Recent Progress on the Resilience of Complex Networks

Many complex systems in the real world can be modeled as complex networks, which has captured in recent years enormous attention from researchers of diverse fields ranging from natural sciences to engineering. The extinction of species in ecosystems and the blackouts of power girds in engineering exhibit the vulnerability of complex networks, investigated by empirical data and analyzed by theoretical models. For studying the resilience of complex networks, three main factors should be focused on: the network structure, the network dynamics and the failure mechanism. In this review, we will introduce recent progress on the resilience of complex networks based on these three aspects. For the network structure, increasing evidence shows that biological and ecological networks are coupled with each other and that diverse critical infrastructures interact with each other, triggering a new research hotspot of “networks of networks” (NON), where a network is formed by interdependent or interconnected networks. The resilience of complex networks is deeply influenced by its interdependence with other networks, which can be analyzed and predicted by percolation theory. This review paper shows that the analytic framework for Energies 2015, 8 12188 NON yields novel percolation laws for n interdependent networks and also shows that the percolation theory of a single network studied extensively in physics and mathematics in the last 60 years is a specific limited case of the more general case of n interacting networks. Due to spatial constraints inherent in critical infrastructures, including the power gird, we also review the progress on the study of spatially-embedded interdependent networks, exhibiting extreme vulnerabilities compared to their non-embedded counterparts, especially in the case of localized attack. For the network dynamics, we illustrate the percolation framework and methods using an example of a real transportation system, where the analysis based on network dynamics is significantly different from the structural static analysis. For the failure mechanism, we here review recent progress on the spontaneous recovery after network collapse. These findings can help us to understand, realize and hopefully mitigate the increasing risk in the resilience of complex networks.

[1]  Shlomo Havlin,et al.  Improved El Niño forecasting by cooperativity detection , 2013, Proceedings of the National Academy of Sciences.

[2]  Massimo Marchiori,et al.  Model for cascading failures in complex networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  S. N. Dorogovtsev,et al.  Avalanche collapse of interdependent networks. , 2012, Physical review letters.

[4]  Kun Zhao,et al.  Percolation on interacting, antagonistic networks , 2012, ArXiv.

[5]  Amir Bashan,et al.  Interdependent Spatially Embedded Networks: Dynamics at Percolation Threshold , 2013, 2013 International Conference on Signal-Image Technology & Internet-Based Systems.

[6]  A. Raftery,et al.  Model‐based clustering for social networks , 2007 .

[7]  Tao Zhou,et al.  Onset of cooperation between layered networks. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Gerhard P. Hancke,et al.  Opportunities and Challenges of Wireless Sensor Networks in Smart Grid , 2010, IEEE Transactions on Industrial Electronics.

[9]  C. Steglich,et al.  DYNAMIC NETWORKS AND BEHAVIOR: SEPARATING SELECTION FROM INFLUENCE: separating selection from influence , 2010 .

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

[11]  K. Aihara,et al.  Efficient recovery of dynamic behavior in coupled oscillator networks. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Michalis Faloutsos,et al.  On power-law relationships of the Internet topology , 1999, SIGCOMM '99.

[13]  Daniel J. Brass,et al.  Network Analysis in the Social Sciences , 2009, Science.

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

[15]  S. Havlin,et al.  The extreme vulnerability of interdependent spatially embedded networks , 2012, Nature Physics.

[16]  E A Leicht,et al.  Suppressing cascades of load in interdependent networks , 2011, Proceedings of the National Academy of Sciences.

[17]  Harry Eugene Stanley,et al.  Percolation of localized attack on complex networks , 2014, ArXiv.

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

[19]  M Barthelemy,et al.  Transport on coupled spatial networks. , 2012, Physical review letters.

[20]  S. Havlin,et al.  Simultaneous first- and second-order percolation transitions in interdependent networks. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Peng Zhang,et al.  The robustness of interdependent transportation networks under targeted attack , 2013, ArXiv.

