Title: Generalized $k$-core pruning process on directed networks,
Abstract: 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.