Northeastern researchers develop a unified framework that captures how both group heterogeneity and overlap drive explosive outbreaks
When ideas, behaviors, or diseases spread through a population, we typically think of transmission happening between two people—one person passing something to another. But in reality, contagion often happens within groups of two or more people—and this is particularly relevant for behavioral spread. Think of a conversation during dinner, a post in a group chat, or a tight group of high-school friends that suddenly adopt a new habit altogether.
New research from Northeastern University reveals that when contagion happens in groups of three or more people, the spread can change dramatically, often leading to explosive outbreaks that seem to come from nowhere. While explosive contagion in group interactions has been observed before, the new study published in Physical Review Letters breaks new ground in understanding how this happens. The team developed a single mathematical framework that simultaneously captures two crucial factors that, until now, have been difficult to study together: how much variation exists in the number of groups people belong to (heterogeneity), and how these groups of different sizes “overlap” with each other —meaning the same individuals can be part of multiple groups of different sizes.
"Previous models have shown that group interactions can cause explosive spreading, but they couldn't properly account for both heterogeneity and overlap in one framework", explains the research team led by Federico Malizia and István Z. Kiss from the Network Science Institute at Northeastern University London. "Our model is the first to analytically disentangle how these two factors work together to shape outbreak dynamics."
A unified mathematical framework
The researchers developed Group-Based Compartmental Modeling (GBCM),which extends classical epidemic models to simultaneously capture both the varying number of groups people belong to and how these groups overlap. What makes this framework unique is its ability to handle both features analytically in a single model. In fact, previous approaches had to choose: either model heterogeneity while ignoring overlap, or assume simplified structures that couldn't fully capture contact patterns characterizing real-world social systems. The GBCM framework overcomes these limitations through a novel mathematical approach that tracks how contagion flows through groups of different sizes. In their model, individuals can be in one of three states: susceptible (S), infected (I), or recovered (R). For behavioral or social contagion, these states could represent those who haven't adopted a behavior, those actively spreading it, and those who have moved past the spreading phase.
The spring-loading mechanism
The team's unified approach revealed how heterogeneity and overlap work together to create what they call a "spring-loading" effect. When groups of three people have no overlap with pairwise connections—imagine friends that always meet in group and never in pairs—a group-level adoption is hard to reach, as per standard models there should be a critical mass of adopters to convince the third.
"It's like loading a spring," the researchers explain. "Person-to-person interactions slowly seed infection into different groups, compressing the spring bit by bit. You could think of these cases as the 'innovators'. Nothing dramatic happens at first. Then, once enough groups reach the critical threshold needed for contagion to take off, the spring releases and the spread suddenly explodes all across the system"—in an extreme example, the entire high-school ends up wearing the same pair of sneakers.
But here's where their unified model reveals something new: the degree of heterogeneity determines how tightly the spring can be wound, while overlap determines how quickly it starts to compress. Previous models could see one or the other effect, but not how they interact. "What's powerful about having both in one model is we can see how they interact,"notes the team. "High heterogeneity creates the potential for explosion but overlap determines the pathway to get there."
Surprising interactions between structure and dynamics
One of the study's most important findings is that heterogeneity and overlap don't just add their effects—they interact in complex ways. When overlap between pairwise and three-person groups is high, group transmission can dominate from the very beginning, even though it has a higher infection threshold. But this effect is amplified or dampened depending on the level of heterogeneity in the network.
The unified model shows that in highly heterogeneous networks, even small amounts of overlap can trigger early group-dominated spreading. Conversely, in more homogeneous networks, high overlap is needed to achieve the same effect. This interaction between structural features couldn't be captured by previous models that treated them separately. "Having an analytical framework that handles both features means we can actually predict thresholds and outbreak sizes, not just observe them in simulations," explains the team. "This is crucial for understanding real systems where both heterogeneity and overlap are always present."
Broader implications
The ability to analytically treat heterogeneity and overlap together opens new possibilities for understanding complex spreading processes. While the model uses epidemiological language, its mathematical framework applies to any spreading process where group interactions matter—from innovation adoption to social movements. The work provides the first rigorous mathematical framework that can predict not just whether explosive transitions will occur, but also which structural features— heterogeneity, overlap, or their interaction—drive the explosion. This level of analytical detail was impossible when these features had to be studied separately.
The team's unified framework represents a significant advance in network science, providing tools to analyze complex spreading processes while accounting for the full richness of group interaction structures. As systems become increasingly interconnected, having models that can handle multiple structural features simultaneously becomes essential for understanding and predicting spreading dynamics in the real world.
Physical Review Letters study: Disentangling the Role of Heterogeneity and Hyperedge Overlap in Explosive Contagion on Higher-Order Networks
The research was conducted at Northeastern University London's Network Science Institute, with collaborators from Northeastern's Boston campus.
