A Unified Approach to Percolation Processes on Multiplex Networks
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Many real complex systems cannot be represented by a single network, but due to multiple sub-systems and types of interactions, must be represented as a multiplex network. This is a set of nodes which exist in several layers, with each layer having its own kind of edges, represented by different colors. An important fundamental structural feature of networks is their resilience to damage, the percolation transition. Generalization of these concepts to multiplex networks requires careful definition of what we mean by connected clusters. We consider two different definitions. One, a rigorous generalization of the single-layer definition leads to a strong non-local rule, and results in a dramatic change in the response of the system to damage. The giant component collapses discontinuously in a hybrid transition characterized by avalanches of diverging mean size. We also consider another definition, which imposes weaker conditions on percolation and allows local calculation, and also leads to different sized giant components depending on whether we consider an activation or pruning process. This ‘weak’ process exhibits both continuous and discontinuous transitions.
KeywordsGiant Component Hybrid Transition Percolation Transition Pruning Process Multiplex Network
This work was partially supported by the FET IP Project MULTIPLEX 317532 and by the FCT projects EXPL/FIS-NAN/1275/2013 and PEst-C/CTM/LA0025/2011, and post-doctoral fellowship SFRH/BPD/74040/2010, Science Foundation Ireland, Grant No. 11/PI/1026 and the FET-Proactive project PLEXMATH (FP7-ICT-2011-8; Grant No. 317614).
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