Fractional core-based collapse mechanism and structural optimization in complex systems

Catastrophic and major disasters in real-world systems ranging from financial markets and ecosystems, often show generic early-warning signals that may indicate a collapse. Hence, understanding the collapse mechanism of a complex network and predicting its process are of uttermost importance. However, these challenges are often hindered by the extremely high dimensionality of the underlying system. We present here the concept of the fractional core (F-core) that considers the contribution of the network topology and dynamics to systematically analyze the collapse process in such networks, and encompass a broad range of dynamical systems, from mutualistic ecosystems to regulatory dynamics. We offer testable predictions on the tipping point, and, in particular, prove that the extinction of the maximum F-core of a network is an efficient indicator of whether a system completely collapses. The results show that the death of species or cells in a low-order F-core may improve the average density and have little influence on the tipping point. Generally, the principle of the F-core demonstrates how complex systems collapse and opens an innovative optimization strategy to uncover the optimal structure of systems.