N.N. Semenov Institute of Chemical Physics RAS, 119991 Moscow, Russia.
Department of Applied Mathematics, National Research University Higher School of Economics, 101000 Moscow, Russia.
Phys Rev E. 2017 Dec;96(6-1):062309. doi: 10.1103/PhysRevE.96.062309. Epub 2017 Dec 15.
We consider critical behavior in the ensemble of polychromatic Erdős-Rényi networks and regular random graphs, where network vertices are painted in different colors. The links can be randomly removed and added to the network subject to the condition of the vertex degree conservation. In these constrained graphs we run the Metropolis procedure, which favors the connected unicolor triads of nodes. Changing the chemical potential, μ, of such triads, for some wide region of μ, we find the formation of a finite plateau in the number of intercolor links, which exactly matches the finite plateau in the network algebraic connectivity (the value of the first nonvanishing eigenvalue of the Laplacian matrix, λ_{2}). We claim that at the plateau the spontaneously broken Z_{2} symmetry is restored by the mechanism of modes collectivization in clusters of different colors. The phenomena of a finite plateau formation holds also for polychromatic networks with M≥2 colors. The behavior of polychromatic networks is analyzed via the spectral properties of their adjacency and Laplacian matrices.
我们研究了多色 Erdős-Rényi 网络和正则随机图组合中的临界行为，其中网络顶点被涂成不同的颜色。可以根据顶点度守恒的条件随机删除和添加网络中的链接。在这些受约束的图中，我们运行 Metropolis 过程，该过程有利于节点的连通单色三联体。对于 μ 的某些宽区域，改变这样的三联体的化学势 μ，我们发现网络代数连通性（拉普拉斯矩阵的第一个非零特征值 λ_{2} 的值）中的互色彩链接数量形成有限平台。我们声称，在平台处，通过不同颜色簇中模式集体化的机制，自发破缺的 Z_{2} 对称性得到恢复。具有 M≥2 种颜色的多色网络也存在有限平台形成的现象。通过分析它们的邻接和拉普拉斯矩阵的谱性质来分析多色网络的行为。
