Eradicating abrupt collapse on single network with dependency groups.

The dependency among nodes has significant effects on the cascading failures of complex networks. Although the prevention of cascading failures on multilayered networks in which the failures of nodes in one layer affect the functioning of nodes in other layers has been widely investigated, the prevention of catastrophic cascade has rarely been addressed to single-layer networks where nodes are grouped and nodes within the same group are dependent on each other. For such networks, we find that it is already enough to prevent abrupt catastrophic collapses by randomly reinforcing a constant density of nodes. More importantly, we give the analytical solutions to the proportion of needed reinforced nodes for three typical networks, i.e., dependent Erdős-Rényi (ER), random regular (RR), and scale-free (SF) networks. Interestingly, the density of reinforced nodes is a constant 0.1756, which holds true for ER networks with group size 2 regardless of average degree, RR, and SF networks with a large average degree. Also, we find the elegant expression of the density with any group size. In addition, we find a hybrid phase transition behavior, which is present in RR and SF networks while absent in ER networks. Our findings might shed some new light on designing more resilient infrastructure networks.

[1]  Kari Alanne,et al.  Distributed energy generation and sustainable development , 2006 .

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

[3]  Ronnie Belmans,et al.  Distributed generation: definition, benefits and issues , 2005 .

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

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

[6]  J. S. Andrade,et al.  Avoiding catastrophic failure in correlated networks of networks , 2014, Nature Physics.

[7]  Pol Colomer-de-Simon,et al.  Double percolation phase transition in clustered complex networks , 2014, ArXiv.

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

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

[10]  K-I Goh,et al.  Network robustness of multiplex networks with interlayer degree correlations. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  Yoed N. Kenett,et al.  Critical tipping point distinguishing two types of transitions in modular network structures. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[13]  Mason A. Porter,et al.  Dynamics on Modular Networks with Heterogeneous Correlations , 2012, Chaos.

[14]  Michael Szell,et al.  Multirelational organization of large-scale social networks in an online world , 2010, Proceedings of the National Academy of Sciences.

[15]  Randy H. Katz,et al.  Transport protocols for Internet-compatible satellite networks , 1999, IEEE J. Sel. Areas Commun..

[16]  Shuliang Wang,et al.  Multiple perspective vulnerability analysis of the power network , 2018 .

[17]  Endre Csóka,et al.  Core percolation on complex networks , 2012, Physical review letters.

[18]  Peng Zhang,et al.  Comparative definition of community and corresponding identifying algorithm. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[20]  Shlomo Havlin,et al.  Eradicating catastrophic collapse in interdependent networks via reinforced nodes , 2016, Proceedings of the National Academy of Sciences.

[21]  Abbas Mohammed,et al.  The Role of High-Altitude Platforms (HAPs) in the Global Wireless Connectivity , 2011, Proceedings of the IEEE.

[22]  Xiaobin Li,et al.  Bootstrap Percolation on Complex Networks with Community Structure , 2014, ArXiv.

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

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

[25]  Stefano Panzieri,et al.  Failures propagation in critical interdependent infrastructures , 2008, Int. J. Model. Identif. Control..

[26]  Alexandre Arenas,et al.  Bond percolation on multiplex networks , 2015, ArXiv.

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

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

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

[30]  Ming Li,et al.  Robustness of networks with assortative dependence groups , 2018, Physica A: Statistical Mechanics and its Applications.

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

[32]  Nicholas Jenkins,et al.  Embedded generation. Part 1 , 1995 .

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

[34]  H. Stanley,et al.  Extreme risk induced by communities in interdependent networks , 2019, Communications Physics.

[35]  Ming Li,et al.  Cascading failures in coupled networks: The critical role of node-coupling strength across networks , 2016, Scientific reports.

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

[37]  Lixin Tian,et al.  Resilience of networks with community structure behaves as if under an external field , 2018, Proceedings of the National Academy of Sciences.

[38]  Chuansheng Shen,et al.  Double phase transition of the Ising model in core–periphery networks , 2018, Journal of Statistical Mechanics: Theory and Experiment.

[39]  Xiao Ma,et al.  Lower bound of network dismantling problem. , 2018, Chaos.

[40]  Shlomo Havlin,et al.  Local structure can identify and quantify influential global spreaders in large scale social networks , 2015, Proceedings of the National Academy of Sciences.

[41]  Ling Feng,et al.  The simplified self-consistent probabilities method for percolation and its application to interdependent networks , 2015 .

[42]  Dong Zhou,et al.  Group percolation in interdependent networks , 2018, Physical review. E.