Resilience of antagonistic networks with regard to the effects of initial failures and degree-degree correlations

In this study we investigate the resilience of duplex networked layers alpha and beta coupled with antagonistic interlinks, each layer of which inhibits its counterpart at the microscopic level, changing the following factors: whether the influence of the initial failures in alpha remains [quenched (case Q)] or not [free (case F)]; the effect of intralayer degree-degree correlations in each layer and interlayer degree-degree correlations; and the type of the initial failures, such as random failures or targeted attacks (TAs). We illustrate that the percolation processes repeat in both cases Q and F, although only in case F are nodes that initially failed reactivated. To analytically evaluate the resilience of each layer, we develop a methodology based on the cavity method for deriving the size of a giant component (GC). Strong hysteresis, which is ignored in the standard cavity analysis, is observed in the repetition of the percolation processes particularly in case F. To handle this, we heuristically modify interlayer messages for macroscopic analysis, the utility of which is verified by numerical experiments. The percolation transition in each layer is continuous in both cases Q and F. We also analyze the influences of degree-degree correlations on the robustness of layer alpha, in particular for the case of TAs. The analysis indicates that the critical fraction of initial failures that makes the GC size in layer alpha vanish depends only on its intralayer degree-degree correlations. Although our model is defined in a somewhat abstract manner, it may have relevance to ecological systems that are composed of endangered species (layer alpha) and invaders (layer beta), the former of which are damaged by the latter whereas the latter are exterminated in the areas where the former are active.