Topology Drives Complexity in Brain, Climate, and AI: A Groundbreaking Study

How topology drives complexity in brain, climate and AI

A study led by Professor Ginestra Bianconi from Queen Mary University of London, in collaboration with international researchers, has unveiled a transformative framework for understanding complex systems. Published in Nature Physics, this paper establishes the new field of higher-order topological dynamics, revealing how the hidden geometry of networks shapes everything from brain activity to artificial intelligence. "Complex systems like the brain, climate, and next-generation artificial intelligence rely on interactions that extend beyond simple pairwise relationships. Our study reveals the critical role of higher-order networks, structures that capture multi-body interactions, in shaping the dynamics of such systems," said Professor Bianconi. By integrating discrete topology with non-linear dynamics, the research highlights how topological signals, dynamical variables defined on nodes, edges, triangles, and other higher-order structures, drive phenomena such as topological synchronization, pattern formation, and triadic percolation. These findings not only advance the understanding of the underlying mechanisms in neuroscience and climate science but also pave the way for revolutionary machine learning algorithms inspired by theoretical physics. "The surprising result that emerges from this research," Professor Bianconi added, "is that topological operators, including the Topological Dirac operator, offer a common language for treating complexity, AI algorithms, and quantum physics. " From the synchronized rhythms of brain activity to the dynamic patterns of the climate system, the study establishes a connection between topological structures and emergent behavior. For instance, researchers demonstrate how higher-order holes in networks can localize dynamical states, offering potential applications in information storage and neural control. In artificial intelligence, this approach may lead to the development of algorithms that mimic the adaptability and efficiency of natural systems. "The ability of topology to both structure and drive dynamics is a game-changer," Professor Bianconi added. "This research sets the stage for further exploration of dynamic topological systems and their applications, from understanding brain research to formulating new AI algorithms." This study brings together leading minds from institutions across Europe, the United States, and Japan, showcasing the power of interdisciplinary research. "Our work demonstrates that the fusion of topology, higher-order networks, and non-linear dynamics can provide answers to some of the most pressing questions in science today," Professor Bianconi concludes.

Groundbreaking study reveals how topology drives complexity in brain, climate, and AI

A groundbreaking study led by Professor Ginestra Bianconi from Queen Mary University of London, in collaboration with international researchers, has unveiled a transformative framework for understanding complex systems. Published in Nature Physics, this pioneering study establishes the new field of higher-order topological dynamics, revealing how the hidden geometry of networks shapes everything from brain activity to artificial intelligence. "Complex systems like the brain, climate, and next-generation artificial intelligence rely on interactions that extend beyond simple pairwise relationships. Our study reveals the critical role of higher-order networks, structures that capture multi-body interactions, in shaping the dynamics of such systems," said Professor Bianconi. By integrating discrete topology with non-linear dynamics, the research highlights how topological signals, dynamical variables defined on nodes, edges, triangles, and other higher-order structures, drive phenomena such as topological synchronization, pattern formation, and triadic percolation. These findings not only advance the understanding of the underlying mechanisms in neuroscience and climate science but also pave the way for revolutionary machine learning algorithms inspired by theoretical physics. "The surprising result that emerges from this research" Professor Bianconi added, is that topological operators including the Topological Dirac operator, offer a common language for treating complexity, AI algorithms, and quantum physics. " From the synchronised rhythms of brain activity to the dynamic patterns of the climate system, the study establishes a connection between topological structures and emergent behaviour. For instance, researchers demonstrate how higher-order holes in networks can localise dynamical states, offering potential applications in information storage and neural control. In artificial intelligence, this approach may lead to the development of algorithms that mimic the adaptability and efficiency of natural systems. "The ability of topology to both structure and drive dynamics is a game-changer," Professor Bianconi added. This research sets the stage for further exploration of dynamic topological systems and their applications, from understanding brain research to formulate new AI algorithms. " This study brings together leading minds from institutions across Europe, the United States, and Japan, showcasing the power of interdisciplinary research. "Our work demonstrates that the fusion of topology, higher-order networks, and non-linear dynamics can provide answers to some of the most pressing questions in science today," Professor Bianconi remarked.

