模网络的特征值谱
Eigenvalue Spectra of Modular Networks
摘要：
网络上发生的大量动力学过程可以用某些线性算子的谱性质来表示，这些线性算子反映了动力学规则如何依赖于网络拓扑。通常，这样的谱特征在理论上是通过只考虑局部节点属性（例如度分布）来获得的。然而，许多网络具有大规模的模块化结构，可以极大地影响它们的光谱特性，并且在这种简化的描述中被忽略了。在这里，我们得到一个统一的方式的频谱的一个大家庭的运营商，包括邻接，拉普拉斯和规范化拉普拉斯矩阵，网络与通用模块化结构，在大的程度的限制。我们专注于合并的孤立的特征值与连续频带的频谱，之后种植的模块化结构可以不再容易地检测到的频谱方法的必要条件。这是一个关键的转变点，它决定了一个模块结构何时足够强大，足以影响一个给定的动态过程。我们表明，这种过渡一般发生在不同的矩阵的不同点，因此，检测阈值可以根据所选择的运营商显着不同。同样地，给定网络中存在相同的大尺度结构，与每个矩阵相关联的不同动力学过程的模块结构的灵敏度将是不同的。此外，我们表明，与例外的拉普拉斯矩阵，不同的过渡合并到同一点的特殊情况下，模块是同质的，但分开，否则。
英文摘要：
A large variety of dynamical processes that take place on networks can be expressed in terms of the spectral properties of some linear operator which reflects how the dynamical rules depend on the network topology. Often such spectral features are theoretically obtained by considering only local node properties, such as degree distributions. Many networks, however, possess large-scale modular structures that can drastically influence their spectral characteristics, and which are neglected in such simplified descriptions. Here we obtain in a unified fashion the spectrum of a large family of operators, including the adjacency, Laplacian and normalized Laplacian matrices, for networks with generic modular structure, in the limit of large degrees. We focus on the conditions necessary for the merging of the isolated eigenvalues with the continuous band of the spectrum, after which the planted modular structure can no longer be easily detected by spectral methods. This is a crucial transition point which determines when a modular structure is strong enough to affect a given dynamical process. We show that this transition happens in general at different points for the different matrices, and hence the detectability threshold can vary significantly depending on the operator chosen. Equivalently, the sensitivity to the modular structure of the different dynamical processes associated with each matrix will be different, given the same large-scale structure present in the network. Furthermore, we show that, with the exception of the Laplacian matrix, the different transitions coalesce into the same point for the special case where the modules are homogeneous, but separate otherwise.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
