Understanding the localization properties of eigenvectors of complex networks is important to get insight into various structural and dynamical properties of the corresponding systems. Here, we analytically develop a scheme to construct a highly localized network for a given set of networks parameters that is the number of nodes and the number of interactions. We find that the localization behavior of the principal eigenvector (PEV) of such a network is sensitive against a single edge rewiring. We find evidences for eigenvalue crossing phenomena as a consequence of the single edge rewiring, in turn providing an origin to the sensitive behavior of the PEV localization. These insights were then used to analytically construct the highly localized network for a given set of networks parameters. The analysis provides fundamental insight into relationships between the structural and the spectral properties of networks for PEV localized networks. Further, we substantiate the existence of the eigenvalue crossing phenomenon by considering a linear-dynamical process, namely the ribonucleic acid (RNA) neutral network population dynamical model. The analysis presented here on model networks aids in understanding the steady-state behavior of a broad range of linear-dynamical processes, from epidemic spreading to biochemical dynamics associated with the adjacency matrices.

中文翻译：

---

从光谱到局部网络：逆向工程方法

了解复杂网络特征向量的定位特性对于深入了解相应系统的各种结构和动力学特性非常重要。在这里，我们分析开发了一种方案，用于为给定的网络参数集构建高度本地化的网络，该网络参数是节点数和交互数。我们发现这种网络的主特征向量 (PEV) 的定位行为对单边重新布线很敏感。我们发现作为单边重新布线的结果的特征值交叉现象的证据，反过来提供了 PEV 定位的敏感行为的起源。然后使用这些见解为给定的网络参数集分析构建高度本地化的网络。该分析提供了对 PEV 局部网络的网络结构和光谱特性之间关系的基本见解。此外，我们通过考虑线性动力学过程，即核糖核酸 (RNA) 中性网络种群动力学模型，证实了特征值交叉现象的存在。这里介绍的模型网络分析有助于理解广泛的线性动力学过程的稳态行为，从流行病传播到与邻接矩阵相关的生化动力学。即核糖核酸 (RNA) 中性网络种群动力学模型。这里介绍的模型网络分析有助于理解广泛的线性动力学过程的稳态行为，从流行病传播到与邻接矩阵相关的生化动力学。即核糖核酸 (RNA) 中性网络种群动力学模型。这里介绍的模型网络分析有助于理解广泛的线性动力学过程的稳态行为，从流行病传播到与邻接矩阵相关的生化动力学。