1. Spanning connectivity in a multilayer network and its relationship to site-bond percolation
- Author
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Don Towsley, Philippe Nain, Cagatay Capar, Saikat Guha, Ananthram Swami, Prithwish Basu, Raytheon BBN Technologies, Department of Computer Science [Amherst], University of Massachusetts System (UMASS), Dynamic Networks : Temporal and Structural Capture Approach (DANTE), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de l'Informatique du Parallélisme (LIP), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Institut Rhône-Alpin des systèmes complexes (IXXI), École normale supérieure de Lyon (ENS de Lyon)-Université Lumière - Lyon 2 (UL2)-Université Jean Moulin - Lyon 3 (UJML), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Université Lumière - Lyon 2 (UL2)-Université Jean Moulin - Lyon 3 (UJML), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Ericsson Research, U.S. Army Research Laboratory [Adelphi, MD] (ARL), United States Army (U.S. Army), Centre National de la Recherche Scientifique (CNRS)-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS)-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-École normale supérieure - Lyon (ENS Lyon)-Institut Rhône-Alpin des systèmes complexes (IXXI), École normale supérieure - Lyon (ENS Lyon)-Université Lumière - Lyon 2 (UL2)-Université Jean Moulin - Lyon 3 (UJML), Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Université Lumière - Lyon 2 (UL2)-Université Jean Moulin - Lyon 3 (UJML), and Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])
- Subjects
Physics::Physics and Society ,Random graph ,Discrete mathematics ,Connected component ,Statistical Mechanics (cond-mat.stat-mech) ,FOS: Physical sciences ,multilayer graph ,Percolation threshold ,Condensed Matter::Disordered Systems and Neural Networks ,01 natural sciences ,Upper and lower bounds ,010305 fluids & plasmas ,Combinatorics ,percolation ,[INFO.INFO-NI]Computer Science [cs]/Networking and Internet Architecture [cs.NI] ,Distribution (mathematics) ,networks ,Percolation ,Lattice (order) ,0103 physical sciences ,Condensed Matter::Statistical Mechanics ,Connection (algebraic framework) ,010306 general physics ,Condensed Matter - Statistical Mechanics ,Mathematics - Abstract
We analyze the connectivity of an $M$-layer network over a common set of nodes that are active only in a fraction of the layers. Each layer is assumed to be a subgraph (of an underlying connectivity graph $G$) induced by each node being active in any given layer with probability $q$. The $M$-layer network is formed by aggregating the edges over all $M$ layers. We show that when $q$ exceeds a threshold $q_c(M)$, a giant connected component appears in the $M$-layer network---thereby enabling far-away users to connect using `bridge' nodes that are active in multiple network layers---even though the individual layers may only have small disconnected islands of connectivity. We show that $q_c(M) \lesssim \sqrt{-\ln(1-p_c)}\,/{\sqrt{M}}$, where $p_c$ is the bond percolation threshold of $G$, and $q_c(1) \equiv q_c$ is its site percolation threshold. We find $q_c(M)$ exactly for when $G$ is a large random network with an arbitrary node-degree distribution. We find $q_c(M)$ numerically for various regular lattices, and find an exact lower bound for the kagome lattice. Finally, we find an intriguingly close connection between this multilayer percolation model and the well-studied problem of site-bond percolation, in the sense that both models provide a smooth transition between the traditional site and bond percolation models. Using this connection, we translate known analytical approximations of the site-bond critical region, which are functions only of $p_c$ and $q_c$ of the respective lattice, to excellent general approximations of the multilayer connectivity threshold $q_c(M)$., Comment: 17 pages, 12 figures, to appear in Physical Review E
- Published
- 2016
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