%0 Generic %1 Guha2014Layered %A Guha, Saikat %A Towsley, Donald %A Capar, Cagatay %A Swami, Ananthram %A Basu, Prithwish %D 2014 %K preprint percolation multiplex-networks bond-percolation %T Layered Percolation %U http://arxiv.org/abs/1402.7057 %X We study the emergence of long-range connectivity in multilayer networks (also termed multiplex, composite and overlay networks) obtained by merging the connectivity subgraphs of multiple percolating instances of an underlying backbone network. Multilayer networks have applications ranging from studying long-range connectivity in a communication or social network formed with hybrid technologies, a transportation network connecting the same cities via rail, road and air, in studying outbreaks of flu epidemics involving multiple viral strains, studying temporal flow of information in dynamic networks, and potentially in studying conductivity properties of graphene-like stacked lattices. For a homogenous multilayer network---formed via merging \$M\$ random site-percolating instances of the same graph \$G\$ with single-layer site-occupation probability \$q\$---we argue that when \$q\$ exceeds a threshold \$q\_c(M) = \Theta(1/M)\$, a spanning cluster appears in the multilayer network. Using a configuration model approach, we find \$q\_c(M)\$ exactly for random graphs with arbitrary degree distributions, which have many applications in mathematical sociology. For multilayer percolation in a general graph \$G\$, we show that \$q\_c/M < q\_c(M) < -łn(1-p\_c)/M, M Z^+\$, where \$q\_c\$ and \$p\_c\$ are the site and bond percolation thresholds of \$G\$, respectively. We show a close connection between multilayer percolation and mixed (site-bond) percolation, since both provide a smooth bridge between pure-site and pure-bond percolations. We find excellent approximations and bounds on layered percolation thresholds for regular lattices using the aforesaid connection, and provide several exact results (via numerical simulations), and a specialized bound for the multilayer kagome lattice using a site-to-bond transformation technique.

@misc{Guha2014Layered, abstract = {We study the emergence of long-range connectivity in multilayer networks (also termed multiplex, composite and overlay networks) obtained by merging the connectivity subgraphs of multiple percolating instances of an underlying backbone network. Multilayer networks have applications ranging from studying long-range connectivity in a communication or social network formed with hybrid technologies, a transportation network connecting the same cities via rail, road and air, in studying outbreaks of flu epidemics involving multiple viral strains, studying temporal flow of information in dynamic networks, and potentially in studying conductivity properties of graphene-like stacked lattices. For a homogenous multilayer network---formed via merging \$M\$ random site-percolating instances of the same graph \$G\$ with single-layer site-occupation probability \$q\$---we argue that when \$q\$ exceeds a threshold \$q\_c(M) = \Theta(1/\sqrt{M})\$, a spanning cluster appears in the multilayer network. Using a configuration model approach, we find \$q\_c(M)\$ exactly for random graphs with arbitrary degree distributions, which have many applications in mathematical sociology. For multilayer percolation in a general graph \$G\$, we show that \$q\_c/\sqrt{M} \< q\_c(M) \< \sqrt{-\ln(1-p\_c)}/{\sqrt{M}}, \forall M \in {\mathbb Z}^+\$, where \$q\_c\$ and \$p\_c\$ are the site and bond percolation thresholds of \$G\$, respectively. We show a close connection between multilayer percolation and mixed (site-bond) percolation, since both provide a smooth bridge between pure-site and pure-bond percolations. We find excellent approximations and bounds on layered percolation thresholds for regular lattices using the aforesaid connection, and provide several exact results (via numerical simulations), and a specialized bound for the multilayer kagome lattice using a site-to-bond transformation technique.}, added-at = {2019-06-10T14:53:09.000+0200}, archiveprefix = {arXiv}, author = {Guha, Saikat and Towsley, Donald and Capar, Cagatay and Swami, Ananthram and Basu, Prithwish}, biburl = {https://www.bibsonomy.org/bibtex/20e324fafde932b3b18d69638c454a062/nonancourt}, citeulike-article-id = {13074854}, citeulike-linkout-0 = {http://arxiv.org/abs/1402.7057}, citeulike-linkout-1 = {http://arxiv.org/pdf/1402.7057}, day = 27, eprint = {1402.7057}, interhash = {40ca25d339031247c4c959f890cfc5d4}, intrahash = {0e324fafde932b3b18d69638c454a062}, keywords = {preprint percolation multiplex-networks bond-percolation}, month = feb, posted-at = {2014-02-28 11:23:56}, priority = {2}, timestamp = {2019-08-23T11:01:11.000+0200}, title = {{Layered Percolation}}, url = {http://arxiv.org/abs/1402.7057}, year = 2014 }