Emergent interactions in complex networks: Beyond pairwise parts in systems ranging from drugs and microbes to consumers and resources

Interactions and emergent behavior are at the heart of complex systems research. The formation of complex structures and the evolution of computationally complex dynamics are considered emergent if the overall behavior of the system is difficult to derive, calculate, or intuit based on the component parts. This emergence typically results from interactions such that the whole is other than the sum of its parts.

This raises the questions: What are the parts, and how do we sum them? More precisely, can parts be defined at multiple levels, and do some levels better correspond to the emergent behavior of the whole? For instance, in searching for indivisible parts of matter, physicists have looked further and further down to quarks and potentially strings. In contrast, in the search for emergence, we are led to looking at higher levels, and perhaps collections of objects or the interactions themselves are best understood as the parts. Here, I focus on interactions among drugs and among consumers as specific systems to gain traction on these questions.

Great progress in understanding and predicting the behavior of complex systems has been achieved by combining network theory with dynamical systems theory. Much of network theory is founded on pairwise interactions. Typically, even when considering collections of multiple nodes (objects) and edges (interactions), each edge is defined by the presence of a pairwise interaction. Nevertheless, some interactions may only emerge when three or more objects are present, even though no interaction exists between each isolated pair. More generally, an interaction among multiple objects may be due to the sum of its pairwise parts, or may only emerge when all objects are considered together. Standard network representations do not make these distinctions or have symbols that capture these higher-order, emergent interactions. Moreover, standard interaction metrics either assume the interactions are the sum of the pairwise parts or measure the total interaction without distinguishing pairwise from emergent contributions.

Here, I outline a plan for addressing these limitations and for tackling the questions above. I suggest directions for novel theory that are tied to experiments and grounded in empirical results. I propose metrics for measuring emergent interactions and a general mathematical framework for defining these emergent metrics relative to many notions of interaction—additivity, covariance, mutual information. I present data that show emergent interactions are common among drugs being used to kill microbes. I argue for similar patterns among consumers and resources. These emergent interactions exhibit some salient and systematic features, such as higher degrees of antagonism. Understanding these features could lead to the development of better tools for aggregating components and interactions in complex networks. Exploiting these systematic patterns and how they change in frequency and magnitude across levels of emergence could also provide clues to novel methods for taming the combinatorial complexity that is usually viewed as the Gordian knot of higher-order interactions. Such an advance could help combat the evolution of resistance to antibiotics and mitigate detrimental impacts of climate change on the diversity and stability of ecological communities.