Network of Interdependent Networks: Overview of Theory and Applications

Complex networks appear in almost every aspect of science and technology. Previous work in network theory has focused primarily on analyzing single networks that do not interact with other networks, despite the fact that many real-world networks interact with and depend on each other. Very recently an analytical framework for studying the percolation properties of interacting networks has been introduced. Here we review the analytical framework and the results for percolation laws for a network of networks (NON) formed by \(n\) interdependent random networks. The percolation properties of a network of networks differ greatly from those of single isolated networks. In particular, although networks with broad degree distributions, e.g., scale-free networks, are robust when analyzed as single networks, they become vulnerable in a NON. Moreover, because the constituent networks of a NON are connected by node dependencies, a NON is subject to cascading failure. When there is strong interdependent coupling between networks, the percolation transition is discontinuous (is a first-order transition), unlike the well-known continuous second-order transition in single isolated networks. We also review some possible real-world applications of NON theory.

[1]  E. B. Wilson.  PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES. , 1919, Science.

[2]  O. Bagasra,et al.  Proceedings of the National Academy of Sciences , 1914, Science.

[3]  E. Hill.  Journal of Theoretical Biology , 1961, Nature.

[4]  Physical Review , 1965, Nature.

[5]  Frank Harary,et al.  Graph Theory , 2016 .

[6]  M. D. Wilkinson,et al.  Management science , 1989, British Dental Journal.

[7]  Securities,et al.  Financial stability review , 2011 .

[8]  G. Vojta.  Fractals and Disordered Systems , 1997 .

[9]  Bruce A. Reed,et al.  The Size of the Giant Component of a Random Graph with a Given Degree Sequence , 1998, Combinatorics, Probability and Computing.

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

[11]  A. Châtelain,et al.  The European Physical Journal D , 1999 .

[12]  D S Callaway,et al.  Network robustness and fragility: percolation on random graphs. , 2000, Physical review letters.

[13]  Albert-László Barabási,et al.  Error and attack tolerance of complex networks , 2000, Nature.

[14]  Cohen,et al.  Resilience of the internet to random breakdowns , 2000, Physical review letters.

[15]  S. Havlin,et al.  Breakdown of the internet under intentional attack. , 2000, Physical review letters.

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

[17]  Christian Upper,et al.  Estimating Bilateral Exposures in the German Interbank Market: Is There a Danger of Contagion? , 2002, SSRN Electronic Journal.

[18]  M. Newman,et al.  The structure of scientific collaboration networks. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[19]  James P. Peerenboom,et al.  Identifying, understanding, and analyzing critical infrastructure interdependencies , 2001 .

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

[21]  Adilson E Motter,et al.  Cascade-based attacks on complex networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Duncan J Watts,et al.  A simple model of global cascades on random networks , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[23]  Beom Jun Kim,et al.  Attack vulnerability of complex networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  H. Stanley,et al.  Optimal paths in disordered complex networks. , 2003, Physical review letters.

[25]  F. Lillo,et al.  Topology of correlation-based minimal spanning trees in real and model markets. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Reuven Cohen,et al.  Efficient immunization strategies for computer networks and populations. , 2002, Physical review letters.

[27]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[28]  M E J Newman,et al.  Finding and evaluating community structure in networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  A. Vespignani,et al.  The architecture of complex weighted networks. , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[30]  H. Shin,et al.  Liquidity Risk and Contagion , 2005 .

[31]  S. Havlin,et al.  Self-similarity of complex networks , 2005, Nature.

[32]  T. Vicsek,et al.  Uncovering the overlapping community structure of complex networks in nature and society , 2005, Nature.

[33]  Reuven Cohen,et al.  Stability and topology of scale-free networks under attack and defense strategies. , 2005, Physical review letters.

[34]  Massimo Marchiori,et al.  Vulnerability and protection of infrastructure networks. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  Shlomo Havlin,et al.  Anomalous transport in scale-free networks. , 2005, Physical review letters.

[36]  Gueorgi Kossinets,et al.  Empirical Analysis of an Evolving Social Network , 2006, Science.

[37]  B. Bollobás,et al.  Combinatorics, Probability and Computing , 2006 .

[38]  Marián Boguñá,et al.  Clustering in complex networks. I. General formalism. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  H. Stanley,et al.  Optimal paths in complex networks with correlated weights: the worldwide airport network. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  Helmut Elsinger,et al.  Risk Assessment for Banking Systems , 2003, Manag. Sci..

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

[42]  Mark E. J. Newman,et al.  Structure and Dynamics of Networks , 2009 .

[43]  New Journal of Physics The , 2007 .

[44]  Winand Emons,et al.  European Economic Review , 2007 .

[45]  Jing Yang,et al.  Network Models and Financial Stability , 2007 .

[46]  Julio M. Ottino,et al.  Cascading failure and robustness in metabolic networks , 2008, Proceedings of the National Academy of Sciences.

[47]  George Sugihara,et al.  Complex systems: Ecology for bankers , 2008, Nature.

[48]  Vittorio Rosato,et al.  Modelling interdependent infrastructures using interacting dynamical models , 2008, Int. J. Crit. Infrastructures.

[49]  Shlomo Havlin,et al.  Finding a better immunization strategy. , 2008, Physical review letters.

[50]  H. Stanley,et al.  Structure of shells in complex networks. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[51]  Alessandro Vespignani,et al.  Multiscale mobility networks and the spatial spreading of infectious diseases , 2009, Proceedings of the National Academy of Sciences.

[52]  Marián Boguñá,et al.  Navigating ultrasmall worlds in ultrashort time. , 2008, Physical review letters.

