Percolation on Networks with Conditional Dependence Group

Recently, the dependence group has been proposed to study the robustness of networks with interdependent nodes. A dependence group means that a failed node in the group can lead to the failures of the whole group. Considering the situation of real networks that one failed node may not always break the functionality of a dependence group, we study a cascading failure model that a dependence group fails only when more than a fraction β of nodes of the group fail. We find that the network becomes more robust with the increasing of the parameter β. However, the type of percolation transition is always first order unless the model reduces to the classical network percolation model, which is independent of the degree distribution of the network. Furthermore, we find that a larger dependence group size does not always make the networks more fragile. We also present exact solutions to the size of the giant component and the critical point, which are in agreement with the simulations well.

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

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

[3]  Bing-Hong Wang,et al.  Cascading failures on networks with asymmetric dependence , 2014 .

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

[5]  Bing-Hong Wang,et al.  Critical effects of overlapping of connectivity and dependence links on percolation of networks , 2013 .

[6]  Amir Bashan,et al.  The Combined Effect of Connectivity and Dependency Links on Percolation of Networks , 2011, ArXiv.

[7]  Sergey N. Dorogovtsev,et al.  Critical phenomena in complex networks , 2007, ArXiv.

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

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

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

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

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

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

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

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

[16]  P. Lambin,et al.  Optical simulations of electron diffraction by carbon nanotubes , 2002 .

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

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

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

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

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

[22]  Peter Grassberger,et al.  Percolation transitions are not always sharpened by making networks interdependent. , 2011, Physical review letters.

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