Impact of graph energy on a measurement of resilience for tipping points in complex systems

Abstract

Societies depend on various complex and highly interconnected systems, leading to increasing interest in methods for managing the resilience of these complex systems and the risks associated with their disruption or failure. Identifying and localizing tipping points, or phase transitions, in complex systems is essential for predicting system behavior but a difficult challenge when there are many interacting elements. Systems may transition from stable to unstable at critical tipping‐point thresholds and potentially collapse. One of the suggested approaches in literature is to measure a complex system's resilience to collapse by modeling the system as a network, reducing the network behavior to a simpler model, and then measuring the resulting model's stability. In particular, Gao and colleagues introduced a methodology in 2016 that introduces a resilience index to measure precariousness (the distance to tipping points). However, those mathematical reductions can cause information loss from reducing the topological complexity of the system. Herein, the authors introduce a new methodology that more‐accurately predicts the location of tipping points in networked systems and their precariousness with respect to those tipping points by integrating two approaches: (1) a new measurement of a system's topological complexity using graph energy (created based on molecular orbital theory) and; (2) the resilience index method from Gao et al. This new approach is tested in three separate case studies involving ecosystem collapse, supply chain sustainability, and disruptive technology. Results show a shift in tipping‐point locations correlated with graph energy. The authors present an equation that corrects errors introduced as a result of the model reduction, providing a measurement of precariousness that gives insight into how a complex system's topology affects the location of its tipping points.