Scientists Uncover How Hidden Interactions Shape Complex Networks

Traditional networks fail to capture multi-agent interactions in ecosystems, neuroscience, and physics. A new study shows how higher-order structures like hypergraphs and simplicial complexes revolutionize our understanding of real-world complexity.

Perspective: Topology shapes dynamics of higher-order networks. Image Credit: Shutterstock AI

Networks are powerful tools for modeling complex systems, which are composed of many parts interacting with each other. They have numerous applications, ranging from neuroscience to epidemiology, computer science, engineering, and ecology.

About twenty-five years ago, networks became popular in modeling complex systems because of 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 network formalism, despite its ductility, fails to capture interactions that involve multiple units simultaneously. This behavior is typical of a wide range of phenomena, such as ecosystems, where the simultaneous interactions of multiple species affect their respective behavior. Networks cannot capture these relations; we need higher-order structures to model such interactions.

While classical networks represent relationships using nodes and edges, higher-order structures such as simplicial complexes and hypergraphs account for multi-agent interactions. These structures are 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, mainly through studying 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—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 than 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 eight countries (Japan, UK, Spain, Italy, Belgium, Germany, Sweden, and the US), including the Institute of Science Tokyo (Science Tokyo), has gathered the main progress made in this field over the past few years and the most exciting challenges ahead. These challenges include 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 of which Science Tokyo researchers' expertise has played an important role. Lastly, the authors have studied the effects of triadic interactions, a kind of higher-order effect 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 the journal Nature Physics as a perspective article. The authors have also made available a Git Repository with supplementary materials and videos, including the simulation codes.

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.

Professor Bianconi's visit is a great opportunity not only for researchers within our institution but also 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.