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Real-world complex systems always interact with each other, which causesthese systems to collapse in an avalanche or cascading manner in the case ofrandom failures or malicious attacks. The robustness of multilayer networks hasattracted great interest, where the modeling and theoretical studies of whichalways rely on the concept of multilayer networks and percolation methods. Astraightforward and tacit assumption is that the interdependence across networklayers is strong, which means that a node will fail entirely with the removalof all links if one of its interdependent neighbours fails. However, thisoversimplification cannot describe the general form of interactions across thenetwork layers in a real-world multilayer system. In this paper, we reveal thenature of the avalanche disintegration of general multilayer networks witharbitrary interdependency strength across network layers. Specifically, weidentify that the avalanche process of the whole system can essentially bedecomposed into two microscopic cascading dynamics in terms of the propagationdirection of the failures: depth penetration and scope extension. In theprocess of depth penetration, the failures propagate from layer to layer, wherethe greater the number of failed nodes is, the greater the destructive powerthat will emerge in an interdependency group. In the process of scopeextension, failures propagate with the removal of connections in each networklayer. Under the synergy of the two processes, we find that the percolationtransition of the system can be discontinuous or continuous with changes in theinterdependency strength across network layers, which means that suddensystem-wide collapse can be avoided by controlling the interdependency strengthacross network layers.

Depth penetration and scope extension of failures in the cascading of multilayer networks