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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.


Epidemic outbreaks and prediction in an adaptive SIS model.(a) Normalized time series for the infected density I (red) and the susceptible-infected link density SI (blue). (b) Logarithmic distances for I and SI are shown as well (see Supplementary Information, Section 2). The linear interpolations (black) indicate the expected linear upward trend before a saddle-escape; the slopes of the four black lines (from left to right) are approximately 7.364, 12.516, 5.466 and 3.461 respectively. The three ellipses in (a2) highlight the three typical regimes between spikes discussed in the text and are there to guide the eye as in (Fig. 2). Parameter values for this figure are p = 0.0058, r = 0.002 and w0 = 0.6.
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f3: Epidemic outbreaks and prediction in an adaptive SIS model.(a) Normalized time series for the infected density I (red) and the susceptible-infected link density SI (blue). (b) Logarithmic distances for I and SI are shown as well (see Supplementary Information, Section 2). The linear interpolations (black) indicate the expected linear upward trend before a saddle-escape; the slopes of the four black lines (from left to right) are approximately 7.364, 12.516, 5.466 and 3.461 respectively. The three ellipses in (a2) highlight the three typical regimes between spikes discussed in the text and are there to guide the eye as in (Fig. 2). Parameter values for this figure are p = 0.0058, r = 0.002 and w0 = 0.6.

Mentions: In simulations of this system the number of infectious agents shows distinguished peaks in time, which can be interpreted as epidemic outbreaks (Fig. 3). In (Fig. 3(b)) the logarithmic distances between consecutive points are shown for the density of I-nodes and S-I-links. Despite the strong fluctuations away from the peaks, both warning signals show three phases after a peak: (1) strong stabilization, (2) plateau- or noise-type behavior and (3) a trend towards instability before the next spike (see Fig. 3(a2)). In fact, similar phases can also be observed for the first model in (Fig. 2(b3)).


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

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

Epidemic outbreaks and prediction in an adaptive SIS model.(a) Normalized time series for the infected density I (red) and the susceptible-infected link density SI (blue). (b) Logarithmic distances for I and SI are shown as well (see Supplementary Information, Section 2). The linear interpolations (black) indicate the expected linear upward trend before a saddle-escape; the slopes of the four black lines (from left to right) are approximately 7.364, 12.516, 5.466 and 3.461 respectively. The three ellipses in (a2) highlight the three typical regimes between spikes discussed in the text and are there to guide the eye as in (Fig. 2). Parameter values for this figure are p = 0.0058, r = 0.002 and w0 = 0.6.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f3: Epidemic outbreaks and prediction in an adaptive SIS model.(a) Normalized time series for the infected density I (red) and the susceptible-infected link density SI (blue). (b) Logarithmic distances for I and SI are shown as well (see Supplementary Information, Section 2). The linear interpolations (black) indicate the expected linear upward trend before a saddle-escape; the slopes of the four black lines (from left to right) are approximately 7.364, 12.516, 5.466 and 3.461 respectively. The three ellipses in (a2) highlight the three typical regimes between spikes discussed in the text and are there to guide the eye as in (Fig. 2). Parameter values for this figure are p = 0.0058, r = 0.002 and w0 = 0.6.
Mentions: In simulations of this system the number of infectious agents shows distinguished peaks in time, which can be interpreted as epidemic outbreaks (Fig. 3). In (Fig. 3(b)) the logarithmic distances between consecutive points are shown for the density of I-nodes and S-I-links. Despite the strong fluctuations away from the peaks, both warning signals show three phases after a peak: (1) strong stabilization, (2) plateau- or noise-type behavior and (3) a trend towards instability before the next spike (see Fig. 3(a2)). In fact, similar phases can also be observed for the first model in (Fig. 2(b3)).

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.