Generalized $k$-core pruning process on directed networks

The resilience of a complex interconnected system concerns the size of the macroscopic functioning node clusters after external perturbations based on a random or designed scheme. For a representation of the interconnected systems with directional or asymmetrical interactions among constituents, the directed network is a convenient choice. Yet how the interaction directions affect the network resilience still lacks thorough exploration. Here, we study the resilience of directed networks with a generalized $k$-core pruning process as a simple failure procedure based on both the in- and out-degrees of nodes, in which any node with an in-degree $< k_{in}$ or an out-degree $< k_{ou}$ is removed iteratively. With an explicitly analytical framework, we can predict the relative sizes of residual node clusters on uncorrelated directed random graphs. We show that the discontinuous transitions rise for cases with $k_{in} \geq 2$ or $k_{ou} \geq 2$, and the unidirectional interactions among nodes drive the networks more vulnerable against perturbations based on in- and out-degrees separately.