Collapses are sudden rapid changes, resulting in a significant reduction of the structural or dynamical complexity within a system. Economies, ecosystems or social systems exhibit collapses. Due to the large societal impact as well as for the theoretical ramifications it is important to understand collapses from a structural and dynamical perspective and to study precursor signals that precede them.

We base our discussion on the class of adaptive network models. These are models in which the dynamics of the network structure evolves dynamically depending on the states of the nodes and vice versa, where the dynamics of the node states depends on the network structure.

In this chapter we consider an adaptive network model developed by Jain and Krishna as our starting point to shine light on the collapse transition in adaptive networks.

We show that there is a quantization phenomenon that can be used as a precursor for an impending collapse. This phenomenon is also shown to be present in SIS processes.

L. Horstmeyer, Chapter 2: Insights into the Collapse of Adaptive Complex Systems, In: Yurij Holovatch (ed.), Order, Distorder and Criticality 7 (2023), 79-116.