Distinct types of eigenvector localization in networks

Abstract

The spectral properties of the adjacency matrix provide a trove of information about the structure and function of complex networks. In particular, the largest eigenvalue and its associated principal eigenvector are crucial in the understanding of nodes centrality and the unfolding of dynamical processes. Here we show that two distinct types of localization of the principal eigenvector may occur in heterogeneous networks. For synthetic networks with degree distribution $P(q)\sim q^{-\gamma }$, localization occurs on the largest hub if $\gamma >5/2$; for $\gamma <5/2$ a new type of localization arises on a mesoscopic subgraph associated with the shell with the largest index in the $K$-core decomposition. Similar evidence for the existence of distinct localization modes is found in the analysis of real-world networks. Our results open a new perspective on dynamical processes on networks and on a recently proposed alternative measure of node centrality based on the non-backtracking matrix.

Final version: 16 pages, 8 figures. Open access article available online at http://www.nature.com/articles/srep18847

References in corpus (7)

• Critical phenomena in complex networks
• Thresholds for epidemic spreading in networks
• Percolation on sparse networks
• Epidemic thresholds of the Susceptible-Infected-Susceptible model on networks: A comparison of numerical and theoretical results
• Spectra of Sparse Random Matrices
• Langevin approach for the dynamics of the contact process on annealed scale-free networks
• Spectra of random graphs with arbitrary expected degrees

Cited by in corpus (50)

• Unification of theoretical approaches for epidemic spreading on complex networks
• Turing patterns mediated by network topology in homogeneous active systems
• Leveraging percolation theory to single out influential spreaders in networks
• Relating Topological Determinants of Complex Networks to Their Spectral Properties: Structural and Dynamical Effects
• Eigenvector localization in real networks and its implications for epidemic spreading
• Predicting the epidemic threshold of the susceptible-infected-recovered model
• Hyper-cores promote localization and efficient seeding in higher-order processes
• Master equation analysis of mesoscopic localization in contagion dynamics on higher-order networks
• The localization of non-backtracking centrality in networks and its physical consequences
• Metastable Localization of Diseases in Complex Networks
• Does the brain behave like a (complex) network? I. Dynamics
• Impact of the distribution of recovery rates on disease spreading in complex networks
• Griffiths effects of the susceptible-infected-susceptible epidemic model on random power-law networks
• Spectral properties and the accuracy of mean-field approaches for epidemics on correlated networks
• Super-resolution community detection for layer-aggregated multilayer networks
• Stochasticity in pandemic spread over the World Airline Network explained by local flight connections
• Tune the topology to create or destroy patterns
• Localization and universality of eigenvectors in directed random graphs
• Principal eigenvector localization and centrality in networks: revisited
• Optimized evolution of networks for principal eigenvector localization
• The nature of hypergraph $k$-core percolation problems
• Phase transition of the susceptible-infected-susceptible dynamics on time-varying configuration model networks
• Localization of multilayer networks by the optimized single-layer rewiring
• Dimension reduction of dynamical systems on networks with leading and non-leading eigenvectors of adjacency matrices
• Quantum centrality testing on directed graphs via PT-symmetric quantum walks
• Physical networks as network-of-networks
• Comparative analysis on the selection of number of clusters in community detection
• Topological dimension tunes activity patterns in hierarchical modular network models
• Localization of eigenvector centrality in networks with a cut vertex
• Large order fluctuations, switching, and control in complex networks
• Network localization is unalterable by infections in bursts
• Universality of eigenvector delocalization and the nature of the SIS phase transition in multiplex networks
• From Spectra to Localized Networks: A Reverse Engineering Approach
• Graph theoretical approaches for the characterization of damage in hierarchical materials
• Multifractal analysis of eigenvectors of smallworld networks
• Nonlinear signaling on biological networks: the role of stochasticity and spectral clustering
• Condensation induced by coupled transport processes
• Localization transition in non-Hermitian systems depending on reciprocity and hopping asymmetry
• Strongly clustered random graphs via triadic closure: An exactly solvable model
• Susceptible-infected-susceptible model on networks with eigenvector localization
• Observing network dynamics through sentinel nodes
• Exact evaluation of the causal spectrum and localization properties of electronic states on a scale-free network
• Influence of individual nodes for continuous-time Susceptible-Infected-Susceptible dynamics on synthetic and real-world networks
• Epidemic threshold and localization of the SIS model on directed complex networks
• Localization in boundary-driven lattice models
• Comparison of theoretical approaches for epidemic processes with waning immunity in complex networks
• Critical avalanches of Susceptible-Infected-Susceptible dynamics in finite networks
• Approximating nonbacktracking centrality and localization phenomena in large networks
• Random Matrix Analysis of Multiplex Networks
• Influence of transition mutations and disorder on charge localization and transfer along B-DNA sequences