Azimi-Tafreshi N, Gómez-Gardeñes J, Dorogovtsev S N
Physics Department, Institute for Advanced Studies in Basic Sciences, 45195-1159 Zanjan, Iran.
Departamento de Física de la Materia Condensada, Universidad de Zaragoza, E-50009 Zaragoza, Spain and Instituto de Biocomputación y Física de los Sistemas Complejos (BIFI), Universidad de Zaragoza, E-50018 Zaragoza, Spain.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Sep;90(3):032816. doi: 10.1103/PhysRevE.90.032816. Epub 2014 Sep 29.
We generalize the theory of k-core percolation on complex networks to k-core percolation on multiplex networks, where k≡(k(1),k(2),...,k(M)). Multiplex networks can be defined as networks with vertices of one kind but M different types of edges, representing different types of interactions. For such networks, the k-core is defined as the largest subgraph in which each vertex has at least k(i) edges of each type, i=1,2,...,M. We derive self-consistency equations to obtain the birth points of the k-cores and their relative sizes for uncorrelated multiplex networks with an arbitrary degree distribution. To clarify our general results, we consider in detail multiplex networks with edges of two types and solve the equations in the particular case of Erdős-Rényi and scale-free multiplex networks. We find hybrid phase transitions at the emergence points of k-cores except the (1,1)-core for which the transition is continuous. We apply the k-core decomposition algorithm to air-transportation multiplex networks, composed of two layers, and obtain the size of (k(1),k(2))-cores.
我们将复杂网络上的k核渗流理论推广到多重网络上的k核渗流，其中k≡(k(1),k(2),...,k(M))。多重网络可定义为具有一类顶点但有M种不同类型边的网络，代表不同类型的相互作用。对于此类网络，k核被定义为最大子图，其中每个顶点对于每种类型的边至少有k(i)条边，i = 1,2,...,M。我们推导出自洽方程，以获得具有任意度分布的不相关多重网络的k核的诞生点及其相对大小。为了阐明我们的一般结果，我们详细考虑具有两种类型边的多重网络，并在Erdős-Rényi和无标度多重网络的特定情况下求解方程。我们发现在k核的出现点处存在混合相变，但(1,1)核的相变是连续的。我们将k核分解算法应用于由两层组成的航空运输多重网络，并获得(k(1),k(2))核的大小。
