A comparative approach to quantify the heterarchical structures of complex systems

Abstract

The dynamics and adaptive capacity of social-ecological systems are heavily contingent on system structure, which is established through geography, institutions, interactions, and movement. Contrasting views of system structure, as hierarchies and single-level networks respectively, have tended to emphasize the role of either top-down or lateral (peer-to-peer) connections. The concept of a heterarchy aims to capture both top-down and lateral connections on orthogonal axes and has been proposed as a way of unifying alternative approaches to measuring structure, but it has not been fully operationalized for quantifying and comparing system structures. We developed a simple approach to consistently quantifying heterarchical structure across different kinds of networks. We first calculated suitable metrics, including modularity and a hierarchy score, for a wide range of both simulated and real-world systems including food webs, biological, infrastructure, and social networks. Metrics were corrected for differences in size and magnitude. The results were then visualized as a heterarchy matrix. We compared the angle (degrees) and Euclidian distance of each simulated and real-world network from the center of the matrix between network groups. All networks showed distinct placement on the heterarchy matrix. Relative to one another, food webs were laterally polycentric, social networks were mainly pyramidal and coordinated polycentric, and biological networks were pyramidal and laterally polycentric. Our test of concept, although relatively basic, provides strong evidence that system structure cannot be fully understood as purely laterally connected or purely hierarchical. System resilience requires a trade-off between modularity, aiding redundancy and collaboration; and hierarchy, aiding efficient action. Our approach has the potential to provide a robust, accessible methodology to quantify system structure that allows for universal contextualization, a key step within fields such as resilience and sustainability science.