The study of complex systems published in Physical Reviews Letters supports a growing trend that focuses more on analyzing the collective behavior of a system rather than discovering the underlying mechanisms of interaction.

When we observe a flock of starlings swirling through the sky in perfect coordination - a phenomenon known as murmuration - we witness the elegant interplay of individual actions creating collective behavior. In trying to understand these fascinating patterns, researchers can isolate simple rules based on an individual bird's field of view and the distance separating it from its neighbors, but there's always the question of whether the model actually captures the processes driving interactions between birds (Fig. 1).

This is a general problem in complex systems research, which comes down to distinguishing mechanisms (the rules governing interactions) from behaviors (the observable patterns that emerge).

Representative networks of interacting individuals, or nodes, are a good way to study mechanisms versus behaviors. Until now, researchers have focused on pairwise interactions, but many systems also include higher-order interactions between several nodes. The impact of these higher-order mechanisms on behavior has not been clearly established. Thomas Robiglio, from the Central European University in Vienna, and his colleagues, including Maxime Lucas (CR FNRS - UNamur) addressed this question. They considered networks with higher-order interactions and evaluated the resulting behaviors in terms of statistical dependencies between node values.

The researchers identified higher-order behavioral signatures which, unlike their pairwise counterparts, reveal the presence of higher-order mechanisms. Their findings open up new avenues for distinguishing mechanisms and behaviors when studying complex systems - a distinction that is crucial for the study of inference in network science, neuroscience, the social sciences and beyond.

This study is also the subject of a "Featured in Physics" and "Editor's suggestion" article, and a "commentary" article at the journal's request, available on their website in English in full.

The study, entitled "Topology shapes dynamics of higher-order networks" proposes a theoretical framework specifically designed to understand complex higher-order networks where several agents interact at the same time and thus generalize networks with their interactions in pairs. More precisely, the study shows how topology shapes dynamics, how dynamics learns topology and how topology evolves dynamically.

The aim of this work is to introduce physicists, mathematicians, computer scientists and network scientists to this emerging research field, as well as to define future research challenges where discrete topology and nonlinear dynamics mix.

With the data in their possession, the researchers show that real-life complex systems such as the brain, chemical reactions and neural networks can be easily modeled as higher-order networks, characterized by multi-body connections indicating the fact that several elements of the system interact simultaneously.

This international team is convinced that the visibility of their work through this publication in Nature Physics will open the door to new collaborations with other disciplines that rely on network analysis to study real complex systems.

Kudos to the team for this publication!

After a Master's degree in physics (University of Florence, June 1995), Timoteo Carletti pursued his doctoral studies in Florence (Italy) and Paris (France) at IMCCE, finally defending his doctoral thesis in mathematics in February 2000.

He moved to Belgium in 2005, and was hired at the University of Namur as a lecturer, then as a professor (2008), and finally as a full professor (2011) in the Mathematics Department of the Faculty of Science. In 2010, he was one of the founders of the Namur Center for Complex Systems (now the Namur Institute for Complex Systems - naXys), which he headed until December 2014.

This article is taken from the "Impact" section of the December 2024 issue of Omalius magazine.

On the night of January 31, 1953, the North Sea suddenly rose by almost four meters, submerging parts of the Netherlands and Belgium. The disaster caused the death of over 2,500 people, as well as considerable damage. According to Anna Kiriliouk, lecturer in statistics at UNamur's Department of Mathematics and EMCP Faculty, this exceptional event truly marked "the beginning of the development of extreme value theory, with the development of the first extreme value construction project"

The Delta Plan, as it is called, is a system of dikes that protects the Netherlands against the risk of flooding, with these dikes overtopping once every 10,000 years. A rare danger, certainly, but not zero, which "could not have been calculated using conventional statistics, which are very poorly adapted to rare events", believes the mathematician.

While climate change is often discussed in terms of averages, such as rising temperatures and sea levels, it also has the consequence of increasing the frequency of extreme weather events, with significant repercussions for our societies. "In other words, the risk increases along with the concentration of greenhouse gases (GHGs) in the atmosphere", summarizes the researcher. "Thus, a flood calculated in 1953 to occur only every 10,000 years does not have the same significance as today. The latter could happen more frequently, for example every 1,000 years."

