image: In the dynamics of network framework, the dynamics take place on the nodes, as depicted in the Figure by the variables u and v, while the interactions between the dynamical units are modeled via the links. When dealing with topological signals, the dynamics take place not only on the nodes, but also on the links, triangles and higher-order simplexes. As an example, we show a 2-simplicial complex, i.e., a simplicial complex made of nodes, links and triangles, where the dynamical variables are u, v and w which are defined on the nodes, links and triangles, respectively, and are coupled via the Bianconi-Dirac operator.
Credit: Science Tokyo
Networks are a powerful tool in the modeling of complex systems, which are composed of a large number of parts interacting with each other, with numerous applications ranging from neuroscience, to epidemiology, computer science, engineering and ecology, to name a few. About twenty five years ago, networks became popular in the modeling of complex systems for their ability to capture the interactions between the parts of a system in a simple way and with a universal and powerful formalism. As research is still evolving, scholars have found that the network formalism, despite its ductility, fails to capture interactions that involve multiple units at the same time, which is a behavior typical of a wide range of phenomena, e.g., in ecosystems, where the simultaneous interactions of multiple species affect their respective behavior. Networks cannot capture these relations and to model such interactions we need higher-order structures.
While classical networks represent relationships using nodes and edges, higher-order structures such as simplicial complexes and hypergraphs account for multi-agent interactions, making them particularly useful in modeling biological, social, and physical systems. Professor Bianconi has played an important role in advancing the mathematical foundations of higher-order networks, particularly through the study of topological signals and the Dirac-Bianconi operator. Topological signals extend the concept of graph signals, which are typically defined on nodes, to higher-dimensional structures such as edges, triangles, and beyond. This shift is crucial because many natural and artificial systems — ranging from brain networks to social interactions — exhibit dynamics that cannot be fully described by node-based models alone. The Dirac-Bianconi operator, inspired by quantum mechanics and differential geometry, provides a powerful generalization of the graph Laplacian. It encodes both local and global interactions across different topological dimensions, making it a valuable tool for studying higher-order diffusion, synchronization, and pattern formation. This approach has broad applications, particularly in neuroscience, where brain activity unfolds across networks of interconnected regions, and in climate science, where edge variables, such as the wind direction, can offer a more accurate description that traditional models fail to capture. Further applications come from data analysis and machine learning, in particular simplicial neural networks.
Now, a team led by Professor Bianconi, involving Universities from 8 different countries (Japan, UK, Spain, Italy, Belgium, Germany, Sweden and US), including Institute of Science Tokyo (Science Tokyo), has gathered the main progresses of this field of the past few years and the most exciting challenges ahead, in particular those involving nonlinear dynamics, signal processing and machine learning, which are among the most promising tools for applications in data analysis. The team has highlighted the interplay between topology and dynamics in the context of synchronization and Dirac-Turing pattern formation (see Figure), part to which the expertise of Science Tokyo researchers has played an important role. Lastly, the authors have studied the effects of triadic interactions, a kind of higher-order effects common in neuroscience and ecology, which causes a time-varying behavior of the network that can be chaotic and periodic. This research has been published in Nature Physics as a perspective article. The authors have also made available a Git Repository with supplementary materials and videos, including the codes for the simulations.
The group of Professor Hiroya Nakao, at the Department of Systems and Control Engineering of Science Tokyo, is involved in several projects regarding this cutting-edge research and, indeed, Professor Bianconi will visit his group next May to further enhance the collaboration between our institutions, thanks to a generous grant from the World Research Hub of Science Tokyo. The visit of Professor Bianconi is a great opportunity not only for researchers within our institution, but, more in general, for the Japanese community of complex systems. This newly published work highlights the strength of interdisciplinary cooperation in advancing scientific knowledge and addressing future challenges on a global scale.
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About Institute of Science Tokyo (Science Tokyo)
Institute of Science Tokyo (Science Tokyo) was established on October 1, 2024, following the merger between Tokyo Medical and Dental University (TMDU) and Tokyo Institute of Technology (Tokyo Tech), with the mission of “Advancing science and human wellbeing to create value for and with society.”
Journal
Nature Physics
Method of Research
Experimental study
Subject of Research
Not applicable
Article Title
Topology shapes dynamics of higher-order networks
Article Publication Date
19-Feb-2025