An international team including researchers from the UNamur publish in the prestigious journal Communications Physics of the Nature group

Networks are fundamental in modeling complex systems, systems composed of an incredibly large number of interacting parts. The applications are numerous, whether in neuroscience, epidemiology, but also in computer science and engineering. A collaboration between the University of Catania (Italy) and the University of Namur, guided by Professor Timoteo Carletti of the Department of Mathematics (naXys Institute), has developed a new formalism that allows the modeling of systems where several parts interact at the same time (multi-body) and asymmetrically. This research was published in the prestigious journal Communication Physics, from the Nature group.

This article is also available in French here...

What is mathematical formalism? It is a set of equations which aims to represent in an unambiguous way an object of study and its temporal evolution to make a model of it. About twenty years ago, networks became popular in the modeling of complex systems for their ability to capture the interactions between parts of a system in a simple way with a universal formalism. If we think of social networks, we can consider people as the nodes of the network and their friendships as the edges of this network. But this formalism is much more general because the nodes can be airports and the edges the aircraft connections, or molecules with their reactions. In many contexts, the information contained in the edges is not symmetrical, so we speak of directed edges and directed networks. To take the example of social networks, it would be similar to Twitter, where we can follow a person who does not necessarily follow us in return. The comparison can also be made with the functioning of our brain in which the nodes are the neurons and the axons, and the dendrites are the edges, the flow of information between neurons often has a privileged direction.

As research is still evolving, researchers have found that network formalism fails to capture interactions that involve multiple bodies at the same time, a typical behaviour of a wide variety of phenomena. Think, for example, of how you behave with one person when you are together. Do you act the same way as when you are in a group? The reason for this limitation of networks is quite simple: in a network, an edge can only connect to two nodes, so there is no room for interactions with three parts or more. Moreover, quite often these multiple interactions have a privileged direction. For example, the phenomena of peer pressure and bullying in sociology are group interactions but directed towards one or more people. In the field of chemical reactions, several components must react together, but there is a preferred direction given by thermodynamics.

The idea developed by researchers from Namur together with their Italian colleagues is to consider a formalism that simultaneously considers multiple interactions and their direction. The UNamur research group was already studying asymmetric interactions, but within the framework of networks. On the other hand, the research group from the University of Catania had recently developed a formalism to deal with multiple but symmetric interactions. Their work entitled “Stability of synchronization in simplicial complexes” was published in Nature Communications.

Thanks to the collaboration between these two groups, the researchers were able to pool their skills and extend the Italian formalism to the Namur approach, making possible the development of an even more general theory, including the asymmetry in interactions at several bodies. The key idea is the following: one can decompose an interaction with several symmetrical bodies as the sum of the elementary interactions with several asymmetrical ones.

In the figure below, adapted from the article, we can consider the interaction between Huey, Dewey and Louie as the sum of the interaction of Huey and Dewey towards Louie, of Huey and Louie towards Dewey or even of Dewey and Louie towards Huey.

Figure adapted from the article

The new formalism made it possible to study the dynamics of the systems represented by these hypergraphs, which they called M-directed (where M is the number of nodes which receive the information, M=1 in the example in the figure), and extend classical results in this new context, opening the door to new and exciting future research. The theory can be used in different disciplines. From social sciences to epidemiological studies, there are many disciplines where there is experimental evidence of directed many-body interactions. From now on, researchers will be able to add one more tool to their toolbox to build more accurate models.

This collaboration was made possible thanks to the Erasmus+ program, which enabled Luca Gallo, doctoral student from the University of Catania and first author of the article, to stay at the UNamur for 9 months, and Riccardo Muolo, FNRS FRIA doctoral student at the UNamur and co-author of the article, to stay with the Catania group for two weeks. Supervised by Professor Carletti in Namur and by Professor Frasca in Catania, the two doctoral students were able to work Erasmus+ together and learn the analytical and numerical techniques of the two research groups.

These results once again show the importance of collaborative projects in research, which promote the exchange of know-how and the emergence of new ideas.

Congratulations to both teams for this publication!

Link to the article in Communications Physics : "Synchronization induced by directed higher-order interactions"