Percolation on networks with antagonistic and dependent interactions

The dynamic nature of percolation on networks with triadic interactions

Percolation establishes the connectivity of complex networks and is one of the most fundamental critical phenomena for the study of complex systems. On simple networks, percolation displays a second-order phase transition; on multiplex networks, the percolation transition can become discontinuous. However, little is known about percolation in networks with higher-order interactions. Here, we show that percolation can be turned into a fully fledged dynamical process when higher-order interactions are taken into account. By introducing signed triadic interactions, in which a node can regulate the interactions between two other nodes, we define triadic percolation. We uncover that in this paradigmatic model the connectivity of the network changes in time and that the order parameter undergoes a period doubling and a route to chaos. We provide a general theory for triadic percolation which accurately predicts the full phase diagram on random graphs as confirmed by extensive numerical simulations. We find that triadic percolation on real network topologies reveals a similar phenomenology. These results radically change our understanding of percolation and may be used to study complex systems in which the functional connectivity is changing in time dynamically and in a non-trivial way, such as in neural and climate networks.

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 α and β 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 α 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 α, in particular for the case of TAs. The analysis indicates that the critical fraction of initial failures that makes the GC size in layer α 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 α) and invaders (layer β), the former of which are damaged by the latter whereas the latter are exterminated in the areas where the former are active.