Spectral properties of complex networks.

This review presents an account of the major works done on spectra of adjacency matrices drawn on networks and the basic understanding attained so far. We have divided the review under three sections: (a) extremal eigenvalues, (b) bulk part of the spectrum, and (c) degenerate eigenvalues, based on the intrinsic properties of eigenvalues and the phenomena they capture. We have reviewed the works done for spectra of various popular model networks, such as the Erdős-Rényi random networks, scale-free networks, 1-d lattice, small-world networks, and various different real-world networks. Additionally, potential applications of spectral properties for natural processes have been reviewed.

[1]  A multilayer PPI network analysis of different life stages in C. elegans , 2016, 1602.08314.

[2]  A. Barabasi,et al.  Spectra of "real-world" graphs: beyond the semicircle law. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  C. Y. Ung,et al.  Spectral analysis of gene co-expression network of Zebrafish , 2012, 1208.4668.

[4]  K. Goh,et al.  Spectra and eigenvectors of scale-free networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  J. Kurths,et al.  Network synchronization, diffusion, and the paradox of heterogeneity. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[7]  Jun Ma,et al.  Spectral properties of the temporal evolution of brain network structure. , 2015, Chaos.

[8]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[9]  S. Teichmann,et al.  Gene regulatory network growth by duplication , 2004, Nature Genetics.

[10]  Feng Luo,et al.  Constructing gene co-expression networks and predicting functions of unknown genes by random matrix theory , 2007, BMC Bioinformatics.

[11]  Gergely Palla,et al.  Spectral transitions in networks , 2006 .

[12]  Alessandro Vespignani,et al.  Cut-offs and finite size effects in scale-free networks , 2003, cond-mat/0311650.

[13]  Duc Thi Luu,et al.  Portfolio Correlations in the Bank-Firm Credit Market of Japan , 2019, Computational Economics.

[14]  F. Chung,et al.  Spectra of random graphs with given expected degrees , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[15]  Christel Kamp,et al.  Spectral analysis of protein-protein interactions in Drosophila melanogaster. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  E. Ott,et al.  Approximating the largest eigenvalue of network adjacency matrices. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Z. Wang,et al.  The structure and dynamics of multilayer networks , 2014, Physics Reports.

[18]  Raffaello Potestio,et al.  Random matrix approach to collective behavior and bulk universality in protein dynamics. , 2009, Physical review letters.

[19]  V. Latora,et al.  Complex networks: Structure and dynamics , 2006 .

[20]  ROBERT M. MAY,et al.  Will a Large Complex System be Stable? , 1972, Nature.

[21]  Sergey N. Dorogovtsev,et al.  Localization and Spreading of Diseases in Complex Networks , 2012, Physical review letters.

[22]  Sarika Jalan,et al.  A multilayer protein-protein interaction network analysis of different life stages in Caenorhabditis elegans , 2015 .

[23]  Camellia Sarkar,et al.  Uncovering Randomness and Success in Society , 2014, PloS one.

[24]  A. Barabasi,et al.  Hierarchical Organization of Modularity in Metabolic Networks , 2002, Science.

[25]  Changsong Zhou,et al.  Universality in the synchronization of weighted random networks. , 2006, Physical review letters.

[26]  Sarika Jalan,et al.  Optimized evolution of networks for principal eigenvector localization. , 2017, Physical review. E.

[27]  S. Jalan,et al.  Extreme-value statistics of networks with inhibitory and excitatory couplings. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  Camellia Sarkar,et al.  Deciphering versatility and cooperation in multilayer social networks , 2015, ArXiv.

[29]  Christos Faloutsos,et al.  Epidemic thresholds in real networks , 2008, TSEC.

[30]  Si Tang,et al.  Stability criteria for complex ecosystems , 2011, Nature.

[31]  R. Kuehn.  Spectra of sparse random matrices , 2008, 0803.2886.

[32]  T. Carroll,et al.  Master Stability Functions for Synchronized Coupled Systems , 1998 .

[33]  Sarika Jalan,et al.  Random matrix analysis of complex networks. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  T. A. Brody.  A statistical measure for the repulsion of energy levels , 1973 .

[35]  E. Ott,et al.  Onset of synchronization in large networks of coupled oscillators. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  J. Kurths,et al.  Interplay of degree correlations and cluster synchronization. , 2016, Physical review. E.

[37]  P. Van Mieghem,et al.  Virus Spread in Networks , 2009, IEEE/ACM Transactions on Networking.

[38]  K. Dessouky,et al.  Network synchronization , 1985, Proceedings of the IEEE.

[39]  Christos Faloutsos,et al.  Epidemic spreading in real networks: an eigenvalue viewpoint , 2003, 22nd International Symposium on Reliable Distributed Systems, 2003. Proceedings..

[40]  S. N. Dorogovtsev,et al.  Evolution of networks , 2001, cond-mat/0106144.

[41]  Alessandro Vespignani,et al.  Epidemic spreading in scale-free networks. , 2000, Physical review letters.

