Recently, eigenvector localization of complex network has seen a spurt in activities due to its versatile applicability in many different areas which includes networks centrality measure, spectral partitioning, development of approximation algorithms and disease spreading phenomenon. For a network, an eigenvector is said to be localized when most of its components are near to zero, with few taking very high values. Here, we develop three different randomized algorithms, which by using edge rewiring method, can evolve a random network having a delocalized principal eigenvector to a network having a highly localized principal eigenvector. We discuss drawbacks and advantages of these algorithms. Additionally, we show that the construction of such networks corresponding to the highly localized principal eigenvector is a non-convex optimization problem when the objective function is the inverse participation ratio.
It has been shown that a network with localized PEV explains disease propagation in SIS model @cite_28 and reduces perturbation propagation in mutualistic ecological networks @cite_40 . used PEV localization to analyze the brain network dynamics @cite_3 . Moreover, dynamics over the networks have been analyzed through eigenvector localization @cite_4 @cite_43 . Low order eigenvectors have also been studied to develop machine learning tools @cite_8 . They used IPR as well as another kind of measure for the eigenvector localization, called as statistical leverage scores @cite_39 which has an impact on statistics and modern big data analysis. Eigenvector localization has been applied to cryptographic applications @cite_48 , development of spectral clustering algorithm @cite_45 , and analyzing of Google matrix for better performance @cite_42 . It has been observed that sometimes eigenvector centrality @cite_24 and spectral partitioning @cite_32 method fail due to eigenvector localization. Note that eigenvector centrality and eigenvector localization are different. Recently, network properties corresponding to highly localized PEV has been investigated for single layer network and multiplex networks @cite_15 @cite_37 .
@cite_37 @cite_4 @cite_8 @cite_28 @cite_48 @cite_42 @cite_32 @cite_3 @cite_39 @cite_24 @cite_43 @cite_40 @cite_45 @cite_15
2772041599 2614256707 2017990598 1714924059
We study localization properties of principal eigenvector (PEV) of multiplex networks. Starting with a multiplex network corresponding to a delocalized PEV, we rewire the network edges using an optimization technique such that the PEV of the rewired multiplex network becomes more localized. The framework allows us to scrutinize structural and spectral properties of the networks at various localization points during the rewiring process. We show that by rewiring only one layer of the multiplex network, we can achieve the PEV localization of the entire system. Our investigations reveal the sensitivity of the PEV to a single edge rewiring in the optimized multiplex network structure corresponding to the most localized PEV providing the deeper insight into the localization behavior of the multiplex networks. Furthermore, by constructing multiplex network using real-world social and biological data, we show that our simulation results for model multiplex network are in good agreement with the properties of these real-world multiplex network. The study is relevant to applications that require understanding of how perturbation propagates in multiplex networks. In this paper, we describe a novel deep convolutional neural network (CNN) that is deeper and wider than other existing deep networks for hyperspectral image classification. Unlike current state-of-the-art approaches in CNN-based hyperspectral image classification, the proposed network, called contextual deep CNN, can optimally explore local contextual interactions by jointly exploiting local spatio-spectral relationships of neighboring individual pixel vectors. The joint exploitation of the spatio-spectral information is achieved by a multi-scale convolutional filter bank used as an initial component of the proposed CNN pipeline. The initial spatial and spectral feature maps obtained from the multi-scale filter bank are then combined together to form a joint spatio-spectral feature map. The joint feature map representing rich spectral and spatial properties of the hyperspectral image is then fed through a fully convolutional network that eventually predicts the corresponding label of each pixel vector. The proposed approach is tested on three benchmark data sets: the Indian Pines data set, the Salinas data set, and the University of Pavia data set. Performance comparison shows enhanced classification performance of the proposed approach over the current state-of-the-art on the three data sets. Using the SIS model on unweighted and weighted networks, we consider the disease localization phenomenon. In contrast to the well-recognized point of view that diseases infect a finite fraction of vertices right above the epidemic threshold, we show that diseases can be localized on a finite number of vertices, where hubs and edges with large weights are centers of localization. Our results follow from the analysis of standard models of networks and empirical data for real-world networks. Eigenvector localization refers to the situation when most of the components of an eigenvector are zero or near-zero. This phenomenon has been observed on eigenvectors associated with extremal eigenvalues, and in many of those cases it can be meaningfully interpreted in terms of "structural heterogeneities" in the data. For example, the largest eigenvectors of adjacency matrices of large complex networks often have most of their mass localized on high-degree nodes; and the smallest eigenvectors of the Laplacians of such networks are often localized on small but meaningful community-like sets of nodes. Here, we describe localization associated with low-order eigenvectors, i.e., eigenvectors corresponding to eigenvalues that are not extremal but that are "buried" further down in the spectrum. Although we have observed it in several unrelated applications, this phenomenon of low-order eigenvector localization defies common intuitions and simple explanations, and it creates serious difficulties for the applicability of popular eigenvector-based machine learning and data analysis tools. After describing two examples where low-order eigenvector localization arises, we present a very simple model that qualitatively reproduces several of the empirically-observed results. This model suggests certain coarse structural similarities among the seemingly-unrelated applications where we have observed low-order eigenvector localization, and it may be used as a diagnostic tool to help extract insight from data graphs when such low-order eigenvector localization is present.