In particular, while keeping the ability to control for mediating variables or common causes, in case of purely synergetic interactions such as the exclusive disjunction, it ascribes the causal role to the multivariate causal set but \emph{not} to individual inputs, distinguishing it thus from the case of e.g. two additive univariate causes. We demonstrate this concept by application to illustrative theoretical examples as well as a biophysically realistic simulation of biological neuronal dynamics recently reported to employ synergetic computations.

This paper extends the possibility to examine the underlying curvature of data through the lens of topology by using the Betti curves, tools of Persistent Homology, as key topological descriptors, building on the clique topology approach. It was previously shown that Betti curves distinguish random from Euclidean geometric matrices - i.e. distance matrices of points randomly distributed in a cube with Euclidean distance. In line with previous experiments, we consider their low-dimensional approximations named integral Betti values, or signatures that effectively distinguish not only Euclidean, but also spherical and hyperbolic geometric matrices, both from purely random matrices as well as among themselves. To prove this, we analyse the behaviour of Betti curves for various geometric matrices -- i.e. distance matrices of points randomly distributed on manifolds of constant sectional curvature, considering the classical models of curvature 0, 1, -1, given by the Euclidean space, the sphere, and the hyperbolic space. We further investigate the dependence of integral Betti signatures on factors including the sample size and dimension. This is important for assessment of real-world connectivity matrices, as we show that the standard approach to network construction gives rise to (spurious) spherical geometry, with topology dependent on sample dimensions. Finally, we use the manifolds of constant curvature as comparison models to infer curvature underlying real-world datasets coming from neuroscience, finance and climate. Their associated topological features exhibit a hyperbolic character: the integral Betti signatures associated to these datasets sit in between Euclidean and hyperbolic (of small curvature). The potential confounding ``hyperbologenic effect'' of intrinsic low-rank modular structures is also evaluated through simulations.

The symmetry of complex networks is a global property that has recently gained attention since MacArthur et al. 2008 showed that many real-world networks contain a considerable number of symmetries. These authors work with a very strict symmetry definition based on the network's automorphism. The potential problem with this approach is that even a slight change in the graph's structure can remove or create some symmetry. Recently, Liu 2020 proposed to use an approximate automorphism instead of strict automorphism. This method can discover symmetries in the network while accepting some minor imperfections in their structure. The proposed numerical method, however, exhibits some performance problems and has some limitations while it assumes the absence of fixed points. In this work, we exploit alternative approaches recently developed for treating the Graph Matching Problem and propose a method, which we will refer to as Quadratic Symmetry Approximator (QSA), to address the aforementioned shortcomings. To test our method, we propose a set of random graph models suitable for assessing a wide family of approximate symmetry algorithms. The performance of our method is also demonstrated on brain networks.

Sparse linear regression methods including the well-known LASSO and the Dantzig selector have become ubiquitous in the engineering practice, including in medical imaging. Among other tasks, they have been successfully applied for the estimation of neuronal activity from functional magnetic resonance data without prior knowledge of the stimulus or activation timing, utilizing an approximate knowledge of the hemodynamic response to local neuronal activity. These methods work by generating a parametric family of solutions with different sparsity, among which an ultimate choice is made using an information criteria. We propose a novel approach, that instead of selecting a single option from the family of regularized solutions, utilizes the whole family of such sparse regression solutions. Namely, their ensemble provides a first approximation of probability of activation at each time-point, and together with the conditional neuronal activity distributions estimated with the theory of mixtures with varying concentrations, they serve as the inputs to a Bayes classifier eventually deciding on the verity of activation at each time-point. We show in extensive numerical simulations that this new method performs favourably in comparison with standard approaches in a range of realistic scenarios. This is mainly due to the avoidance of overfitting and underfitting that commonly plague the solutions based on sparse regression combined with model selection methods, including the corrected Akaike Information Criterion. This advantage is finally documented in selected fMRI task datasets.

