Structure and dynamical behaviour of non-normal networks.

We show that many real-world networks are non-normal, with important consequences for their stability and dynamical behavior. We analyze a collection of empirical networks in a wide spectrum of disciplines and show that strong non-normality is ubiquitous in network science. Dynamical processes evolving on non-normal networks exhibit a peculiar behavior, as initial small disturbances may undergo a transient phase and be strongly amplified in linearly stable systems. In addition, eigenvalues may become extremely sensible to noise and have a diminished physical meaning. We identify structural properties of networks that are associated with non-normality and propose simple models to generate networks with a tunable level of non-normality. We also show the potential use of a variety of metrics capturing different aspects of non-normality and propose their potential use in the context of the stability of complex ecosystems.

[1]  T. Tao,et al.  RANDOM MATRICES: THE CIRCULAR LAW , 2007, 0708.2895.

[2]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[3]  D. Baird,et al.  Assessment of spatial and temporal variability in ecosystem attributes of the St Marks national wildlife refuge, Apalachee bay, Florida , 1998 .

[4]  Michael Ley,et al.  The DBLP Computer Science Bibliography: Evolution, Research Issues, Perspectives , 2002, SPIRE.

[5]  A. Barabasi,et al.  Network medicine : a network-based approach to human disease , 2010 .

[6]  Wiley Interscience.  Journal of the American Society for Information Science and Technology , 2013 .

[7]  Brian A. Worley.  KNOWLEDGE DISCOVERY FROM DYNAMIC, DISPARATE DATA , 2012 .

[8]  Lada A. Adamic,et al.  The political blogosphere and the 2004 U.S. election: divided they blog , 2005, LinkKDD '05.

[9]  Cynthia M. Webster,et al.  Exploring social structure using dynamic three-dimensional color images , 1998 .

[10]  A. Rbnyi.  ON THE EVOLUTION OF RANDOM GRAPHS , 2001 .

[11]  Albert-László Barabási,et al.  Controllability of complex networks , 2011, Nature.

[12]  S. Holmes,et al.  TRACKING NETWORK DYNAMICS : A SURVEY OF DISTANCES AND SIMILARITY METRICS , 2018 .

[13]  J. Coleman.  Introduction to Mathematical Sociology , 1965 .

[14]  Neo D. Martinez.  Artifacts or Attributes? Effects of Resolution on the Little Rock Lake Food Web , 1991 .

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

[16]  Joaquín Goñi,et al.  On the origins of hierarchy in complex networks , 2013, Proceedings of the National Academy of Sciences.

[17]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

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

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

[20]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[21]  P. Cochat,et al.  Et al , 2008, Archives de pediatrie : organe officiel de la Societe francaise de pediatrie.

[22]  R W Eash,et al.  EQUILIBRIUM TRAFFIC ASSIGNMENT ON AN AGGREGATED HIGHWAY NETWORK FOR SKETCH PLANNING , 1983 .

[23]  Marcus Kaiser,et al.  Nonoptimal Component Placement, but Short Processing Paths, due to Long-Distance Projections in Neural Systems , 2006, PLoS Comput. Biol..

[24]  J. Coleman,et al.  The Diffusion of an Innovation Among Physicians , 1957 .

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

[26]  S. Shen-Orr,et al.  Network motifs in the transcriptional regulation network of Escherichia coli , 2002, Nature Genetics.

[27]  Harry Eugene Stanley,et al.  Catastrophic cascade of failures in interdependent networks , 2009, Nature.

[28]  Thomas E. Gorochowski,et al.  Organization of feed-forward loop motifs reveals architectural principles in natural and engineered networks , 2017, Science Advances.

[29]  S. Shen-Orr,et al.  Network motifs: simple building blocks of complex networks. , 2002, Science.

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

[31]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[32]  K. Miller,et al.  Balanced Amplification: A New Mechanism of Selective Amplification of Neural Activity Patterns , 2016, Neuron.

[33]  Claire Donnat,et al.  Tracking Network Dynamics: a review of distances and similarity metrics , 2018, ArXiv.

[34]  Tore Opsahl,et al.  Clustering in weighted networks , 2009, Soc. Networks.

[35]  Jure Leskovec,et al.  Predicting positive and negative links in online social networks , 2010, WWW '10.

[36]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[37]  S. Swain.  Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences , 1984 .

[38]  Munmun De Choudhury,et al.  Social Synchrony: Predicting Mimicry of User Actions in Online Social Media , 2009, 2009 International Conference on Computational Science and Engineering.

[39]  M. Moran,et al.  Large-scale mapping of human protein–protein interactions by mass spectrometry , 2007, Molecular systems biology.

[40]  B. Bollobás.  The evolution of random graphs , 1984 .

