In dynamical networks, the presence of adaptation establishing the relationship between the coherence of local populations and unit’s effective coupling provides the explosive transition — an abrupt transition from incoherence to coherence and vice versa through the hysteresis loop. Explosive transition is even possible under the coexistence of two opposite types of adaptation – interdependence and competition, wherein growing the competitive population dramatically narrows the area of hysteresis. Here, we demonstrate that considering a mixed adaptive model from a multilayer perspective expands the hysteresis region and shifts both forward and backward transition boundaries to the higher values of coupling strength as compared with a monolayer case. We show that this is due to greater robustness of the multilayer network against the intralayer topology and lower sensitivity to the amplification of the pre-bifurcation noise, i.e., spurious fluctuations of local coherence, in the vicinity of a tipping point as opposed to a single-layer network.

在动态网络中，适应的存在建立了本地人口的连贯性与单位有效耦合之间的关系，从而提供了爆炸性的过渡-从不连贯到连贯的突然过渡，并且通过磁滞回线反之亦然。在相互适应和竞争这两种相反类型的适应并存的情况下，爆炸性过渡甚至是可能的，其中增加竞争人口会大大缩小滞后区域。在这里，我们证明了从多层的角度考虑混合自适应模型可以扩展滞后区域，并且与单层情况相比，可以将向前和向后的过渡边界都移至更高的耦合强度值。