We analyze partial synchronization patterns in a network of FitzHugh-Nagumo oscillators with empirical structural connectivity measured in healthy human subjects. We report a dynamical asymmetry between the hemispheres, induced by the natural structural asymmetry. We show that the dynamical asymmetry can be enhanced by introducing the inter-hemispheric coupling strength as a control parameter for partial synchronization patterns. We specify the possible modalities for existence of unihemispheric sleep in human brain, where one hemisphere sleeps while the other remains awake. In fact, this state is common among migratory birds and mammals like aquatic species., Comment: 7 pages, 6 figures

Stock networks, constructed from stock price time series, are a well-established tool for the characterization of complex behavior in stock markets. Following Mantegna's seminal paper, the linear Pearson's correlation coefficient between pairs of stocks has been the usual way to determine network edges. Recently, possible effects of nonlinearity on the graph-theoretical properties of such networks have been demonstrated when using nonlinear measures such as mutual information instead of linear correlation. In this paper, we quantitatively characterize the nonlinearity in stock time series and the effect it has on stock network properties. This is achieved by a systematic multi-step approach that allows us to quantify the nonlinearity of coupling; correct its effects wherever it is caused by simple univariate non-Gaussianity; potentially localize in space and time any remaining strong sources of this nonlinearity; and, finally, study the effect nonlinearity has on global network properties. By applying this multi-step approach to stocks included in three prominent indices (NYSE100, FTSE100 and SP500), we establish that the apparent nonlinearity that has been observed is largely due to univariate non-Gaussianity. Furthermore, strong nonstationarity in a few specific stocks may play a role. In particular, the sharp decrease in some stocks during the global financial crisis of 2008 gives rise to apparent nonlinear dependencies among stocks.

Estimating causal interactions in complex dynamical systems is an important problem encountered in many fields of current science. While a theoretical solution for detecting the causal interactions has been previously formulated in the framework of prediction improvement, it generally requires the computation of high-dimensional information functionals -- a situation invoking the curse of dimensionality with increasing network size. Recently, several methods have been proposed to alleviate this problem, based on iterative procedures for assessment of conditional (in)dependences. In the current work, we bring a comparison of several such prominent approaches. This is done both by theoretical comparison of the algorithms using a formulation in a common framework, and by numerical simulations including realistic complex coupling patterns. The theoretical analysis highlights the key similarities and differences between the algorithms, hinting on their comparative strengths and weaknesses. The method assumptions and specific properties such as false positive control and order dependence are discussed. Numerical simulations suggest that while the accuracy of most of the algorithms is almost indistinguishable, there are substantial differences in their computational demands, ranging theoretically from polynomial to exponential complexity, and leading to substantial differences in computation time in realistic scenarios depending on the density and size of networks. Based on analysis of the algorithms and numerical simulations, we propose a hybrid approach providing competitive accuracy with improved computational efficiency.

Complex spatiotemporal patterns, called chimera states, consist of coexisting coherent and incoherent domains and can be observed in networks of coupled oscillators. The interplay of synchrony and asynchrony in complex brain networks is an important aspect in studies of both brain function and disease. We analyse the collective dynamics of FitzHugh-Nagumo neurons in complex networks motivated by its potential application to epileptology and epilepsy surgery. We compare two topologies: an empirical structural neural connectivity derived from diffusion-weighted magnetic resonance imaging and a mathematically constructed network with modular fractal connectivity. We analyse the properties of chimeras and partially synchronized states, and obtain regions of their stability in the parameter planes. Furthermore, we qualitatively simulate the dynamics of epileptic seizures and study the influence of the removal of nodes on the network synchronizability, which can be useful for applications to epileptic surgery.

