Layer degradation triggers an abrupt structural transition in multiplex networks.

Network robustness is a central point in network science, both from a theoretical and a practical point of view. In this paper, we show that layer degradation, understood as the continuous or discrete loss of links' weight, triggers a structural transition revealed by an abrupt change in the algebraic connectivity of the graph. Unlike traditional single layer networks, multiplex networks exist in two phases, one in which the system is protected from link failures in some of its layers and one in which all the system senses the failure happening in one single layer. We also give the exact critical value of the weight of the intralayer links at which the transition occurs for continuous layer degradation and its relation with the value of the coupling between layers. This relation allows us to reveal the connection between the transition observed under layer degradation and the one observed under the variation of the coupling between layers.

[1]  Yamir Moreno,et al.  Contact-based Social Contagion in Multiplex Networks , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Yamir Moreno,et al.  A polynomial eigenvalue approach for multiplex networks , 2018, New Journal of Physics.

[3]  Piet Van Mieghem,et al.  On the Robustness of Complex Networks by Using the Algebraic Connectivity , 2008, Networking.

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

[5]  Piet Van Mieghem,et al.  Exact Coupling Threshold for Structural Transition in Interconnected Networks , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  D. Selkoe.  Alzheimer's Disease Is a Synaptic Failure , 2002, Science.

[7]  Moody T. Chu,et al.  Inverse Eigenvalue Problems , 1998, SIAM Rev..

[8]  Jie Sun,et al.  Approximating spectral impact of structural perturbations in large networks. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Huijuan Wang,et al.  Algebraic Connectivity of Interdependent Networks , 2014 .

[10]  Ziyou Gao,et al.  Cascading failures on weighted urban traffic equilibrium networks , 2007 .

[11]  Yamir Moreno,et al.  Multiplex Networks: Basic Formalism and Structural Properties , 2018 .

[12]  Yamir Moreno,et al.  Dimensionality reduction and spectral properties of multiplex networks , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  Mason A. Porter,et al.  Structure of triadic relations in multiplex networks , 2013, ArXiv.

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

[15]  David Lusseau,et al.  Formalising the multidimensional nature of social networks , 2011, ArXiv.

[16]  Massimiliano Zanin,et al.  Modeling the multi-layer nature of the European Air Transport Network: Resilience and passengers re-scheduling under random failures , 2012, ArXiv.

[17]  M. Fiedler.  Algebraic connectivity of graphs , 1973 .

[18]  Caterina M. Scoglio,et al.  Optimizing algebraic connectivity by edge rewiring , 2013, Appl. Math. Comput..

[19]  Y. Moreno,et al.  Diffusion Dynamics and Optimal Coupling in Multiplex Networks with Directed Layers , 2018, Physical Review X.

[20]  Y. Moreno,et al.  Diffusion Dynamics and Optimal Coupling in Directed Multiplex Networks , 2017, 1708.01951.

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

[22]  Jane Memmott,et al.  Tolerance of pollination networks to species extinctions , 2004, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[23]  Ernesto Estrada,et al.  Network robustness to targeted attacks. The interplay of expansibility and degree distribution , 2006 .

[24]  Ernesto Estrada,et al.  Communicability reveals a transition to coordinated behavior in multiplex networks , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Darren M. Scott,et al.  Network Robustness Index : a new method for identifying critical links and evaluating the performance of transportation networks , 2006 .

[26]  J. A. Almendral,et al.  Dynamical and spectral properties of complex networks , 2007, 0705.3216.

[27]  Y. Moreno,et al.  Characterization of multiple topological scales in multiplex networks through supra-Laplacian eigengaps. , 2016, Physical review. E.

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