Competition in two-layer multiplex networks Sergio Gmez - PowerPoint PPT Presentation

Competition in two-layer multiplex networks Sergio Gómez Universitat Rovira i Virgili, Tarragona (Spain) CCS / DOOCN-XI 2018 Thessaloniki

Competition in two-layer multiplex networks Outline  Motivation  Competition dynamics  Ground State  Optimization  Results 2

Competition in two-layer multiplex networks  Structure  Multiplex network 3 3

Competition in two-layer multiplex networks 4 4

Competition in two-layer multiplex networks 5 5

Competition in two-layer multiplex networks 6 6

Competition in two-layer multiplex networks 7 7

Competition in two-layer multiplex networks 8 8

Competition in two-layer multiplex networks  Dynamics  Individuals can choose between several alternatives 9 9

Competition in two-layer multiplex networks  Dynamics  Individuals can choose between several alternatives  Using several alternatives at once has a cost 10 10

Competition in two-layer multiplex networks  Dynamics  Individuals can choose between several alternatives  Using several alternatives at once has a cost  Sharing an alternative with peers is beneficial 11 11

Competition in two-layer multiplex networks  Dynamics  Individuals can choose between several alternatives  Using several alternatives at once has a cost  Sharing an alternative with peers is beneficial 12 12

Competition in two-layer multiplex networks  Dynamics  Individuals can choose between several alternatives  Using several alternatives at once has a cost  Sharing an alternative with peers is beneficial 13 13

Competition in two-layer multiplex networks  Dynamics  Individuals can choose between several alternatives  Using several alternatives at once has a cost  Sharing an alternative with peers is beneficial benefit benefit cost 14 14

Competition in two-layer multiplex networks  Dynamics  Individuals can choose between several alternatives  Using several alternatives at once has a cost  Sharing an alternative with peers is beneficial benefit benefit cost Competition between layers 15 15

Competition in two-layer multiplex networks Structure Interconnected Multiplex multilayer network network 16

Competition in two-layer multiplex networks  Mathematical description  Adjacency (or weights) tensor node i in layer a connects to node j in layer b  De Domenico et al: Mathematical formulation of multilayer networks Physical Review X 3 (2013) 041022 17 17

Competition in two-layer multiplex networks  Mathematical description  Supra-adjacency matrix adjacency (or weights) matrix of layer a  interaction matrix between layers a and b  Solé-Ribalta et al: Spectral properties of the Laplacian of multiplex networks Physical Review E 88 (2013) 032807 18 18

Competition in two-layer multiplex networks  Mathematical description  Supra-adjacency matrix multiplex network adjacency (or weights) matrix of layer a  interaction strength between layers a and b  Solé-Ribalta et al: Spectral properties of the Laplacian of multiplex networks Physical Review E 88 (2013) 032807 19 19

Competition in two-layer multiplex networks  Mathematical description  Supra-adjacency matrix multiplex network adjacency (or weights) matrix of layer a  interaction strength between layers a and b   Hypotheses  All nodes same interlayer strength  No self-loops  Symmetry Solé-Ribalta et al: Spectral properties of the Laplacian of multiplex networks Physical Review E 88 (2013) 032807 20 20

Competition in two-layer multiplex networks  Mathematical description  Two-layer multiplex networks Gómez et al: Diffusion dynamics on multiplex networks Physical Review Letters 110 (2013) 028701 21 21

Competition in two-layer multiplex networks Related dynamics  Diffusion in multiplex networks Laplacian Diffusion time 22

Competition in two-layer multiplex networks  Results  Superdiffusion! Gómez et al: Diffusion dynamics on multiplex networks Physical Review Letters 110 (2013) 028701 23 23

Competition in two-layer multiplex networks  Results  Superdiffusion! Solé-Ribalta et al: Spectral properties of the Laplacian of multiplex networks Physical Review E 88 (2013) 032807 24 24

Competition in two-layer multiplex networks Competition dynamics  Variables probability of node i being active in layer a  25

Competition in two-layer multiplex networks  Competition  Hamiltonian where Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117 26 26

Competition in two-layer multiplex networks  Competition in two-layer multiplex  Variables  Hamiltonian Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117 27 27

Competition in two-layer multiplex networks  Competition in two-layer multiplex Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117 28 28

Competition in two-layer multiplex networks  Competition in two-layer multiplex Minimum value when all p i = 1 Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117 29 29

Competition in two-layer multiplex networks  Competition in two-layer multiplex Minimum value when all p i = 0 Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117 30 30

Competition in two-layer multiplex networks  Competition in two-layer multiplex Minimum value when all p i = 0.5 Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117 31 31

Competition in two-layer multiplex networks  Magnetization  All p i = 1 M = +1 All nodes in first layer  All p i = 0.5 M = 0 All nodes equally in all layers  All p i = 0 M = -1 All nodes in second layer Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117 32 32

Competition in two-layer multiplex networks Ground state  Minimize with the constraints solution inside the hypercube Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117 33

Competition in two-layer multiplex networks Ground state Quadratic programming problem  Minimize with the constraints solution inside the hypercube Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117 34

Competition in two-layer multiplex networks  Gradient  Zero gradient equation  Hessian Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117 35 35

Competition in two-layer multiplex networks  Ground state conditions  If inside and Hessian positive definite is feasible solution  is the ground state   Else  Ground state lies in one side of the hypercube  If Hessian not positive definite NP-hard problem Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117 36 36

Competition in two-layer multiplex networks  Asymptotic limits  When  When Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117 37 37

Competition in two-layer multiplex networks  Asymptotic limits  When Other terms negligible Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117 38 38

Competition in two-layer multiplex networks  Asymptotic limits  When Minimum value when M = 0 Other terms negligible all p i = 0.5 Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117 39 39

Competition in two-layer multiplex networks  Asymptotic limits  When Other term negligible Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117 40 40

Competition in two-layer multiplex networks  Asymptotic limits  When If s ( 1 ) > s ( 2 ) M = +1 If s ( 1 ) < s ( 2 ) M = -1 Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117 41 41

Competition in two-layer multiplex networks  Asymptotic limits Localized activity in Mixed activity in first layer all layers (supposing s ( 1 ) > s ( 2 ) ) Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117 42 42