Maximum entropy estimation is of broad interest for inferring properties of systems across many different disciplines. In this work, we significantly extend a technique we previously introduced for estimating the maximum entropy of a set of random discrete variables when conditioning on bivariate mutual informations and univariate entropies. Specifically, we show how to apply the concept to continuous random variables and vastly expand the types of information-theoretic quantities one can condition on. This allows us to establish a number of significant advantages of our approach over existing ones. Not only does our method perform favorably in the undersampled regime, where existing methods fail, but it also can be dramatically less computationally expensive as the cardinality of the variables increases. In addition, we propose a nonparametric formulation of connected informations and give an illustrative example showing how this agrees with the existing parametric formulation in cases of interest. We further demonstrate the applicability and advantages of our method to real world systems for the case of resting-state human brain networks. Finally, we show how our method can be used to estimate the structural network connectivity between interacting units from observed activity and establish the advantages over other approaches for the case of phase oscillator networks as a generic example.

Reconstructing the structural connectivity between interacting units from observed activity is a challenge across many different disciplines. The fundamental first step is to establish whether or to what extent the interactions between the units can be considered pairwise and, thus, can be modeled as an interaction network with simple links corresponding to pairwise interactions. In principle this can be determined by comparing the maximum entropy given the bivariate probability distributions to the true joint entropy. In many practical cases this is not an option since the bivariate distributions needed may not be reliably estimated, or the optimization is too computationally expensive. Here we present an approach that allows one to use mutual informations as a proxy for the bivariate distributions. This has the advantage of being less computationally expensive and easier to estimate. We achieve this by introducing a novel entropy maximization scheme that is based on conditioning on entropies and mutual informations. This renders our approach typically superior to other methods based on linear approximations. The advantages of the proposed method are documented using oscillator networks and a resting-state human brain network as generic relevant examples.

Quantification of relations between measured variables of interest by statistical measures of dependence is a common step in analysis of climate data. The term "connectivity" is used in the network context including the study of complex coupled dynamical systems. The choice of dependence measure is key for the results of the subsequent analysis and interpretation. The use of linear Pearson's correlation coefficient is widespread and convenient. On the other side, as the climate is widely acknowledged to be a nonlinear system, nonlinear connectivity quantification methods, such as those based on information-theoretical concepts, are increasingly used for this purpose. In this paper we outline an approach that enables well informed choice of connectivity method for a given type of data, improving the subsequent interpretation of the results. The presented multi-step approach includes statistical testing, quantification of the specific non-linear contribution to the interaction information, localization of nodes with strongest nonlinear contribution and assessment of the role of specific temporal patterns, including signal nonstationarities. In detail we study the consequences of the choice of a general nonlinear connectivity measure, namely mutual information, focusing on its relevance and potential alterations in the discovered dependence structure. We document the method by applying it on monthly mean temperature data from the NCEP/NCAR reanalysis dataset as well as the ERA dataset. We have been able to identify main sources of observed non-linearity in inter-node couplings. Detailed analysis suggested an important role of several sources of nonstationarity within the climate data. The quantitative role of genuine nonlinear coupling at this scale has proven to be almost negligible, providing quantitative support for the use of linear methods for this type of data.

Characterization of real-world complex systems increasingly involves the study of their topological structure using graph theory. Among global network properties, small-world property, consisting in existence of relatively short paths together with high clustering of the network, is one of the most discussed and studied. When dealing with coupled dynamical systems, links among units of the system are commonly quantified by a measure of pairwise statistical dependence of observed time series (functional connectivity). We argue that the functional connectivity approach leads to upwardly biased estimates of small-world characteristics (with respect to commonly used random graph models) due to partial transitivity of the accepted functional connectivity measures such as the correlation coefficient. In particular, this may lead to observation of small-world characteristics in connectivity graphs estimated from generic randomly connected dynamical systems. The ubiquity and robustness of the phenomenon is documented by an extensive parameter study of its manifestation in a multivariate linear autoregressive process, with discussion of the potential relevance for nonlinear processes and measures., Comment: The following article has been submitted to Chaos: An interdisciplinary journal of nonlinear science. After it is published, it will be found at http://chaos.aip.org/