Scientists discuss how higher-order interactions can transform a system on large, even global scales.
Networks, consisting of nodes and their connections, assist researchers in modeling dynamic systems such as disease transmission or how the brain processes information. While pairwise interactions between nodes may illustrate relationships between individuals – like neuron connections in the brain – researchers also examine interactions involving three or more nodes. These higher-order interactions unveil changes and phenomena that are not evident when focusing solely on pairs.
Yuanzhao Zhang, an SFI Complexity Postdoctoral Fellow, has investigated the impact of higher-order interactions on systems at smaller scales. In a recent study published in Science Advances, he shares his findings on how these interactions can alter systems on larger, even global scales. “Our goal was to understand how they transform the entire landscape,” he explains.
Zhang and his team discovered that higher-order interactions can create deeper “basins of attraction,” which represent groups of starting points that eventually reach the same state as time progresses. In a pendulum analogy, the lowest point is considered an attractor, where every potential starting point resides in the basin of attraction because they all ultimately converge there. In the case of a brain working through a complex mathematical problem, the thought processes leading to a solution – ideally the correct one – exist within the basin of attraction. A deeper basin indicates greater stability for these solutions; that is, starting points reach the lowest point more rapidly or recover quickly from minor disturbances.
Interestingly, however, Zhang and his researchers found that as the basins deepen, they also become narrower. While starting points that do make it into the basin arrive more quickly, there are fewer starting points that reach the bottom overall. “When starting from a random point in the landscape, we seem to never arrive at [the basins of attraction],” Zhang notes. Higher-order interactions introduce a type of nonlinearity that has not been widely examined.
The researchers observed this behavior by testing a specific type of network, but Zhang theorizes that the phenomena of shrinking and deepening basins is likely applicable to dynamic systems in general – and that higher-order interactions may facilitate the emergence of new basins. “With their introduction, many new attractors that previously didn’t exist could emerge,” he states.
This new research could be instrumental in exploring complex interactions within real-world systems. “We aim for basins that are deeper yet smaller,” Zhang says. Smaller basins would enable a brain to switch fluidly between different states to tackle complex tasks. Deeper basins would help a conversation partner – or someone reading an SFI article online – maintain their thought process, even if briefly distracted. “This stability allows the brain to recover more quickly from minor disruptions,” Zhang explains.