[42]  Bruno Cessac,et al.  Transmitting a signal by amplitude modulation in a chaotic network , 2005, Chaos.

[43]  Conrado J. Pérez Vicente,et al.  Diffusion dynamics on multiplex networks , 2012, Physical review letters.

[44]  Claudio Castellano,et al.  Relating topological determinants of complex networks to their spectral properties: structural and dynamical effects , 2017, 1703.10438.

[45]  Claudio Castellano,et al.  Thresholds for epidemic spreading in networks , 2010, Physical review letters.

[46]  E. Wigner.  Characteristic Vectors of Bordered Matrices with Infinite Dimensions I , 1955 .

[47]  Alessandro Vespignani,et al.  Absence of epidemic threshold in scale-free networks with degree correlations. , 2002, Physical review letters.

[48]  D. Cvetkovic,et al.  Spectra of Graphs: Theory and Applications , 1997 .

[49]  Potsdam,et al.  Complex networks in climate dynamics. Comparing linear and nonlinear network construction methods , 2009, 0907.4359.

[50]  Piet Van Mieghem,et al.  Graph Spectra for Complex Networks , 2010 .

[51]  Sommers,et al.  Chaos in random neural networks. , 1988, Physical review letters.

[52]  Anirban Banerjee,et al.  On the spectrum of the normalized graph Laplacian , 2007, 0705.3772.

[53]  Y. Bar-Yam,et al.  Spectral analysis and the dynamic response of complex networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[54]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[55]  Raj Kumar Pan,et al.  Modularity produces small-world networks with dynamical time-scale separation , 2008, 0802.3671.

[56]  David Poole,et al.  Linear Algebra: A Modern Introduction , 2002 .

[57]  I. Sokolov,et al.  Reshuffling scale-free networks: from random to assortative. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[58]  Camellia Sarkar,et al.  Randomness and Structure in Collaboration Networks: A Random Matrix Analysis , 2016, IEEE Transactions on Computational Social Systems.

[59]  Sarika Jalan,et al.  Dissortativity and duplications in oral cancer , 2015, 1602.07170.

[60]  Sarika Jalan,et al.  Assortative and disassortative mixing investigated using the spectra of graphs. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[61]  Sarika Jalan,et al.  Optimization of synchronizability in multiplex networks by rewiring one layer. , 2017, Physical review. E.

[62]  Edward Ott,et al.  Characterizing the dynamical importance of network nodes and links. , 2006, Physical review letters.

[63]  D. Cvetkovic,et al.  An Introduction to the Theory of Graph Spectra: References , 2009 .

[64]  Sarika Jalan,et al.  Universality in complex networks: random matrix analysis. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[65]  Sarika Jalan,et al.  Randomness and preserved patterns in cancer network , 2014, Scientific Reports.

[66]  Sanjiv K. Dwivedi,et al.  Quantifying randomness in protein–protein interaction networks of different species: A random matrix approach , 2013, 1312.0711.

[67]  Sarika Jalan,et al.  Analysing degeneracies in networks spectra , 2016, ArXiv.

[68]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[69]  Jürgen Kurths,et al.  Synchronization - A Universal Concept in Nonlinear Sciences , 2001, Cambridge Nonlinear Science Series.

[70]  W. Hager,et al.  and s , 2019, Shallow Water Hydraulics.

[71]  T. Guhr,et al.  RANDOM-MATRIX THEORIES IN QUANTUM PHYSICS : COMMON CONCEPTS , 1997, cond-mat/9707301.

[72]  F. Chung,et al.  Eigenvalues of Random Power law Graphs , 2003 .

[73]  S. N. Dorogovtsev,et al.  Spectra of complex networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[74]  E. Ott,et al.  Spectral properties of networks with community structure. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[75]  L. Abbott,et al.  Neural network dynamics. , 2005, Annual review of neuroscience.

[76]  Sarika Jalan,et al.  Interplay of mutation and disassortativity. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[77]  M. Stephanov,et al.  Random Matrices , 2005, hep-ph/0509286.

[78]  Fan Chung Graham,et al.  The Spectra of Random Graphs with Given Expected Degrees , 2004, Internet Math..

[79]  S. Boccaletti,et al.  Synchronization is enhanced in weighted complex networks. , 2005, Physical review letters.

[80]  P. Van Mieghem,et al.  Influence of assortativity and degree-preserving rewiring on the spectra of networks , 2010 .

[81]  G. J. Rodgers,et al.  INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 38 (2005) 9431–9437 doi:10.1088/0305-4470/38/43/003 Eigenvalue spectra of complex networks , 2005 .

[82]  Sarika Jalan,et al.  Random matrix analysis of localization properties of gene coexpression network. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[83]  Sarika Jalan,et al.  Randomness of random networks: A random matrix analysis , 2009 .

[84]  A. Pikovsky,et al.  Synchronization: Theory and Application , 2003 .

[85]  Sarika Jalan,et al.  Origin and implications of zero degeneracy in networks spectra. , 2014, Chaos.

[86]  Orestis Georgiou,et al.  Spectral statistics of random geometric graphs , 2016, 1608.01154.