Goh K I, Kahng B, Kim D
School of Physics, Center for Theoretical Physics, Seoul National University, Seoul 151-747, Korea.
Phys Rev E Stat Nonlin Soft Matter Phys. 2001 Nov;64(5 Pt 1):051903. doi: 10.1103/PhysRevE.64.051903. Epub 2001 Oct 15.
We study the spectra and eigenvectors of the adjacency matrices of scale-free networks when bidirectional interaction is allowed, so that the adjacency matrix is real and symmetric. The spectral density shows an exponential decay around the center, followed by power-law long tails at both spectrum edges. The largest eigenvalue lambda1 depends on system size N as lambda1 approximately N1/4 for large N, and the corresponding eigenfunction is strongly localized at the hub, the vertex with largest degree. The component of the normalized eigenfunction at the hub is of order unity. We also find that the mass gap scales as N(-0.68).
我们研究了允许双向相互作用时无标度网络邻接矩阵的谱和特征向量，使得邻接矩阵是实对称的。谱密度在中心附近呈指数衰减，随后在谱的两边出现幂律长尾。对于大的(N)，最大特征值(\lambda_1)与系统大小(N)的关系为(\lambda_1\approx N^{1/4})，并且相应的特征函数强烈局域在中心节点，即度数最大的顶点。归一化特征函数在中心节点处的分量为一阶量级。我们还发现质量间隙的标度为(N^{(-0.68)})。
