Network construction: A learning framework through localizing principal eigenvector

Information of localization properties of eigenvectors of the complex network has applicability in many different areas which include networks centrality measures, spectral partitioning, development of approximation algorithms, and disease spreading phenomenon. For linear dynamical process localization of principal eigenvector (PEV) of adjacency matrices infers condensation of the information in the smaller section of the network. 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 provide three different random-sampling-based algorithms which, by using the edge rewiring method, can evolve a random network having a delocalized PEV to a network having a highly localized PEV. In other words, we develop a learning framework to explore the localization of PEV through a random sampling-based optimization method. We discuss the drawbacks and advantages of these algorithms. Additionally, we show that the construction of such networks corresponding to the highly localized PEV is a non-convex optimization problem when the objective function is the inverse participation ratio. This framework is also relevant to construct a network structure for other lower-order eigenvectors.