论文标题
弱的多重渗透
Weak Multiplex Percolation
论文作者
论文摘要
在许多由相互作用子系统组成的系统中，可以使用多层网络表示元素之间的复杂交互。然而，渗透是理解连通性和鲁棒性的关键，并未在多个层上毫无用处地概括。我们描述了对多层网络渗透的概括：弱的多重渗透。一个节点属于连接的组件，如果其每一层中至少一个邻居在此组件中。我们充分描述了此过程的关键现象。特别是，在两层中具有有限的度分布的有限第二矩，这是一个不寻常的连续过渡，二次生长高于阈值。当第二矩差异时，奇异性是由程度分布的渐近学决定的，从而产生了丰富的关键行为。在三层或多个层中，我们发现了一个不连续的混合过渡，即使在高度异构度分布中，它才会持续，只有当PowerLaw指数达到$ 1+ 1/（M-1）$ for $ m $ layers时，才会变得连续。
In many systems consisting of interacting subsystems, the complex interactions between elements can be represented using multilayer networks. However percolation, key to understanding connectivity and robustness, is not trivially generalised to multiple layers. We describe a generalisation of percolation to multilayer networks: weak multiplex percolation. A node belongs to a connected component if at least one of its neighbours in each layer is in this component. We fully describe the critical phenomena of this process. In particular, in two layers, with finite second moments of the degree distributions, an unusual continuous transition with quadratic growth above the threshold. When the second moments diverge, the singularity is determined by the asymptotics of the degree distributions, creating a rich set of critical behaviours. In three or more layers we find a discontinuous hybrid transition which persists even in highly heterogeneous degree distributions, becoming continuous only when the powerlaw exponent reaches $1+ 1/(M-1)$ for $M$ layers.