Topology shapes dynamics of higher-order networks - Nature Physics

Higher-order networks capture the many-body interactions present in complex systems, shedding light on the interplay between topology and dynamics. The theory of higher-order topological dynamics, which combines higher-order interactions with discrete topology and nonlinear dynamics, has the potential to enhance our understanding of complex systems, such as the brain and the climate, and to advance the development of next-generation AI algorithms. This theoretical framework, which goes beyond traditional node-centric descriptions, encodes the dynamics of a network through topological signals -- variables assigned not only to nodes but also to edges, triangles and other higher-order cells. Recent findings show that topological signals lead to the emergence of distinct types of dynamical state and collective phenomena, including topological and Dirac synchronization, pattern formation and triadic percolation. These results offer insights into how topology shapes dynamics, how dynamics learns topology and how topology evolves dynamically. This Perspective primarily aims to guide physicists, mathematicians, computer scientists and network scientists through the emerging field of higher-order topological dynamics, while also outlining future research challenges. Understanding, modelling and predicting the emergent behaviour of complex systems are among the biggest challenges in current scientific research. Major examples include brain function, epidemic spreading and climate change. Through the use of graphs and networks to represent interactions, network science has provided a powerful theoretical framework that has deeply transformed the theory of complex systems. Networks encode relevant information about the complex systems that they represent, and their statistical and combinatorial properties strongly affect the unfolding of dynamical processes and critical phenomena. The success of network science stems from the simplicity of its basic assumption: a complex system can be described in terms of interactions between its elements. However, this assumption also highlights a limitation of conventional network representations, as they encode only pairwise interactions. As a matter of fact, representing a complex system with just pairwise interactions provides only an approximation of reality. Assuming the presence of many-body interactions is certainly more appropriate for many, if not all, systems. In high-energy physics, vertices of Feynman diagrams involve the creation and annihilation of more than two particles, with quantum chromodynamics notably admitting four-gluon vertices. Moreover, quantum many-body wave functions display strong higher-order quantum correlations and, owing to entanglement, cannot be fully described by two-point correlation functions. In inferential problems, a general multivariate distribution must involve higher-order interactions (as in higher-order graphical models, for instance). Likewise, in network science, pairwise networks cannot capture the many-body interactions that are present in the brain, social networks, ecosystems and inferential financial models. Allowing higher-order interactions leads to the formulation of higher-order networks that include interactions between two or more nodes. Building blocks such as triangles, tetrahedra or hypercubes form the backbone of higher-order networks, yielding a topological description of complex systems that significantly alters our understanding of the interplay between structure and dynamics. Topology involves the study of shapes and their invariant properties, such as Betti numbers and Euler characteristics. It plays a core role in topological data analysis, an approach for analysing the shape of high-dimensional and noisy data, and in one of its main tools: persistent homology. In particular, topological data analysis has already been shown to be key to detecting higher-order aspects of brain networks, offering a very powerful set of tools to characterize different states of brain activity. Topology has also been instrumental in the development of topological filtering algorithms, particularly for the analysis of financial data. Higher-order network structure can be investigated under the lens of topological and homological percolation, the latter characterizing the emergence of large cycles and of higher-dimensional holes. Finally, by assigning weights to the higher-order interactions, weighted homology and cohomology have been shown to enhance the representation power of simplicial complexes, which can encode hypergraph data without loss of information. Topology is crucial not only to characterize the structure of complex systems, but also to capture their higher-order dynamics. For the purpose of this Perspective, we are particularly interested in the emergent field of topological dynamics of higher-order networks, which combines topology with nonlinear dynamics. Specifically, higher-order topology unlocks fundamental mechanisms for higher-order topological diffusion, higher-order topological synchronization, topological pattern formation, triadic percolation, triadic neural networks and topological machine learning algorithms. These phenomena may be key to transforming our understanding of complex phenomena in neuroscience and climate change, and to formulating a new generation of physics-inspired machine learning algorithms (Fig. 1). This Perspective focuses on the shift that the adoption of higher-order networks implies for the description of the interplay between topology and dynamics in complex systems. We outline key results and recent developments in the field, and the open challenges that future research must address. Accompanying supporting code, movies and material can be found in ref. .