Collapse process prediction of mutualistic dynamical networks with k-core and dimension reduction method

Network collapse, characterized by the abrupt transition between distinct states within a complex networked system, carries significant implications encompassing catastrophic repercussions and substantial societal costs, including power grid failures and disease outbreaks. Despite notable advancements, the intricate interplay between the hierarchical structure and dynamics of networks in shaping the collapse process remains inadequately understood. In this study, we establish a mathematical framework that facilitates the reduction of the dimension of any N-dimensional dynamical network to a kmax shell-dimensional rendition by imposing the k-core as a governing constraint upon the network’s structure and dynamics. Subsequently, we employ this framework to elucidate the collapse process of the dynamical network. Our investigation underscores that the patterns of network collapse are intrinsically linked to the influence exerted by the network hierarchy upon the collapse process. Notably, dynamic networks may undergo either hierarchical or simultaneous collapse contingent upon the hierarchical impact on the network’s collapse progression. In scenarios where hierarchy holds sway, the network attains the tipping point of collapse upon the collapse of nodes within the maximum k-core. Conversely, when the hierarchy’s influence is absent, nodes within the dynamic network succumb to simultaneous collapse. Furthermore, we introduce a robust criterion for prognosticating the tipping points of dynamic network collapse. Our explorations reveal that it suffices to consider nodes and links within the maximum k-shell and t-shell with βtt<βtk in the network, provided the collapse process is entwined with network hierarchy, to anticipate the tipping point.