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Early warning signs for saddle-escape transitions in complex networks.

Kuehn C, Zschaler G, Gross T - Sci Rep (2015)

Bottom Line: We illustrate our results in two network models and epidemiological data.This work thus establishes a connection from critical transitions to networks and an early warning sign for a new type of critical transition.In complex models and big data we anticipate that saddle-transitions will be encountered frequently in the future.

View Article: PubMed Central - PubMed

Affiliation: Vienna University of Technology, 1040 Vienna, Austria.

ABSTRACT
Many real world systems are at risk of undergoing critical transitions, leading to sudden qualitative and sometimes irreversible regime shifts. The development of early warning signals is recognized as a major challenge. Recent progress builds on a mathematical framework in which a real-world system is described by a low-dimensional equation system with a small number of key variables, where the critical transition often corresponds to a bifurcation. Here we show that in high-dimensional systems, containing many variables, we frequently encounter an additional non-bifurcative saddle-type mechanism leading to critical transitions. This generic class of transitions has been missed in the search for early-warnings up to now. In fact, the saddle-type mechanism also applies to low-dimensional systems with saddle-dynamics. Near a saddle a system moves slowly and the state may be perceived as stable over substantial time periods. We develop an early warning sign for the saddle-type transition. We illustrate our results in two network models and epidemiological data. This work thus establishes a connection from critical transitions to networks and an early warning sign for a new type of critical transition. In complex models and big data we anticipate that saddle-transitions will be encountered frequently in the future.

No MeSH data available.


Related in: MedlinePlus

Measles epidemics in the UK between 1944 and 1966.(a) Typical time series of a city (here: Birmingham) for the infected population I; the series has been normalized by the maximum outbreak3738. (b) ROC curves for a five (dots) and ten (stars) data point prediction averaged over all cities. The diagonal is shown as well. Blue corresponds to a precursor volume with δ > 0 and red to a precursor volume with δ < 0. A prediction time window of 5 months is indicated by ‘stars’ while a time window of 2.5 months is indicated by ‘dots’.
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f4: Measles epidemics in the UK between 1944 and 1966.(a) Typical time series of a city (here: Birmingham) for the infected population I; the series has been normalized by the maximum outbreak3738. (b) ROC curves for a five (dots) and ten (stars) data point prediction averaged over all cities. The diagonal is shown as well. Blue corresponds to a precursor volume with δ > 0 and red to a precursor volume with δ < 0. A prediction time window of 5 months is indicated by ‘stars’ while a time window of 2.5 months is indicated by ‘dots’.

Mentions: Here we focus on repeated measles outbreaks documented biweekly between 1944 and 1966 in 60 cities in the United Kingdom2728. The time series of the proportion of infected individuals shows long repeated periods of low disease prevalence interspersed with large, but very short, outbreaks. This strongly indicates that a saddle-type mechanism may be at work (Fig. 4(a)). We use a receiver-operating-characteristic (ROC) curve2930 to quantify the performance of our warning sign for saddle escapes. We briefly recall, how ROC curves are calculated. First, one defines a scalar precursory variable X to be computed from observations before the transitions and considers a threshold δ such that if X > δ an alarm is given. Then the two ratios of correct alarms to the total number of actual tipping events and false predictions to the total number of non-tipping events are calculated for various thresholds δ. This provides a quantitative indicator for the ability of the precursor variable to detect tipping points (see Supplementary Information, Section 6, for more background on ROC curves and their interpretation).


Early warning signs for saddle-escape transitions in complex networks.

Kuehn C, Zschaler G, Gross T - Sci Rep (2015)

Measles epidemics in the UK between 1944 and 1966.(a) Typical time series of a city (here: Birmingham) for the infected population I; the series has been normalized by the maximum outbreak3738. (b) ROC curves for a five (dots) and ten (stars) data point prediction averaged over all cities. The diagonal is shown as well. Blue corresponds to a precursor volume with δ > 0 and red to a precursor volume with δ < 0. A prediction time window of 5 months is indicated by ‘stars’ while a time window of 2.5 months is indicated by ‘dots’.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4544003&req=5

f4: Measles epidemics in the UK between 1944 and 1966.(a) Typical time series of a city (here: Birmingham) for the infected population I; the series has been normalized by the maximum outbreak3738. (b) ROC curves for a five (dots) and ten (stars) data point prediction averaged over all cities. The diagonal is shown as well. Blue corresponds to a precursor volume with δ > 0 and red to a precursor volume with δ < 0. A prediction time window of 5 months is indicated by ‘stars’ while a time window of 2.5 months is indicated by ‘dots’.
Mentions: Here we focus on repeated measles outbreaks documented biweekly between 1944 and 1966 in 60 cities in the United Kingdom2728. The time series of the proportion of infected individuals shows long repeated periods of low disease prevalence interspersed with large, but very short, outbreaks. This strongly indicates that a saddle-type mechanism may be at work (Fig. 4(a)). We use a receiver-operating-characteristic (ROC) curve2930 to quantify the performance of our warning sign for saddle escapes. We briefly recall, how ROC curves are calculated. First, one defines a scalar precursory variable X to be computed from observations before the transitions and considers a threshold δ such that if X > δ an alarm is given. Then the two ratios of correct alarms to the total number of actual tipping events and false predictions to the total number of non-tipping events are calculated for various thresholds δ. This provides a quantitative indicator for the ability of the precursor variable to detect tipping points (see Supplementary Information, Section 6, for more background on ROC curves and their interpretation).

Bottom Line: We illustrate our results in two network models and epidemiological data.This work thus establishes a connection from critical transitions to networks and an early warning sign for a new type of critical transition.In complex models and big data we anticipate that saddle-transitions will be encountered frequently in the future.

View Article: PubMed Central - PubMed

Affiliation: Vienna University of Technology, 1040 Vienna, Austria.

ABSTRACT
Many real world systems are at risk of undergoing critical transitions, leading to sudden qualitative and sometimes irreversible regime shifts. The development of early warning signals is recognized as a major challenge. Recent progress builds on a mathematical framework in which a real-world system is described by a low-dimensional equation system with a small number of key variables, where the critical transition often corresponds to a bifurcation. Here we show that in high-dimensional systems, containing many variables, we frequently encounter an additional non-bifurcative saddle-type mechanism leading to critical transitions. This generic class of transitions has been missed in the search for early-warnings up to now. In fact, the saddle-type mechanism also applies to low-dimensional systems with saddle-dynamics. Near a saddle a system moves slowly and the state may be perceived as stable over substantial time periods. We develop an early warning sign for the saddle-type transition. We illustrate our results in two network models and epidemiological data. This work thus establishes a connection from critical transitions to networks and an early warning sign for a new type of critical transition. In complex models and big data we anticipate that saddle-transitions will be encountered frequently in the future.

No MeSH data available.


Related in: MedlinePlus