[22]  D. Pe’er,et al.  Module networks: identifying regulatory modules and their condition-specific regulators from gene expression data , 2003, Nature Genetics.

[23]  Stephen P. Borgatti.  Identifying sets of key players in a network , 2003, IEMC '03 Proceedings. Managing Technologically Driven Organizations: The Human Side of Innovation and Change (IEEE Cat. No.03CH37502).

[24]  Sergey N. Dorogovtsev,et al.  Evolution of Networks: From Biological Nets to the Internet and WWW (Physics) , 2003 .

[25]  B.F. Wollenberg,et al.  Toward a smart grid: power delivery for the 21st century , 2005, IEEE Power and Energy Magazine.

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

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

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

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

[30]  Shlomo Havlin,et al.  Complex networks embedded in space: Dimension and scaling relations between mass, topological distance and Euclidean distance , 2012, ArXiv.

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

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

[33]  S. Strogatz.  Exploring complex networks , 2001, Nature.

[34]  D. V. Senthilkumar,et al.  Restoration of rhythmicity in diffusively coupled dynamical networks , 2015, Nature Communications.

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

[36]  Béla Bollobás,et al.  Random Graphs , 1985 .

[37]  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.

[38]  Shlomo Havlin,et al.  Two distinct transitions in spatially embedded multiplex networks , 2015 .

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

[40]  Hans J. Herrmann,et al.  Onion-like network topology enhances robustness against malicious attacks , 2011 .

[41]  H. Farhangi,et al.  The path of the smart grid , 2010, IEEE Power and Energy Magazine.

[42]  Harry Eugene Stanley,et al.  Epidemics on Interconnected Networks , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  Benoit Robert,et al.  The operational tools for managing physical interdependencies among critical infrastructures , 2008, Int. J. Crit. Infrastructures.

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

[45]  K. Goh,et al.  Universal behavior of load distribution in scale-free networks. , 2001, Physical review letters.

[46]  H. Stanley,et al.  Direct molecular dynamics observation of protein folding transition state ensemble. , 2002, Biophysical journal.

[47]  Lidia A. Braunstein,et al.  Multiple tipping points and optimal repairing in interacting networks , 2015, Nature Communications.

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

[49]  Reuven Cohen,et al.  Complex Networks: Structure, Robustness and Function , 2010 .

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

[51]  Michael J. O. Pocock,et al.  The Robustness and Restoration of a Network of Ecological Networks , 2012, Science.

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

[53]  Reuven Cohen,et al.  Percolation in Interdependent and Interconnected Networks: Abrupt Change from Second to First Order Transition , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[54]  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.

[55]  Javier M. Buldú,et al.  Successful strategies for competing networks , 2013, Nature Physics.

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

[57]  Shlomo Havlin,et al.  Robustness of a network formed of spatially embedded networks. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[58]  Peter Grassberger,et al.  Percolation theory on interdependent networks based on epidemic spreading , 2011, 1109.4447.

[59]  W. Wallace,et al.  Impacts of the 2001 World Trade Center Attack on New York City Critical Infrastructures , 2006 .

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

[61]  Alessandro Vespignani,et al.  Dynamical Processes on Complex Networks , 2008 .

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

[63]  R. D’Souza,et al.  Percolation on interacting networks , 2009, 0907.0894.

[64]  Seth A. Myers,et al.  Spontaneous synchrony in power-grid networks , 2013, Nature Physics.

[65]  Kang Rui,et al.  Spatial correlation analysis of cascading failures: Congestions and Blackouts , 2014, Scientific Reports.

[66]  Shlomo Havlin,et al.  Diffusion, annihilation, and chemical reactions in complex networks with spatial constraints. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[67]  Mason A. Porter,et al.  Multilayer networks , 2013, J. Complex Networks.

[68]  K. C. Kapur,et al.  Methodology for Assessing the Resilience of Networked Infrastructure , 2009, IEEE Systems Journal.

[69]  Stephen P. Borgatti,et al.  Identifying sets of key players in a social network , 2006, Comput. Math. Organ. Theory.