While extreme weather events are on the increase, it's difficult in practice to attribute any particular flood or drought to climate change. With this in mind, Anna Kiriliouk has just been awarded an interdisciplinary research project,named EXALT, in collaboration with UCLouvain. "It involves both climatologists and statisticians, she reveals.

In practice, the EXALT project will therefore calculate the probabilities of an extreme event occurring, and compare this probability with that of the same situation in a world where GHG emissions would not have increased. "Of course, we don't have real data from such a world", says Anna Kiriliouk. "We are therefore basing ourselves on alternative climate simulations, the quality of which we will moreover compare, with a focus on extreme events."

Divided into three working groups, the EXALT project will seek in particular to determine the role of climate change in the occurrence of floods, as well as heat waves and drought in Europe. And to do so as realistically as possible: "One of the things we want to incorporate into climate models concerns the dependency between data," explains Anna Kiriliouk. "For example, if a heat wave hits Namur, there's a good chance that the same temperatures will affect Louvain-La-Neuve. We therefore say that there is a strong spatial dependency between these two data. However, this dependence is probably not at all valid for rain, which is much more heterogeneous. By taking into account all these variables, both spatial and temporal, we hope to improve existing models."

A third working group will study much more distant areas, located in Antarctica. "Until 2016, the extent of the Antarctic ice pack was increasing, before abruptly decreasing", the researcher illuminates. "Or, according to the models, this event was considered almost impossible. But with one of EXALT's partners, we began to analyze the evolution of pack ice extent using extreme value theory. With the latter, this sudden drop was no longer so improbable. This gave us confidence in our approach, which is all the more important when the state of the pack ice has such a strong influence on other climate variables."

This interaction between several climatic processes is, moreover, the subject of a second project just obtained by Anna Kiriliouk and funded by an FNRS Mandat d'Impulsion Scientifique. "The aim is to make it possible to study what we call compound events", explains the researcher. "During extreme climatic situations, we usually associate very high or low values simultaneously, such as a lack of rain and high temperature, resulting in an intense drought. But in the case of compound phenomena, we find that the combination of several variables, albeit in a moderate state, results in a severe and unusual event."

In 2017, for example, Hurricane Sandy, which struck the US coastline, is considered a compound event. While North Atlantic hurricanes usually dissipate in mid-ocean, this one coincided with onshore winds and a high tide, leading to massive flooding of New York and the surrounding area.

"In this project, we will therefore try to include more flexibility between the different variables, by introducing different degrees of dependence, the mathematician elaborates. "We're also going to try, as a second step, to group the dependencies together, in order to lighten the models, which become more and more complex as we add nuances to them. And once these models have been modified, we'll apply them to recent events to test their realism."

Mais qu’est-ce que le formalisme mathématique ?  Il s’agit d’un ensemble d’équations qui a pour objectif de représenter de manière non-ambigüe un objet d’étude et son évolution temporelle pour en faire un modèle.  Il y a une vingtaine d’années, les réseaux sont devenus populaires dans la modélisation des systèmes complexes pour leur capacité à capturer les interactions entre les parties d’un système de manière simple avec un formalisme universel.  Si l’on pense aux réseaux sociaux, on peut considérer les personnes comme les nœuds du réseau et leurs relations d’amitié comme les arrêtes de ce réseau. Mais ce formalisme est beaucoup plus général, car les nœuds peuvent être des aéroports et les arrêtes des connexions en avion, ou des molécules avec leurs réactions. Dans beaucoup de contextes, l’information contenue dans les arrêtes n’est pas symétriques, on parlera donc d’arêtes dirigées et de réseaux dirigés. Pour reprendre l’exemple des réseaux sociaux, cela s’apparenterait à Twitter, où l’on peut suivre une personne qui ne nous suit pas forcément en retour. La comparaison peut aussi être faite avec le fonctionnement de notre cerveau dans lequel les nœuds sont les neurones et les axones et les dendrites, eux, sont les arrêtes, le flux d’information entre neurones a souvent une direction privilégiée.