[41]  Nick S Jones,et al.  Looplessness in networks is linked to trophic coherence , 2015, Proceedings of the National Academy of Sciences.

[42]  D J PRICE,et al.  NETWORKS OF SCIENTIFIC PAPERS. , 1965, Science.

[43]  Albert-László Barabási,et al.  Hierarchical organization in complex networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[45]  Higinio Mora-Mora,et al.  μ-MAR: Multiplane 3D Marker based Registration for depth-sensing cameras , 2015, Expert Syst. Appl..

[46]  P. Chesson.  Mechanisms of Maintenance of Species Diversity , 2000 .

[47]  C. Chicone.  Ordinary Differential Equations with Applications , 1999, Texts in Applied Mathematics.

[49]  R. Ulanowicz,et al.  The Seasonal Dynamics of The Chesapeake Bay Ecosystem , 1989 .

[50]  K. Foster,et al.  The ecology of the microbiome: Networks, competition, and stability , 2015, Science.

[51]  S. Holmes,et al.  Tracking network dynamics: A survey using graph distances , 2018, The Annals of Applied Statistics.

[52]  Christos Faloutsos,et al.  Graph evolution: Densification and shrinking diameters , 2006, TKDD.

[53]  Cristopher Moore,et al.  A physical model for efficient ranking in networks , 2017, Science Advances.

[54]  Timoteo Carletti,et al.  Topological resilience in non-normal networked systems , 2017, ALIFE.

[55]  S. L. Wong,et al.  Towards a proteome-scale map of the human protein–protein interaction network , 2005, Nature.

[56]  Anne E. Trefethen,et al.  Hydrodynamic Stability Without Eigenvalues , 1993, Science.

[57]  David E. Boyce,et al.  IMPLEMENTATION AND EVALUATION OF COMBINED MODELS OF URBAN TRAVEL AND LOCATION ON A SKETCH PLANNING NETWORK , 1985 .

[58]  David Liben-Nowell,et al.  The link-prediction problem for social networks , 2007 .

[59]  Y. J. Lee,et al.  Equilibrium Traffic Assignment on an Aggregated Highway Network for Sketch Planning , 2022 .

[60]  Dániel Czégel,et al.  Random walk hierarchy measure: What is more hierarchical, a chain, a tree or a star? , 2015, Scientific Reports.

[61]  Robert E. Ulanowicz,et al.  Benthic-pelagic switching in a coastal subtropical lagoon , 1999 .

[62]  Jure Leskovec,et al.  Signed networks in social media , 2010, CHI.

[63]  R. Ulanowicz.  Growth and development : ecosystems phenomenology , 1988 .

[64]  J. E. Cohen,et al.  Global stability, local stability and permanence in model food webs. , 2001, Journal of theoretical biology.

[65]  Kenneth E. Boulding,et al.  Introduction to Mathematical Sociology. , 1966 .

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

[67]  Hrvoje Štefančić,et al.  Model of Wikipedia growth based on information exchange via reciprocal arcs , 2009, ArXiv.

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

[69]  A. Arenas,et al.  Community detection in complex networks using extremal optimization. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[70]  S. Shen-Orr,et al.  Superfamilies of Evolved and Designed Networks , 2004, Science.

[71]  M. Newman,et al.  Finding community structure in networks using the eigenvectors of matrices. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[72]  Matthew Richardson,et al.  Trust Management for the Semantic Web , 2003, SEMWEB.

[73]  W. Gerstner,et al.  Non-normal amplification in random balanced neuronal networks. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[74]  Shilpa Chakravartula,et al.  Complex Networks: Structure and Dynamics , 2014 .

[75]  Alessandro Vespignani,et al.  The role of the airline transportation network in the prediction and predictability of global epidemics , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[76]  Francesco Picciolo,et al.  Reciprocity of weighted networks , 2012, Scientific Reports.

[77]  Fardad Farokhi,et al.  Application of a time delay neural network for predicting positive and negative links in social networks , 2016 .

[78]  L. Trefethen,et al.  Spectra and Pseudospectra , 2020 .

[79]  Mark Newman,et al.  Networks: An Introduction , 2010 .

[80]  Yiming Yang,et al.  The Enron Corpus: A New Dataset for Email Classi(cid:12)cation Research , 2004 .

[81]  H. Caswell,et al.  ALTERNATIVES TO RESILIENCE FOR MEASURING THE RESPONSES OF ECOLOGICAL SYSTEMS TO PERTURBATIONS , 1997 .

[82]  G. E. Hutchinson,et al.  The Balance of Nature and Human Impact: The paradox of the plankton , 2013 .

[83]  Robert E. Ulanowicz,et al.  Comparative ecosystem trophic structure of three U.S. mid-Atlantic estuaries , 1997 .