[70]  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.

[71]  Gouhei Tanaka,et al.  Dynamical robustness in complex networks: the crucial role of low-degree nodes , 2012, Scientific Reports.

[72]  Huijuan Wang,et al.  Epidemics on interconnected lattices , 2014 .

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

[74]  Yunpeng Wang,et al.  Percolation transition in dynamical traffic network with evolving critical bottlenecks , 2014, Proceedings of the National Academy of Sciences.

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

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

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

[78]  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.

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

[80]  A. Arenas,et al.  Abrupt transition in the structural formation of interconnected networks , 2013, Nature Physics.

[81]  M. Zhan,et al.  Dynamical robustness analysis of weighted complex networks , 2013 .

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

[83]  D. Newth,et al.  Optimizing complex networks for resilience against cascading failure , 2007 .

[84]  Duncan J. Watts,et al.  The Structure and Dynamics of Networks: (Princeton Studies in Complexity) , 2006 .

[85]  Yang-Yu Liu,et al.  Inducing effect on the percolation transition in complex networks , 2013, Nature Communications.

[86]  David Bawden,et al.  Book Review: Evolution and Structure of the Internet: A Statistical Physics Approach. , 2006 .

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

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

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

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

[91]  R. Thottappillil,et al.  Methodology for Classifying Facilities With Respect to Intentional EMI , 2009, IEEE Transactions on Electromagnetic Compatibility.

[92]  Katherine Faust,et al.  Comparing Social Networks: Size, Density, and Local Structure , 2006 .

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

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

[95]  L. Summers,et al.  Hysteresis and the European Unemployment Problem , 1986, NBER Macroeconomics Annual.

[96]  Amir Bashan,et al.  Percolation in networks composed of connectivity and dependency links , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[97]  Junshan Zhang,et al.  Optimal Allocation of Interconnecting Links in Cyber-Physical Systems: Interdependence, Cascading Failures, and Robustness , 2012, IEEE Transactions on Parallel and Distributed Systems.

[98]  Dong Zhou,et al.  Percolation of interdependent networks with intersimilarity. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[99]  Brian W. Rogers,et al.  Meeting Strangers and Friends of Friends: How Random are Social Networks? , 2007 .

[100]  Massimo Marchiori,et al.  Error and attacktolerance of complex network s , 2004 .

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

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

[103]  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.

[104]  S. Havlin,et al.  Fractals and Disordered Systems , 1991 .

[105]  Amir Bashan,et al.  Localized attacks on spatially embedded networks with dependencies , 2015, Scientific Reports.

[106]  A-L Barabási,et al.  Structure and tie strengths in mobile communication networks , 2006, Proceedings of the National Academy of Sciences.

[107]  Rae Zimmerman.  Decision-making and the vulnerability of interdependent critical infrastructure , 2004, 2004 IEEE International Conference on Systems, Man and Cybernetics (IEEE Cat. No.04CH37583).

[108]  M. Newman,et al.  Random graphs with arbitrary degree distributions and their applications. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[109]  Antonio Scala,et al.  Networks of Networks: The Last Frontier of Complexity , 2014 .

[110]  Márk Jelasity,et al.  Gossip-based aggregation in large dynamic networks , 2005, TOCS.

[111]  L. Tian,et al.  Percolation of partially interdependent networks under targeted attack. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[112]  U. Alon.  Biological Networks: The Tinkerer as an Engineer , 2003, Science.

[113]  H E Stanley,et al.  Finite-size scaling investigation of the liquid-liquid critical point in ST2 water and its stability with respect to crystallization. , 2013, The Journal of chemical physics.

[114]  Marc Timme,et al.  Self-organized adaptation of a simple neural circuit enables complex robot behaviour , 2010, ArXiv.

[115]  Hans J. Herrmann,et al.  Towards designing robust coupled networks , 2011, Scientific Reports.

[116]  Daqing Li,et al.  From a single network to a network of networks , 2014 .

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

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