Researchers use networks to model the dynamics of coupled systems ranging from food webs to neurological processes. Those models originally focused on peer-to-peer interactions, or behaviors that arise from interactions between two entities. But in recent years, network theorists have been asking, what about phenomena involving three or more? In medicine, antibiotic combinations can fight a bacterial infection in a different way than they would on their own. In ecology, survival strategies can arise from three competing species that are not observable when observing individual pairs.
Network theorists call these phenomena “higher order interactions.” Understanding them can be tricky, says Yuanzhao Zhang, an SFI Complexity postdoctoral fellow who uses network theory to study collective behaviors. The way the network is represented, for example, can influence how phenomena emerge.
In a new article on nature Communications, Zhang and colleagues show how the choice of network representation can influence the observed effects. His work focuses on the phenomenon of synchronization, which arises in systems ranging from circadian clocks to vascular networks.
Previous studies have suggested that these behaviors can improve synchronization, but the question of when and why that happens has remained largely unexplored.
“We don’t have a very good understanding of how the higher-order coupling structure influences synchronization,” says Zhang. “For systems with non-pair interactions, we want to know how their representation affects the dynamics.”
Zhang and his colleagues studied two frameworks used to model interactions beyond those of pairs: hypergraphs and simplicial complexes. Hypergraphs use so-called “hyperedges” to connect three or more nodes, analogous to how conventional networks use edges. Simplicial complexes are more structured and use triangles (and higher-dimensional surfaces analogous to triangles) to represent those connections. Simplicial complexes are more specialized than general hypergraphs, Zhang says, meaning that to model higher-order interactions, triangles can only be added in regions that are already well connected. “It is this rich-gets-richer effect that makes simplicial complexes more heterogeneous than hypergraphs in general,” says Zhang.
Researchers generally do not consider the two frameworks to be very different. “People have been using those two frameworks interchangeably, choosing one or the other based on technical convenience,” Zhang says, “but we found that they can be very different” in how they influence timing.
In the paper, Zhang and colleagues reported that networks modeled with hypergraphs easily lead to synchronization, while simple complex ones tend to complicate the process due to their highly heterogeneous structure. That suggests that choices in higher-order representations may influence the outcome, and Zhang suspects that the results may spill over to other dynamic processes such as diffusion or contagion.
“Structural heterogeneity is important not only in timing, but is fundamental to most dynamical processes,” he says. “Whether we model the system as a hypergraph or a simplicial complex can drastically affect our conclusions.”
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