Do group interactions really require higher-order models?

• A new research, published in Physical Review Research, brings a key contribution to a current and intense scientific debate on the modeling of group interactions and the use and abuse of complex mathematical frameworks to predict collective behavior.
• The findings, developed by researchers at IFISC, show that a broad class of group interactions can be exactly reduced to pairwise interactions. The mapping allows researchers to study complex group dynamics using well-established network methods.

How does social pressure shape individual choices? Whether it is a political committee trying to reach a consensus, a circle of friends adopting a new mobile app, or an online community propagating a rumor, human interactions rarely occur in isolated pairs. Over the last decade, network science has experienced a major paradigm shift: moving past conventional complex networks representing pairwise interactions, to embrace hypergraphs.

In a hypergraph, traditional links are replaced by hyperedges, which can group any number of people simultaneously (three, four, or dozens of them). This structure has emerged as a natural way to represent social conformity, collective decision-making, and other processes that involve simultaneous interactions among several individuals. However, the growing popularity of hypergraphs has also fueled a fundamental scientific debate. Do group interactions require an explicitly higher-order description, or can their effects be captured through simpler pairwise interactions? Resolving this question is crucial for understanding when higher-order models are truly needed and when simpler descriptions are sufficient.

Reducing complexity without losing precision

A new study published in Physical Review Research by researchers Jaume Llabrés, Raúl Toral, Maxi San Miguel and Federico Vázquez from the Institute for Cross-Disciplinary Physics and Complex Systems (IFISC, CSIC-UIB) has provided an answer to this problem valid in some general setups. The team has demonstrated that a general class of social impact models on hypergraphs can be represented exactly as pairwise interaction models on a conventional network.

What makes this finding remarkable is that the equivalence is exact at the microscopic level: the probability of any individual changing their opinion or state remains perfectly intact after the reduction. To achieve this, the researchers derived a mathematical framework that encodes the influence of entire groups into weighted pairwise interactions, allowing complex group effects to be represented within a much simpler network structure.

Linear vs. Non-linear

To test their theory, the authors introduced hypergraph-voter models, virtual scenarios where individuals change their opinions based on group pressure. The study revealed that the behavior of the simplified network depends entirely on the nature of that social pressure. In a linear scenario, where a group's influence scales proportionally to its size, the simplified network's links remain static over time. Conversely, when modeling complex non-linear contagion, like peer pressure that requires reaching a specific threshold before someone changes their mind, the network weights become dynamic, constantly fluctuating alongside the shifting opinions of the system.

"Our main result demonstrates that what appears at first glance to be a highly complex, multi-body group interaction can actually be broken down into a combination of effective pairwise interactions" explains Jaume Llabrés, researcher at IFISC. "Once the dynamics is expressed in terms of pairwise interactions, we can use classic, well-established analytical tools in network science, such as pair approximations, to describe the macroscopic evolution of the system with high precision”.

A guide for real-world data analysis

To validate their framework, the scientists carried out extensive numerical simulations across several structural environments, including various types of random and regular hypergraphs. They measured key macroscopic observables, such as the total fixation time required for the population to reach a complete consensus, and the density of active links connecting people with opposing opinions throughout the process.

The simulation results perfectly overlapped with the theoretical predictions. Surprisingly, the study proved that in the linear case, the macroscopic evolution of the system is completely insensitive to higher-order group topologies; the weight heterogeneity becomes irrelevant, and the system behaves exactly like a standard network. Even in the most complex non-linear scenarios with high connectivity, the researchers discovered that a standard, unweighted network is capable of reproducing the main trends of the hypergraph with great fidelity, acting as a minimal model when dealing with empirical real-world data that carries a high margin of measurement uncertainty.

Resolving a long-standing academic debate

This study directly contributes to the ongoing academic discussion regarding when higher-order frameworks are necessary to describe complex systems. By identifying the conditions under which group interactions can be represented exactly through effective pairwise interactions, the IFISC team provides a clearer framework for determining which level of description is required to model collective behavior.

Rather than arguing for or against higher-order approaches, the work helps distinguish situations in which hypergraphs capture genuinely new dynamical effects from those in which simpler network representations are sufficient. This insight may prove valuable across a wide range of disciplines, from statistical physics and network science to social dynamics and collective behavior.

Llabrés, J., Toral, R., San Miguel, M., & Vázquez, F. (2026). Reducibility of higher-order to pairwise interactions: Social impact models on hypergraphs. Physical Review Research, 8(2), 023250. https://doi.org/10.1103/wy1x-3px8