da Costa R A, Baxter G J, Dorogovtsev S N, Mendes J F F
Department of Physics & I3N, University of Aveiro, Aveiro, Portugal.
Sci Rep. 2022 Mar 10;12(1):3973. doi: 10.1038/s41598-022-07913-x.
Weak multiplex percolation generalizes percolation to multi-layer networks, represented as networks with a common set of nodes linked by multiple types (colors) of edges. We report a novel discontinuous phase transition in this problem. This anomalous transition occurs in networks of three or more layers without unconnected nodes, [Formula: see text]. Above a critical value of a control parameter, the removal of a tiny fraction [Formula: see text] of nodes or edges triggers a failure cascade which ends either with the total collapse of the network, or a return to stability with the system essentially intact. The discontinuity is not accompanied by any singularity of the giant component, in contrast to the discontinuous hybrid transition which usually appears in such problems. The control parameter is the fraction of nodes in each layer with a single connection, [Formula: see text]. We obtain asymptotic expressions for the collapse time and relaxation time, above and below the critical point [Formula: see text], respectively. In the limit [Formula: see text] the total collapse for [Formula: see text] takes a time [Formula: see text], while there is an exponential relaxation below [Formula: see text] with a relaxation time [Formula: see text].
弱多重渗流将渗流推广到多层网络，多层网络表示为具有一组共同节点的网络，这些节点由多种类型（颜色）的边相连。我们报道了该问题中一种新的不连续相变。这种异常相变发生在三层或更多层且无孤立节点的网络中，[公式：见原文]。在一个控制参数的临界值之上，移除一小部分[公式：见原文]的节点或边会引发一个故障级联，该级联要么以网络的完全崩溃结束，要么以系统基本完好地恢复稳定结束。与通常在此类问题中出现的不连续混合相变不同，这种不连续性并不伴随着巨分支的任何奇异性。控制参数是每层中具有单个连接的节点的比例，[公式：见原文]。我们分别得到了临界点[公式：见原文]之上和之下的崩溃时间和弛豫时间的渐近表达式。在极限[公式：见原文]下，对于[公式：见原文]，完全崩溃需要时间[公式：见原文]，而在[公式：见原文]之下存在指数弛豫，弛豫时间为[公式：见原文]。
