New mathematics may broaden understanding of the brain, climate, and other complex networks
Research on complex networks and dynamical systems overturns a scientific paradigm concerning synchronization that has prevailed for more than 400 years
The study, described in the article Hypernetworks Induce Stable Hyperlocking, published in Nature Communications, paves the way for the discovery of new forms of interaction. “Rather than being described solely as the sum of pairwise influences, some interactions depend simultaneously on three or more components. The gravitational interaction among the Earth, the Sun, and the Moon, for example, can be broken down into pairwise interactions”, explained Tiago Pereira. “Other natural phenomena, however, do not follow this logic. Evidence indicates that the relationship between heavy rainfall in India and the El Niño phenomenon, for example, also depends on volcanic activity, which acts as a third mediator. In this case, the combined influence cannot be reduced to the simple sum of pairwise interactions.”
In modern science, the first researcher to describe synchronization was the Dutch physicist and mathematician Christiaan Huygens, after observing pendulum clocks hanging from the same beam. The phenomenon was recorded approximately 15 years before the publication of Newton’s laws, and since then, the adjustment of rhythms caused by interaction has remained at the center of mathematical research on dynamical systems and complex networks. The scientific community had consolidated the understanding that, when many systems interact within a network through pairwise interactions, systems with similar rhythms begin to behave in unison, forming a collective group in which every system has the same rhythm. As the strength of the interaction increases, new systems whose rhythms are close to the established collective rhythm join the synchronized group.
The question that motivated the study was precisely how this third mediator changes the collective behavior of these systems.
The article’s first author, mathematician Eddie Nijholt, began developing part of the theory underlying the discovery during his doctoral studies at the University of Amsterdam, in the Netherlands, where his research on complex systems received international recognition. The research advanced during his postdoctoral fellowship at the ICMC and was consolidated after he became a professor at the Institute, when the new phenomenon was identified.
“The result significantly broadens our understanding of how different natural and technological systems can generate collective behaviors”, said Nijholt. “By showing that collective synchronization can emerge exclusively from interactions involving three elements, we realized that many natural and technological phenomena may need to be reinterpreted from this new perspective.”
From theory to the laboratory
To demonstrate that the phenomenon did not exist only in mathematical models, the international team conducted experiments using networks of electrochemical oscillators—small nickel electrodes immersed in an acidic solution that produce oscillatory behavior. The researchers introduced perturbations and time delays specifically to prevent synchronization from emerging between pairs. Even so, the system spontaneously developed synchronization involving only groups of three elements.
“The experiment showed that the collective behavior was not hidden within traditional pairwise connections”, Pereira emphasized. “It was produced exclusively by the simultaneous interaction among three components, revealing a type of organization that classical models are simply unable to capture”. Although the discovery originated in mathematics, it offers a new framework for understanding complex systems across several fields of science.
In neuroscience, different brain regions often depend on the simultaneous activity of other areas to coordinate their functions. The new mathematical model may contribute to a better understanding of how these collective processes emerge. In particular, it shows that research must pay special attention to the formation of collective dynamics that are not present in pairwise interactions and may, in the future, help in the development of more accurate models for phenomena such as epileptic seizures.
In climate science, the theory offers a new way of representing situations in which variables such as temperature, atmospheric pressure, winds, and humidity influence one another simultaneously, a common condition in extreme events. Its applications also extend to technological systems. Power grids, autonomous systems, critical infrastructure, and digital platforms often display collective behaviors that cannot be explained solely by pairwise relationships. The new theory expands the tools available to model, predict, and control these dynamics.
In addition to USP, the study involved researchers from Imperial College London, in the United Kingdom; the Weierstrass Institute and the Potsdam Institute for Climate Impact Research, in Germany; and Saint Louis University, in the United States. Funded by the São Paulo Research Foundation (Fapesp) through RIDC-CeMEAI, by the National Council for Scientific and Technological Development (CNPq), and by the Serrapilheira Institute, the research made all computational codes and experimental data publicly available, allowing research groups around the world to use the findings to investigate new phenomena involving complex systems. Read the full study published in Nature Communications.
*Written by Raquel Vieira, from the ICMC Communications Office. Adapted for Jornal da USP
**Intern under the supervision of Simone Gomes
English version: Nexus Traduções, edited by Denis Pacheco
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