La recherche étant toujours en évolution, les chercheurs ont constaté que le formalisme des réseaux n’arrivait pas à capturer les interactions qui impliquent plusieurs corps en même temps, comportement typique d’une grande variété de phénomènes. Pensez, par exemples, à la manière dont vous vous comportez avec une personne lorsque vous êtes à deux. Agissez-vous de la même manière que lorsque vous êtes en groupe ? La raison de cette limitation des réseaux est assez simple : dans un réseau, une arête ne peut connecter qu’à deux nœuds, donc pas de place pour des interactions avec trois ou plus. En outre, assez souvent ces interactions multiples ont une direction privilégiée.  Par exemple, dans les phénomènes de peer pressure (pression par les pairs) et bullying (intimidation) en sociologie, ce sont des interactions de groupe mais dirigées vers une ou plusieurs personnes. Dans le domaine des réactions chimiques, plusieurs composants doivent réagir ensemble, mais il y a une direction privilégiée donnée par la thermodynamique.

L’idée développée par les chercheurs Namurois avec leurs collègues Italiens est de considérer un formalisme qui tienne compte en même temps des interactions multiples et de leur direction. Le groupe de recherche de l’UNamur étudiait déjà les interactions asymétriques, mais dans le cadre des réseaux. De l'autre côté, le groupe de recherche de l’Université de Catania avait récemment développé un formalisme pour traiter les interactions multiples mais symétriques.  Leur travail intitulé « Stability of synchronization in simplicial complexes » avait été publié dans Nature Communications.

Grâce à la collaboration entre ces deux groupes, les chercheurs ont pu mettre leurs compétences en commun et étendre le formalisme italien à l’approche namuroise, en rendant possible le développement d’une théorie encore plus générale, incluant l’asymétrie dans les interactions à plusieurs corps.

L’idée-clé est la suivante : on peut décomposer une interaction à plusieurs corps symétriques comme la somme des interactions élémentaires à plusieurs asymétriques. Dans la figure ci-contre, adaptée de l’article, on peut considérer l’interaction entre Riri, Fifi et Loulou comme la somme de l’interaction de Riri et Fifi vers Loulou, de Riri et Loulou vers Fifi ou encore de Fifi et Loulou vers Riri.

Le nouveau formalisme a permis d’étudier la dynamique des systèmes représentés par ces hypergraphes, qu’ils ont appelés M-dirigés (M-directed, en anglais, où M est le nombre de nœuds qui reçoivent l’information, M=1 dans l’exemple de la figure), et étendre des résultats classiques dans ce nouveau contexte, en ouvrant la porte à des nouvelles et excitantes recherches futures. La théorie développée pourra être utilisée dans des disciplines différentes.  Des sciences sociales aux études épidémiologiques, nombreuses sont les disciplines où on a des évidences expérimentales des interactions à plusieurs corps dirigées.  Dès maintenant, les chercheurs pourront ajouter un outil de plus à leur boîte à outils pour construire des modèles plus précis. 

Cette collaboration a été rendue possible grâce au programme Erasmus+, qui a permis à Luca Gallo, doctorant de l’Université de Catania et premier auteur de l'article, de séjourner à l’UNamur pendant 9 mois, et à Riccardo Muolo, doctorant FNRS FRIA à l'UNamur et co-auteur de l’article, de séjourner dans le groupe de Catania pendant deux semaines. Supervisés par le Professeur Carletti à Namur et par le Professeur Frasca à Catania, les deux doctorants ont pu travailler ensemble et apprendre les techniques analytiques et numériques des deux groupes de recherche.

Ces résultats montrent encore une fois l’importance des projets collaboratifs dans la recherche, qui favorisent l’échange de savoir-faire et l’émergence de nouvelles idées.

Bravo aux deux équipes pour cette publication !

University of Catania (Italy)

• Prof Mattia Frasca
• Prof Vito Latora (also Queen Mary University - London)
• Dr Valentina Gambuzza
• Luca Gallo

University of Namur (Belgium)  - naXys Institute

• Prof Timoteo Carletti
• Riccardo Muolo (doctorant FNRS FRIA)