Climate change, pandemic or coordinated activity of neurons in the brain: in all of these examples, a transition takes place at some point from the baseline state to a new state. Researchers at the Technical University of Munich (TUM) have discovered a universal mathematical structure at these so-called tipping points. It creates the basis for a better understanding of the behavior of networked systems.
This is a critical question for scientists in all fields: how can we predict and influence changes in a networked system? “In biology, an example is the modeling of coordinated neuronal activity”, explains Christian Kühn, professor of multiscale and stochastic dynamics at TUM. Models of this type are also used in other disciplines, for example when studying the spread of disease or climate change.
All critical changes in networked systems have one thing in common: a tipping point where the system transitions from a baseline state to a new state. It can be a smooth change, where the system can easily go back to baseline. Or it can be a sharp, hard-to-reverse transition where the state of the system can change abruptly or “explosively”. Transitions of this type also occur in the context of climate change, for example with the melting of the polar caps. In many cases, transitions result from variation in a single parameter, such as the increase in greenhouse gas concentrations behind climate change.
Similar structures in many models
In some cases – like climate change – a sharp tipping point would have extremely negative effects, while in others it would be desirable. Therefore, the researchers used mathematical models to study how the type of transition is influenced by the introduction of new parameters or conditions. “For example, you can vary another parameter, maybe related to how people change their behavior during a pandemic. Or you can adjust an input in a neural system, ”says Kühn. “In these examples and in many other cases, we have seen that we can go from a continuous transition to a discontinuous transition or vice versa.”
Kühn and Dr Christian Bick from the Vrije Universiteit Amsterdam studied existing models from various disciplines that were created to understand certain systems. “We found it remarkable that so many mathematical structures related to the tipping point look very similar in these models,” says Bick. “By reducing the problem to the most basic equation possible, we were able to identify a universal mechanism that decides the type of tipping point and that is valid for as many models as possible.”
Universal math tool
Scientists have thus described a new central mechanism which makes it possible to calculate whether a networked system will have a continuous or a discontinuous transition. “We provide a mathematical tool that can be applied universally – in other words, in theoretical physics, climate science, neurobiology and other disciplines – and works regardless of the specific case,” says Kühn.
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Material provided by Technical University of Munich (TUM). Note: Content can be changed for style and length.