Higher-order interactions: what does it mean to be connected?

Network science helps us to better understand the evolution of the highly interconnected world in which we live, and complex networks have become the main paradigm for modelling the dynamics of interacting systems. However, networks are intrinsically limited to describing pairwise interactions, whereas real-world systems are often characterized by higher-order interactions involving groups of three or more units and complex connections. Higher-order mathematical structures, such as hypergraphs and simplicial complexes, are therefore a better tool to map the real organization of many social, biological and man-made systems.

In a new Perspective article out in Nature Physics, an international team of scientists led by ISI Senior Research Scientist Giovanni Petri, and including ISI Principal Scientist Yamir Moreno and ISI Postdoctoral Associate Guilherme Ferraz de Arruda, highlights modern evidence of collective behaviours depicted by higher-order interactions and outlines three key challenges for the physics of higher-order systems.

“The idea that our social interactions, the interplay between regions of our brains, or between species in an ecosystem, are better described in terms of groups is not new. However, it has been gaining traction thanks to a combination of new mathematical and computational tools with new datasets. Like a perfect storm, it brewed for a while and is now unleashing a wave of new ideas, results and - of course - open questions. Here we focused on what these might be.” explains Dr. Giovanni Petri, who led the paper and for the last few years worked at the interface between complex systems and higher-order descriptions.

Among these challenges, the first is understanding how rich and general is the phenomenology of dynamical processes like --social spreading or synchronization-- when higher-order interactions are added to the mix: indeed, network modelling approaches that take higher-order interactions into account suggest that explosive phenomena (like the sudden appearance of new social fads or norms) appear naturally and can therefore be studied more easily within this framework. The second one involves topological dynamical processes: by endowing groups with their own states, it becomes possible to define couplings between higher-order interactions of different dimensions and study how the states of groups of all sizes evolve in time. In this way, a process and the substrate on which it takes place merge together becoming interdependent. Finally, the third challenge refers to a crucial ingredient in modelling real systems: the reconstruction of higher-order interactions from incomplete and lower-order data, e.g. how can we reconstruct the group structure from observations of pairwise interactions.

Each of these points is hard (“and exciting!“, adds Dr. Petri) in itself. Together, the authors conclude, they could create opportunities for a wider dialogue on the physics of dynamical systems, and pave the way forward for network science.

“The physics of higher-order interactions in complex systems”, Federico Battiston, Enrico Amico, Alain Barrat, Ginestra Bianconi, Guilherme Ferraz de Arruda, Benedetta Franceschiello, Iacopo Iacopini, Sonia Kéfi, Vito Latora, Yamir Moreno, Micah M. Murray, Tiago P. Peixoto, Francesco Vaccarino and Giovanni Petri, Nature Physics, 4th October 2021. Link: https://www.nature.com/articles/s41567-021-01371-4.