Competition in two-layer multiplex networks Sergio Gmez - - PowerPoint PPT Presentation
Competition in two-layer multiplex networks Sergio Gmez Universitat Rovira i Virgili, Tarragona (Spain) CCS / DOOCN-XI 2018 Thessaloniki Competition in two-layer multiplex networks Outline Motivation Competition dynamics Ground
CCS / DOOCN-XI 2018 Thessaloniki
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Motivation Competition dynamics Ground State Optimization Results
Competition in two-layer multiplex networks
Structure
Multiplex network
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Competition in two-layer multiplex networks
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Dynamics
Individuals can choose between several alternatives
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Dynamics
Individuals can choose between several alternatives Using several alternatives at once has a cost
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Dynamics
Individuals can choose between several alternatives Using several alternatives at once has a cost Sharing an alternative with peers is beneficial
Competition in two-layer multiplex networks
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Dynamics
Individuals can choose between several alternatives Using several alternatives at once has a cost Sharing an alternative with peers is beneficial
Competition in two-layer multiplex networks
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Dynamics
Individuals can choose between several alternatives Using several alternatives at once has a cost Sharing an alternative with peers is beneficial
Competition in two-layer multiplex networks
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Dynamics
Individuals can choose between several alternatives Using several alternatives at once has a cost Sharing an alternative with peers is beneficial
Competition in two-layer multiplex networks
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Dynamics
Individuals can choose between several alternatives Using several alternatives at once has a cost Sharing an alternative with peers is beneficial
Competition in two-layer multiplex networks
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Competition in two-layer multiplex networks Interconnected multilayer network Multiplex network
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Mathematical description
Adjacency (or weights) tensor
node i in layer a connects to node j in layer b
Competition in two-layer multiplex networks
De Domenico et al: Mathematical formulation of multilayer networks Physical Review X 3 (2013) 041022
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Mathematical description
Supra-adjacency matrix
adjacency (or weights) matrix of layer a
interaction matrix between layers a and b
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Solé-Ribalta et al: Spectral properties of the Laplacian of multiplex networks Physical Review E 88 (2013) 032807
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Mathematical description
Supra-adjacency matrix multiplex network
adjacency (or weights) matrix of layer a
interaction strength between layers a and b
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Solé-Ribalta et al: Spectral properties of the Laplacian of multiplex networks Physical Review E 88 (2013) 032807
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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
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Solé-Ribalta et al: Spectral properties of the Laplacian of multiplex networks Physical Review E 88 (2013) 032807
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Mathematical description
Two-layer multiplex networks
Competition in two-layer multiplex networks
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Gómez et al: Diffusion dynamics on multiplex networks Physical Review Letters 110 (2013) 028701
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Competition in two-layer multiplex networks
Diffusion in multiplex networks
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Results
Superdiffusion!
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Gómez et al: Diffusion dynamics on multiplex networks Physical Review Letters 110 (2013) 028701
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Results
Superdiffusion!
