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Register a new userNetwork construction: A learning framework through localizing principal eigenvector
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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.
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Centrality is widely recognized as one of the most critical measures to provide insight in the structure and function of complex networks. While various centrality measures have been proposed for single-layer networks, a general framework for studying centrality in multilayer networks (i.e., multicentrality) is still lacking. In this study, a tensor-based framework is introduced to study eigenvector multicentrality, which enables the quantification of the impact of interlayer influence on multicentrality, providing a systematic way to describe how multicentrality propagates across different layers. This framework can leverage prior knowledge about the interplay among layers to better characterize multicentrality for varying scenarios. Two interesting cases are presented to illustrate how to model multilayer influence by choosing appropriate functions of interlayer influence and design algorithms to calculate eigenvector multicentrality. This framework is applied to analyze several empirical multilayer networks, and the results corroborate that it can quantify the influence among layers and multicentrality of nodes effectively.
Complex networks or graphs provide a powerful framework to understand importance of individuals and their interactions in real-world complex systems. Several graph theoretical measures have been introduced to access importance of the individual in systems represented by networks. Particularly, eigenvector centrality (EC) measure has been very popular due to its ability in measuring importance of the nodes based on not only number of interactions they acquire but also particular structural positions they have in the networks. Furthermore, the presence of certain structural features, such as the existence of high degree nodes in a network is recognized to induce localization transition of the principal eigenvector (PEV) of the networks adjacency matrix. Localization of PEV has been shown to cause difficulties in assigning centrality weights to the nodes based on the EC. We revisit PEV localization and its relation with failure of EC problem, and by using simple model networks demonstrate that in addition to the localization of the PEV, the delocalization of PEV may also create difficulties for using EC as a measure to rank the nodes. Our investigation while providing fundamental insight to the relation between PEV localization and centrality of nodes in networks, suggests that for the networks having delocalized PEVs, it is better to use degree centrality measure to rank the nodes.
Systematic relations between multiple objects that occur in various fields can be represented as networks. Real-world networks typically exhibit complex topologies whose structural properties are key factors in characterizing and further exploring the networks themselves. Uncertainty, modelling procedures and measurement difficulties raise often insurmountable challenges in fully characterizing most of the known real-world networks; hence, the necessity to predict their unknown elements from the limited data currently available in order to estimate possible future relations and/or to unveil unmeasurable relations. In this work, we propose a deep learning approach to this problem based on Graph Convolutional Networks for predicting networks while preserving their original structural properties. The study reveals that this method can preserve scale-free and small-world properties of complex networks when predicting their unknown parts, a feature lacked by the up-to-date conventional methods. An external validation realized by testing the approach on biological networks confirms the results, initially obtained on artificial data. Moreover, this process provides new insights into the retainability of network structure properties in network prediction. We anticipate that our work could inspire similar approaches in other research fields as well, where unknown mechanisms behind complex systems need to be revealed by combining machine-based and experiment-based methods.
There are different measures to classify a networks data set that, depending on the problem, have different success. For example, the resistance distance and eigenvector centrality measures have been successful in revealing ecological pathways and differentiating between biomedical images of patients with Alzheimers disease, respectively. The resistance distance measures the effective distance between any two nodes of a network taking into account all possible shortest paths between them and the eigenvector centrality measures the relative importance of each node in the network. However, both measures require knowing the networks eigenvalues and eigenvectors -- eigenvectors being the more computationally demanding task. Here, we show that we can closely approximate these two measures using only the eigenvalue spectra, where we illustrate this by experimenting on elemental resistor circuits and paradigmatic network models -- random and small-world networks. Our results are supported by analytical derivations, showing that the eigenvector centrality can be perfectly matched in all cases whilst the resistance distance can be closely approximated. Our underlying approach is based on the work by Denton, Parke, Tao, and Zhang [arXiv:1908.03795 (2019)], which is unrestricted to these topological measures and can be applied to most problems requiring the calculation of eigenvectors.
The spectral properties of the adjacency matrix, in particular its largest eigenvalue and the associated principal eigenvector, dominate many structural and dynamical properties of complex networks. Here we focus on the localization properties of the principal eigenvector in real networks. We show that in most cases it is either localized on the star defined by the node with largest degree (hub) and its nearest neighbors, or on the densely connected subgraph defined by the maximum $K$-core in a $K$-core decomposition. The localization of the principal eigenvector is often strongly correlated with the value of the largest eigenvalue, which is given by the local eigenvalue of the corresponding localization subgraph, but different scenarios sometimes occur. We additionally show that simple targeted immunization strategies for epidemic spreading are extremely sensitive to the actual localization set.
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