Finding the direction of lowest resilience in multivariate complex systems

Abstract

<jats:p>The dynamics of complex systems, such as ecosystems, financial markets and the human brain, emerge from the interactions of numerous components. We often lack the knowledge to build reliable models for the behaviour of such network systems. This makes it difficult to predict potential instabilities. We show that one could use the natural fluctuations in multivariate time series to reveal network regions with particularly slow dynamics. The multidimensional slowness points to the direction of minimal resilience, in the sense that simultaneous perturbations on this set of nodes will take longest to recover. We compare an autocorrelation-based method with a variance-based method for different time-series lengths, data resolution and different noise regimes. We show that the autocorrelation-based method is less robust for short time series or time series with a low resolution but more robust for varying noise levels. This novel approach may help to identify unstable regions of multivariate systems or to distinguish safe from unsafe perturbations.</jats:p>

Anticipating the direction of symptom progression using critical slowing down: a proof-of-concept study

“…1 ). In line with earlier studies, we expected the vector to lengthen prior to symptom transitions, meaning that each window should have a larger vector than the previous windows [ 33 , 34 , 46 , 47 ]. Provided that there is no change in overall variance, this corresponds to an increased symptom covariance (as previously observed in group-level studies [ 17 – 24 , 48 – 50 ]).…”

“…In the context of psychopathology, exposing the direction of critical slowing down may allow for inferring whether an upcoming symptom transition is directed towards worsening or remitting symptoms. The direction of critical slowing down can be monitored using metrics similar to those described in earlier studies, namely symptom covariances (or, more specifically: the eigenvalues of the covariance matrix [ 33 , 34 , 36 , 37 ]). Hence, the hypothesis that follows from a complex dynamic systems approach can be considered an extension of what was reported earlier, namely: a gradual alteration in the structure of psychopathological symptoms prospectively predicts whether a specific individual will experience a symptom transition towards remission (decrease of symptom severity) or worsening (increase of symptom severity).…”

“…To this end, we will repeat the analyses in a larger dataset, which consists of 134 young adults at increased risk for mental health problems who provided daily ratings on their mental states over a period of six months [ 40 ]. We hypothesize that in the first dataset, the direction of critical slowing down [ 33 , 34 ] points towards a relapse of symptoms. Similarly, in the second dataset, we hypothesize that the direction of critical slowing down corresponds to the change in symptoms reported by individuals ( e.g., strong critical slowing down towards improvement in individuals who experienced a large reduction of symptoms, and vice versa).…”

Early Warning Signals in Phase Space: Geometric Resilience Loss Indicators From Multiplex Cumulative Recurrence Networks

“…A related issue concerns the fact that EWS are naive to the direction of change, that is, to be of clinical use it would be helpful to know whether an EWS implies the current state is deteriorating (higher symptom severity) or not. There have been theoretical advances to finding the direction of least resilience in multivariate time series data ( Weinans et al, 2019 ) or based on formal models ( Cui et al, 2021 ). Recently Schreuder et al (2022) used a technique based on two variables obtained from windowed principal component analysis (the variance explained by the first component and the skewness of the scores projected onto this component) to identify the direction of the critical transition in the same data set used in the present paper.…”

Anticipating critical transitions in psychological systems using early warning signals: Theoretical and practical considerations.

“…Even if the target system falls into the class of systems that show critical slowing down before critical transitions, early warning signals may only be observable in a small number of variables of the system (Boerlijst et al, 2013;Patterson et al, 2021). While recent work tries to find the system components that most strongly express critical slowing down (Dakos, 2018;Weinans et al, 2019), purely statistical work will not put early warning signals on a solid footing. Instead, we need to build a basic understanding of the system under study.…”