One of the outstanding problems in complexity science and engineering is the study of high-dimensional networked systems and of their susceptibility to transitions to undesired states as a result of changes in external drivers or in the structural properties. Because of the incredibly large number of parameters controlling the state of such complex systems and the heterogeneity of its components, the study of their dynamics is extremely difficult. Here we propose an analytical framework for collapsing complex N-dimensional networked systems into an S+1-dimensional manifold as a function of S effective control parameters with S << N. We test our approach on a variety of real-world complex problems showing how this new framework can approximate the system's response to changes and correctly identify the regions in the parameter space corresponding to the system's transitions. Our work offers an analytical method to evaluate optimal strategies in the design or management of networked systems.

The year 2017 saw the rise and fall of the crypto-currency market, followed by high variability in the price of all crypto-currencies. In this work, we study the abrupt transition in crypto-currency residuals, which is associated with the critical transition (the phenomenon of critical slowing down) or the stochastic transition phenomena. We find that, regardless of the specific crypto-currency or rolling window size, the autocorrelation always fluctuates around a high value, while the standard deviation increases monotonically. Therefore, while the autocorrelation does not display the signals of critical slowing down, the standard deviation can be used to anticipate critical or stochastic transitions. In particular, we have detected two sudden jumps in the standard deviation, in the second quarter of 2017 and at the beginning of 2018, which could have served as the early warning signals of two major price collapses that have happened in the following periods. We finally propose a mean-field phenomenological model for the price of crypto-currency to show how the use of the standard deviation of the residuals is a better leading indicator of the collapse in price than the time-series' autocorrelation. Our findings represent a first step towards a better diagnostic of the risk of critical transition in the price and/or volume of crypto-currencies.

Cointegration focuses on whether the long-term linear relationship between two or more time series is stationary even if this linear relationship does not exist or is not strong for the short term. Identifying the potential cointegration is important for economics, ecology, meteorology, neuroscience, and much more. Classic methods only considered or restricted in cointegration where the order of integration of all time series is 1. We introduce a method based on searching the vector to minimize the absolute correlation of convergent cross-mapping that can explore the universal cointegration and its extent. The proposed method can be applied to time series whose order of integration is not 1, cases that are not covered by classic cointegration. The proposed method is first illustrated and validated through time series generated by mathematical models in which the underlying relationships are known and then applied to three real-world examples.

Pine wilt disease (PWD) is a highly destructive disease in forest ecosystems, resulting in extensive forest decline and substantial economic losses. As soil pH plays a critical role in soil microbial activity and significantly impacts the prevalence and severity of diseases, we conducted an experiment to regulate soil pH for alleviating PWD in a black pine (Pinus thunbergii) forest. The result reveals that: (1) The pH of the soil under a P. thunbergii forest was 5.19 ± 0.40, which was significantly lower than that of soils under other vegetation types at 8.53 ± 0.44. (2) Finely ground shell powder (F-SP) was the optimal size for long-term and efficient regulation, but quicklime (QL) exhibited the strongest efficacy in raising soil pH, followed by F-SP and plant ash. The regulation effect strengthened with the dosage amount. (3) In the situ experiments, part of symptomatic black pine in F-SP or QL plots were apparently improved and converted to asymptomatic trees separately by 15.9% and 5.4%. Applying F-SP can alleviate PWD in a sustainable way. This paper presents the first investigation to assess the effects of regulating soil pH for controlling PWD. It holds significant practical value for the rational planning and the sustainable development of artificial forests in coastal regions.