一类具有介观非均匀性的稀疏网络的谱和动力学性质
Spectral and Dynamical Properties in Classes of Sparse Networks with Mesoscopic Inhomogeneities
摘要：
我们研究结构，特征值谱和扩散动力学在一个广泛的类网络的子图（模块）在介观尺度。网络在模型中生长，三个参数控制模块的数量，它们的内部结构为无标度和相关子图，以及连接网络的拓扑结构。在详尽的谱分析的邻接矩阵和规范化的拉普拉斯矩阵，我们确定的频谱特性，表征稀疏循环图和树的介观结构。最小连接节点、聚类和平均连接影响频谱的中心部分。不同模块的数量导致在循环图的拉普拉斯谱的较低部分处的额外峰值。这样的峰值不会发生在拓扑不同的树子图连接在树上的情况下。而在树和循环图中，相关的特征向量仍然局限于子图上。我们还发现了一个特征模式的周期性本地化沿链上的树的特征向量分量与最大的特征值等于2的拉普拉斯算子。我们证实了几种类型的网络上的随机游走模拟的结果。我们的结果的返回时间的分布的步行到原点（自相关器）同意最近的解析解树，它似乎是独立的介观和全球结构。对于循环图，我们发现了新的结果与两倍大的拉伸指数的尾部的分布，这是几乎无关的大小的周期。模块化和集群有助于在短返回时间内呈现幂律衰减。
英文摘要：
We study structure, eigenvalue spectra and diffusion dynamics in a wide class of networks with subgraphs (modules) at mesoscopic scale. The networks are grown within the model with three parameters controlling the number of modules, their internal structure as scale-free and correlated subgraphs, and the topology of connecting network. Within the exhaustive spectral analysis for both the adjacency matrix and the normalized Laplacian matrix we identify the spectral properties which characterize the mesoscopic structure of sparse cyclic graphs and trees. The minimally connected nodes, clustering, and the average connectivity affect the central part of the spectrum. The number of distinct modules leads to an extra peak at the lower part of the Laplacian spectrum in cyclic graphs. Such a peak does not occur in the case of topologically distinct tree-subgraphs connected on a tree. Whereas the associated eigenvectors remain localized on the subgraphs both in trees and cyclic graphs. We also find a characteristic pattern of periodic localization along the chains on the tree for the eigenvector components associated with the largest eigenvalue equal 2 of the Laplacian. We corroborate the results with simulations of the random walk on several types of networks. Our results for the distribution of return-time of the walk to the origin (autocorrelator) agree well with recent analytical solution for trees, and it appear to be independent on their mesoscopic and global structure. For the cyclic graphs we find new results with twice larger stretching exponent of the tail of the distribution, which is virtually independent on the size of cycles. The modularity and clustering contribute to a power-law decay at short return times.
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