Competition in two-layer multiplex networks
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Solé-Ribalta et al: Spectral properties of the Laplacian of multiplex networks Physical Review E 88 (2013) 032807
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Competition in two-layer multiplex networks
Variables
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Competition
Hamiltonian
where
Competition in two-layer multiplex networks
Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117
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Competition in two-layer multiplex
Variables Hamiltonian
Competition in two-layer multiplex networks
Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117
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Competition in two-layer multiplex
Competition in two-layer multiplex networks
Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117
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Competition in two-layer multiplex
Competition in two-layer multiplex networks
Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117
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Minimum value when all pi = 1
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Competition in two-layer multiplex
Competition in two-layer multiplex networks
Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117
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Minimum value when all pi = 0
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Competition in two-layer multiplex
Competition in two-layer multiplex networks
Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117
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Minimum value when all pi = 0.5
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Magnetization
All pi = 1
M = +1 All nodes in first layer
All pi = 0.5
M = 0 All nodes equally in all layers
All pi = 0
M = -1 All nodes in second layer
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Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117
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Competition in two-layer multiplex networks
Minimize
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
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Competition in two-layer multiplex networks
Minimize
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
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Gradient Zero gradient equation Hessian
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Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117
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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
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Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117
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Asymptotic limits
When When
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Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117
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Asymptotic limits
When
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Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117
Other terms negligible
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Asymptotic limits
When
Competition in two-layer multiplex networks
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Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117
Other terms negligible Minimum value when all pi = 0.5 M = 0
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Asymptotic limits
When
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Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117
Other term negligible
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Asymptotic limits
When
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Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117
If s(1) > s(2) M = +1 If s(1) < s(2) M = -1
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Asymptotic limits
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Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117
Localized activity in first layer (supposing s(1) > s(2) ) Mixed activity in all layers
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Gradient at
If
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Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117
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Gradient at
If below
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Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117
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Solutions diagram pi* = 1 pi* analytic M = +1 M → 0
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Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117
feasible solution
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Solutions diagram pi* = 1 Optimization pi* analytic M = +1 heuristics M → 0
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Gómez-Gardeñes et al: Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117
feasible solution
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NP-complete / NP-hard optimization problems
Many variables Huge search space No known polynomial time algorithms
Algorithms
Local search Collective search Hybrid search
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Local search
Characteristics
One individual moves in the search state Travel guided by local information Short-term memory Try to avoid local optima
Some methods
Gradient descent Simulated annealing Tabu search Extremal optimization
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Local search methods
Gradient descent
Continuous variables Needs the gradient Easily stacked in local minima Add noise or inertia to improve search
Simulated annealing
Inspired by physics at equilibrium Adequate for discrete variables Explore neighbors Allow uphill moves with certain probability (temperature)
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Local search methods
Tabu search
Adequate for discrete variables Explore neighbors Forbid uphill moves in a certain tabu list
Extremal optimization
Inspired by physics out of equilibrium Adequate for discrete variables Explore neighbors Objective function sum of one-variable terms Improve the worst contribution
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Collective search
Characteristics
Several individuals move in the search state Communication between individuals Travel guided by local and global information Long-term memory, swarm intelligence, diversity
Some methods
Evolutionary computation Genetic algorithms Evolution strategies Swarm intelligence Particle swarm optimization Ant colony systems Artificial bee colony
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Collective search methods
Genetic algorithms
Inspired by evolution and natural selection Adequate for discrete binary variables Population of individuals, each with a chromosome Iteration over generations Selection Reproduction (crossover) Mutation Elitism
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Collective search methods
Particle swarm optimization (PSO)
Inspired by bird flocks and fish schools Adequate for continuous variables Set of particles Each particle has position and velocity Each particle remembers its best position Inertia Approach local best Approach global best
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Hybrid search
Collective search + Local optimization Memetic algorithms
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Ground state search
Standard optimization package
METIS, failed
Local search
Simulated annealing, failed
Collective search
Particle swarm optimization, success
selected
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Model of competition between layers Analytically tractable in part of the phase diagram Optimization heuristics required
There is life beyond simulated annealing! Use the most appropriate Try several Check always the natural candidate solutions
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Contact
sergio.gomez@urv.cat http://deim.urv.cat/~sergio.gomez/
References
J Gómez-Gardeñes, M De Domenico, G Gutiérrez, A Arenas, S Gómez
Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117 Competition in two-layer multiplex networks
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CompleNet 2019
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https://complenet.weebly.com
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CompleNet 2019
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Contact
sergio.gomez@urv.cat http://deim.urv.cat/~sergio.gomez/
References
J Gómez-Gardeñes, M De Domenico, G Gutiérrez, A Arenas, S Gómez
Layer-layer competition in multiplex complex networks Philosophical Transactions of the Royal Society A 373 (2015) 20150117 Competition in two-layer multiplex networks