[53]  H. Herrmann,et al.  How to Make a Fragile Network Robust and Vice Versa , 2009, Physical review letters.

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

[55]  S. Melnik,et al.  Analytical results for bond percolation and k-core sizes on clustered networks. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[56]  Joel C. Miller,et al.  Percolation and epidemics in random clustered networks. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[57]  M E J Newman,et al.  Random graphs with clustering. , 2009, Physical review letters.

[58]  S. Havlin,et al.  Interdependent networks: reducing the coupling strength leads to a change from a first to second order percolation transition. , 2010, Physical review letters.

[59]  Cesar Ducruet,et al.  Inter-similarity between coupled networks , 2010, ArXiv.

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

[61]  Mark E. J. Newman,et al.  Random graphs containing arbitrary distributions of subgraphs , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[62]  Alessandro Vespignani,et al.  Complex networks: The fragility of interdependency , 2010, Nature.

[63]  Sergey V. Buldyrev,et al.  Critical effect of dependency groups on the function of networks , 2010, Proceedings of the National Academy of Sciences.

[64]  Lawrence J. White,et al.  Déjà Vu All Over Again: The Causes of U.S. Commercial Bank Failures This Time Around , 2011, Journal of Financial Services Research.

[65]  Marc Barthelemy,et al.  Spatial Networks , 2010, Encyclopedia of Social Network Analysis and Mining.

[66]  E. Ben-Jacob,et al.  Dominating Clasp of the Financial Sector Revealed by Partial Correlation Analysis of the Stock Market , 2010, PloS one.

[67]  Marco Tomassini,et al.  Worldwide spreading of economic crisis , 2010, 1008.3893.

[68]  Sergey Melnik,et al.  How clustering affects the bond percolation threshold in complex networks. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[69]  Chris Arney,et al.  Complex Webs: Anticipating the Improbable , 2011 .

[70]  Jürgen Kurths,et al.  Investigating the topology of interacting networks , 2011, 1102.3067.

[71]  Thomas Lux,et al.  Financial systems: Ecology and economics , 2011, Nature.

[72]  S. Buldyrev,et al.  Interdependent networks with identical degrees of mutually dependent nodes. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[73]  Hans J. Herrmann,et al.  Mitigation of malicious attacks on networks , 2011, Proceedings of the National Academy of Sciences.

[74]  R. May,et al.  Systemic risk in banking ecosystems , 2011, Nature.

[75]  S. Havlin,et al.  Dimension of spatially embedded networks , 2011 .

[76]  Eshel Ben-Jacob,et al.  Analyses of antigen dependency networks unveil immune system reorganization between birth and adulthood. , 2011, Chaos.

[77]  Harry Eugene Stanley,et al.  Robustness of interdependent networks under targeted attack , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[78]  Harry Eugene Stanley,et al.  Robustness of a Network of Networks , 2010, Physical review letters.

[79]  Harry Eugene Stanley,et al.  Cascade of failures in coupled network systems with multiple support-dependent relations , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[80]  Yoed N. Kenett,et al.  Global and Local Features of Semantic Networks: Evidence from the Hebrew Mental Lexicon , 2011, PloS one.

[81]  Harry Eugene Stanley,et al.  The robustness of interdependent clustered networks , 2012, ArXiv.

[82]  Thomas Lux,et al.  Evolvement of Uniformity and Volatility in the Stressed Global Financial Village , 2012, PloS one.

[83]  H. Stanley,et al.  Networks formed from interdependent networks , 2011, Nature Physics.

[84]  Wei Li,et al.  Cascading Failures in Interdependent Lattice Networks: The Critical Role of the Length of Dependency Links , 2012, Physical review letters.

[85]  G. Caldarelli,et al.  DebtRank: Too Central to Fail? Financial Networks, the FED and Systemic Risk , 2012, Scientific Reports.

[86]  Harry Eugene Stanley,et al.  Robustness of onion-like correlated networks against targeted attacks , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[87]  Amir Bashan,et al.  Network physiology reveals relations between network topology and physiological function , 2012, Nature Communications.

[88]  E. Ben-Jacob,et al.  Challenges in network science: Applications to infrastructures, climate, social systems and economics , 2012 .

[89]  Thomas Lux,et al.  CORRELATIONS AND DEPENDENCIES IN THE GLOBAL FINANCIAL VILLAGE , 2012 .

[90]  S. Havlin,et al.  Robustness of a network formed by n interdependent networks with a one-to-one correspondence of dependent nodes. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[91]  Harry Eugene Stanley,et al.  Assortativity Decreases the Robustness of Interdependent Networks , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[92]  Eshel Ben-Jacob,et al.  Dependency Network and Node Influence: Application to the Study of Financial Markets , 2012, Int. J. Bifurc. Chaos.

[93]  Eshel Ben-Jacob,et al.  Individual and meta-immune networks , 2013, Physical biology.

[94]  Ginestra Bianconi,et al.  Percolation in multiplex networks with overlap. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[95]  H. Stanley,et al.  Robustness of network of networks under targeted attack. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[96]  Xiaoming Xu,et al.  Percolation of a general network of networks. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[97]  Kathy P. Wheeler,et al.  Reviews of Modern Physics , 2013 .

[98]  H. Stanley,et al.  Spontaneous recovery in dynamical networks , 2013, Nature Physics.

[99]  Web Appendix to “Unstable Inflation Targets”∗ Appearing in the Journal of Money, Credit and Banking , 2016 .

[100]  Physics Reports , 2022 .

[101]  October I.  Physical Review Letters , 2022 .