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.
We solve the weak percolation problem for multiplex networks with overlapping edges. In weak percolation, a vertex belongs to a connected component if at least one of its neighbors in each of the layers is in this component. This is a weaker condition than for a mutually connected component in interdependent networks, in which any two vertices must be connected by a path within each of the layers. The effect of the overlaps on weak percolation turns out to be opposite to that on the giant mutually connected component. While for the giant mutually connected component, overlaps do not change the…
Higher order interactions are increasingly recognised as a fundamental aspect of complex systems ranging from the brain to social contact networks. Hypergraph as well as simplicial complexes capture the higher-order interactions of complex systems and allow to investigate the relation between their higher-order structure and their function. Here we establish a general framework for assessing hypergraph robustness and we characterize the critical properties of simple and higher-order percolation processes. This general framework builds on the formulation of the random multiplex hypergraph ensem…
We describe the critical behavior of weak multiplex percolation, a generalization of percolation to multiplex or interdependent networks. A node can determine its active or inactive status simply by referencing neighboring nodes. This is not the case for the more commonly studied generalization of percolation to multiplex networks, the mutually connected clusters, which requires an interconnecting path within each layer between any two vertices in the giant mutually connected component. We study the emergence of a giant connected component of active nodes under the weak percolation rule, findi…
Percolation transitions in the survival of interdependent agents on multiplex networks, catastrophic cascades, and SOS
标题中的“SOS”不是指国际求救信号，而是指“固体对固体”（SOS）表面生长。Buldyrev {\it et al.}观察到了灾难性级联。在相互依赖的网络中，我们重新解释为多路复用网络，只有在相互支持的情况下才能生存，并且我们将其生存斗争映射到SOS类型的增长模型。这种映射不仅揭示了模型相空间中的非平凡结构，而且还导致了一种新的和非常有效的仿真算法。我们使用该算法来研究双工Erdös-Rényi（ER）网络和2，3，4和5维格上的相互依赖的代理。在所有这些情况下，我们获得了新的和令人惊讶的结果，我们纠正ER网络和2-D晶格文献中的陈述。特别是，我们发现，$d=4$是上临界尺寸，即渗流过渡是连续的$d\leq 4$，但-至少为$d\neq 3$-不是在普通渗流的普适类。对于ER网络，我们验证了集群统计是准确描述的平均场论，但发现证据表明，级联过程是不是。对于$d=5$，我们发现一个一阶过渡的ER网络，但我们也发现，小集群有一个非平凡的质量分布，在过渡点的规模。最后，对于具有中间范围依赖链路的$d=2$，我们提出了一种不同于W. Li {\it等人}，PRL {\bf 108}，228702（2012）。
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, $P(0)=0$. Above a critical value of a control parameter, the removal of a tiny fraction $Δ$ 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 discont…
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 th…
We study the current flow paths between two edges in a random resistor network on a $L\times L$ square lattice. Each resistor has resistance $e^{ax}$, where $x$ is a uniformly-distributed random variable and $a$ controls the broadness of the distribution. We find (a) the scaled variable $u\equiv L/a^ν$, where $ν$ is the percolation connectedness exponent, fully determines the distribution of the current path length $\ell$ for all values of $u$. For $u\gg 1$, the behavior corresponds to the weak disorder limit and $\ell$ scales as $\ell\sim L$, while for $u\ll 1$, the behavior corresponds to th…
Intrinsic mixed state topological order in a stabilizer system under stochastic decoherence: Strong-to-weak spontaneous symmetry breaking from percolation point of view
Universitat de Barcelona（巴塞罗那大学）；Institute of Complex Systems (UBICS)（复杂系统研究所（UBICS））；Northeastern University（东北大学）；Institució Catalana de Recerca i Estudis Avançats (ICREA)（加泰罗尼亚研究与高级研究学院（ICREA））
