Node Immunization with Non-backtracking Eigenvalues

The non-backtracking matrix and its eigenvalues have many applications in network science and graph mining, such as node and edge centrality, community detection, length spectrum theory, graph distance, and epidemic and percolation thresholds. Moreover, in network epidemiology, the reciprocal of the largest eigenvalue of the non-backtracking matrix is a good approximation for the epidemic threshold of certain network dynamics. In this work, we develop techniques that identify which nodes have the largest impact on the leading non-backtracking eigenvalue. We do so by studying the behavior of the spectrum of the non-backtracking matrix after a node is removed from the graph. From this analysis we derive two new centrality measures: X-degree and X-non-backtracking centrality. We perform extensive experimentation with targeted immunization strategies derived from these two centrality measures. Our spectral analysis and centrality measures can be broadly applied, and will be of interest to both theorists and practitioners alike.

[1]  Daniel Massey,et al.  Collecting the internet AS-level topology , 2005, CCRV.

[2]  Lenka Zdeborová,et al.  Percolation on sparse networks , 2014, Physical review letters.

[3]  Andrew Mellor,et al.  Graph Comparison via the Non-backtracking Spectrum , 2018, Physical review. E.

[4]  Ulrik Brandes,et al.  What is network science? , 2013, Network Science.

[5]  Claudio Castellano,et al.  Leveraging percolation theory to single out influential spreaders in networks , 2016, Physical review. E.

[6]  Claudio Castellano,et al.  Relevance of backtracking paths in recurrent-state epidemic spreading on networks , 2018, Physical Review. E.

[7]  Jure Leskovec,et al.  Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters , 2008, Internet Math..

[8]  M E J Newman,et al.  Community structure in social and biological networks , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[9]  Mark E. J. Newman,et al.  Stochastic blockmodels and community structure in networks , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  Tamara G. Kolda,et al.  A Scalable Generative Graph Model with Community Structure , 2013, SIAM J. Sci. Comput..

[11]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[12]  Peter Grindrod,et al.  On the exponential generating function for non-backtracking walks , 2018, Linear Algebra and its Applications.

[13]  Hernán A. Makse,et al.  Collective Influence Algorithm to find influencers via optimal percolation in massively large social media , 2016, Scientific Reports.

[14]  Romualdo Pastor-Satorras,et al.  Influential spreaders for recurrent epidemics on networks , 2019, Physical Review Research.

[15]  Peter Grindrod,et al.  Non-backtracking walk centrality for directed networks , 2018, J. Complex Networks.

[16]  Laurent Massoulié,et al.  Non-backtracking Spectrum of Random Graphs: Community Detection and Non-regular Ramanujan Graphs , 2014, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[17]  Wei Chen,et al.  Assessing Percolation Threshold Based on High-Order Non-Backtracking Matrices , 2016, WWW.

[18]  Yizhe Zhu,et al.  Eigenvalues of the non-backtracking operator detached from the bulk , 2019, Random Matrices: Theory and Applications.

[19]  Pan Zhang.  Nonbacktracking operator for the Ising model and its applications in systems with multiple states. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Martin G. Everett,et al.  Induced, endogenous and exogenous centrality , 2010, Soc. Networks.

[21]  Kathleen E. Hamilton,et al.  Tight lower bound for percolation threshold on an infinite graph. , 2014, Physical review letters.

[22]  Willem H. Haemers,et al.  Spectra of Graphs , 2011 .

[23]  M. De Domenico,et al.  The Anatomy of a Scientific Rumor , 2013, Scientific Reports.

[24]  Piet Van Mieghem,et al.  Epidemic processes in complex networks , 2014, ArXiv.

[25]  N. Alon,et al.  Non-backtracking random walks mix faster , 2006, math/0610550.

[26]  Lev Muchnik,et al.  Identifying influential spreaders in complex networks , 2010, 1001.5285.

[27]  Albert-Lszl Barabsi,et al.  Network Science , 2016, Encyclopedia of Big Data.

[28]  Mark Kempton.  Non-backtracking random walks and a weighted Ihara's theorem , 2016, 1603.05553.

[29]  M. Newman.  Spread of epidemic disease on networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  M. Lelarge,et al.  Nonbacktracking spectrum of random graphs: Community detection and nonregular Ramanujan graphs , 2018 .

[31]  Tina Eliassi-Rad,et al.  Non-backtracking cycles: length spectrum theory and graph mining applications , 2019, Appl. Netw. Sci..

[32]  Cristopher Moore,et al.  A message-passing approach for recurrent-state epidemic models on networks , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[33]  Xiao Zhang,et al.  Localization and centrality in networks , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  Hernán A. Makse,et al.  Influence maximization in complex networks through optimal percolation , 2015, Nature.

[35]  Aric Hagberg,et al.  Exploring Network Structure, Dynamics, and Function using NetworkX , 2008, Proceedings of the Python in Science Conference.

[36]  Elchanan Mossel,et al.  Spectral redemption in clustering sparse networks , 2013, Proceedings of the National Academy of Sciences.

[37]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[38]  Tamara G. Kolda,et al.  Community structure and scale-free collections of Erdös-Rényi graphs , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.