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Eigenvector Localization in Real Networks and Its Implications for Epidemic Spreading

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Abstract

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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Author information

Authors and Affiliations

  1. Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Campus Nord B4, 08034, Barcelona, Spain

    Romualdo Pastor-Satorras

  2. Istituto dei Sistemi Complessi (ISC-CNR), via dei Taurini 19, 00185, Rome, Italy

    Claudio Castellano

  3. Dipartimento di Fisica, “Sapienza” Università di Roma, P.le A. Moro 2, 00185, Rome, Italy

    Claudio Castellano

Authors
  1. Romualdo Pastor-Satorras
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  2. Claudio Castellano
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Correspondence to Claudio Castellano.

Additional information

We acknowledge financial support from the Spanish MINECO, under Projects No. FIS2013-47282-C2-2 and No. FIS2016-76830-C2-1-P. R. P.-S. acknowledges additional financial support from ICREA Academia, funded by the Generalitat de Catalunya.

Appendix

Appendix

In this appendix we present data about the real-world networks considered in the analysis.

Table 1 Topological and spectral properties of the real networks considered: ratio of the distance between the actual largest eigenvalue \(\varLambda _M\) and the LEV’s of the relevant subgraphs (star centered at the hub and max K-core), \(d_\mathrm {star} / d_\mathrm {K_M}\); network size N, relative weight of the PEV on the star centered in the hub, \(W_\mathrm {H+L}\); relative weight of the PEV on the max K-core, \(W_\mathrm {K_M}\); largest eigenvalue \(\varLambda _M\); square root of the maximum degree, \(\sqrt{q_\mathrm {max}}\); average degree of the max K-core, \(\langle {q} \rangle _\mathrm {K_M}\); size of the max K-core, \(N_\mathrm {K_M}\)
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Pastor-Satorras, R., Castellano, C. Eigenvector Localization in Real Networks and Its Implications for Epidemic Spreading. J Stat Phys 173, 1110–1123 (2018). https://doi.org/10.1007/s10955-018-1970-8

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