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We review a class of energy landscape analysis method that uses the Ising model and takes multivariate time series data as input. The method allows one to capture dynamics of the data as trajectories of a ball from one basin to a different basin to yet another, constrained on the energy landscape specified by the estimated Ising model. While this energy landscape analysis has mostly been applied to functional magnetic resonance imaging (fMRI) data from the brain for historical reasons, there are emerging applications outside fMRI data and neuroscience. To inform such applications in various research fields, this review paper provides a detailed tutorial on each step of the analysis, terminologies, concepts underlying the method, and validation, as well as recent developments of extended and related methods. 

We study synthetic temporal networks whose evolution is determined by stochastically evolving node variables—synthetic analogues of, e.g., temporal proximity networks of mobile agents. We quantify the long-timescale correlations of these evolving networks by an autocorrelative measure of network-structural memory. Several distinct patterns of autocorrelation arise, including power-law decay and exponential decay, depending on the choice of node-variable dynamics and connection probability function. Our methods are also applicable in wider contexts; our temporal network models are tractable mathematically and in simulation, and our long-term memory quantification is analytically tractable and straightforwardly computable from temporal network data. Published by the American Physical Society2025 

Abstract Real systems showing regime shifts, such as ecosystems, are often composed of many dynamical elements interacting on a network. Various early warning signals have been proposed for anticipating regime shifts from observed data. However, it is unclear how one should combine early warning signals from different nodes for better performance. Based on theory of stochastic differential equations, we propose a method to optimize the node set from which to construct an early warning signal. The proposed method takes into account that uncertainty as well as the magnitude of the signal affects its predictive performance, that a large magnitude or small uncertainty of the signal in one situation does not imply the signal’s high performance, and that combining early warning signals from different nodes is often but not always beneficial. The method performs well particularly when different nodes are subjected to different amounts of dynamical noise and stress. 

Abstract Electroencephalogram (EEG) microstate analysis entails finding dynamics of quasi-stable and generally recurrent discrete states in multichannel EEG time series data and relating properties of the estimated state-transition dynamics to observables such as cognition and behavior. While microstate analysis has been widely employed to analyze EEG data, its use remains less prevalent in functional magnetic resonance imaging (fMRI) data, largely due to the slower timescale of such data. In the present study, we extend various data clustering methods used in EEG microstate analysis to resting-state fMRI data from healthy humans to extract their state-transition dynamics. We show that the quality of clustering is on par with that for various microstate analyses of EEG data. We then develop a method for examining test–retest reliability of the discrete-state transition dynamics between fMRI sessions and show that the within-participant test–retest reliability is higher than between-participant test–retest reliability for different indices of state-transition dynamics, different networks, and different data sets. This result suggests that state-transition dynamics analysis of fMRI data could discriminate between different individuals and is a promising tool for performing fingerprinting analysis of individuals. 

The population structure often impacts evolutionary dynamics. In constant-selection evolutionary dynamics between two types, amplifiers of selection are networks that promote the fitter mutant to take over the entire population, and suppressors of selection do the opposite. It has been shown that most undirected and unweighted networks are amplifiers of selection under a common updating rule and initial condition. Here, we extensively investigate how edge weights influence selection on undirected networks. We show that random edge weights make small networks less amplifying than the corresponding unweighted networks in a majority of cases and also make them suppressors of selection (i.e. less amplifying than the complete graph, or equivalently, the Moran process) in many cases. Qualitatively, the same result holds true for larger empirical networks. These results suggest that amplifiers of selection are not as common for weighted networks as for unweighted counterparts.