Mitrović Marija, Tadić Bosiljka
Scientific Computing Laboratory, Institute of Physics, 11000 Belgrade, Serbia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Aug;80(2 Pt 2):026123. doi: 10.1103/PhysRevE.80.026123. Epub 2009 Aug 24.
We study structure, eigenvalue spectra, and random-walk 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, the 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 lambda(L)=2 of the Laplacian. Further differences between the cyclic modular graphs and trees are found by the statistics of random walks return times and hitting patterns at nodes on these graphs. The distribution of first-return times averaged over all nodes exhibits a stretched exponential tail with the exponent sigma approximately 1/3 for trees and sigma approximately 2/3 for cyclic graphs, which is independent of their mesoscopic and global structure.
我们研究了一类广泛的网络中的结构、特征值谱和随机游走动力学，这类网络在介观尺度上具有子图（模块）。这些网络是在一个具有三个参数的模型中生长的，这三个参数分别控制模块的数量、它们作为无标度和相关子图的内部结构，以及连接网络的拓扑结构。在对邻接矩阵和归一化拉普拉斯矩阵进行详尽的谱分析过程中，我们确定了谱性质，这些性质表征了稀疏循环图和树的介观结构。连接最少的节点、聚类和平均连通性会影响谱的中心部分。不同模块的数量在循环图的拉普拉斯谱的下部导致一个额外的峰值。而在树形结构上连接的拓扑结构不同的树子图的情况下，不会出现这样的峰值，不过相关的特征向量在树和循环图中都仍然局限于子图上。我们还发现，对于与拉普拉斯矩阵的最大特征值λ(L)=2相关的特征向量分量，沿着树中的链存在一种周期性局域化的特征模式。通过对这些图上节点的随机游走返回时间和命中模式的统计，发现了循环模块化图和树之间的进一步差异。在所有节点上平均的首次返回时间分布呈现出拉伸指数尾部，对于树，指数σ约为1/3，对于循环图，指数σ约为2/3，这与它们的介观和全局结构无关。
