Interplay between social influence and competitive strategical games in multiplex networks
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The document discusses the interplay between social influence and competitive strategic games on multiplex networks. It shows that an opinion dynamics model with pro-cooperation bias can transform a prisoner's dilemma game into a snowdrift game. Considering multiplex topology is important, as correlations between network layers can have an even bigger impact on cooperation than individual layer topologies alone. When similarity correlations are present between layers, cooperative clusters can form across both layers through self-organization.
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Interplay between social influence and competitive strategical games in multiplex networks
- 1. Interplay between social influence and competitive strategical games in multiplex networks Kaj Kolja Kleineberg | kkleineberg@ethz.ch @KoljaKleineberg | koljakleineberg.wordpress.com
- 2. Topologies Do they matter? From simple to complex to multiplex
- 3. Evolutionary games on networks: from simple to complex topologies Topology Scale-free & clusteredNone Fixed neighbors, dynamical correlations Static networks Complex networks Multiplex networks No static neighbors, no dynamical correlations Replicator equation (or other dynamics) MechanismInsight Grid/"simple" networks "Spatial selection" "Network reciprocity" Hubs play special role, clustering leads to patterns Game dynamics Everything so far + more complex dynamics + multiplex topology Multiple layers Heterogeneity can favor cooperation Very rich behavior, sometime cooperation is favored, sometimes not Am. Math. Soc. 40, 479-519 Nature 359, 826–829 PRL 95, 098104 Nat. Comm. 8, 1888 New J. Phys. 19 073017 New J. Phys. 20 053030 Sci. Rep. 2, 620 Physics Reports 687, 1-51 Game dynamics
- 4. Evolutionary games on networks: from simple to complex topologies Topology Scale-free & clusteredNone Fixed neighbors, dynamical correlations Static networks Complex networks Multiplex networks No static neighbors, no dynamical correlations Replicator equation (or other dynamics) MechanismInsight Grid/"simple" networks "Spatial selection" "Network reciprocity" Hubs play special role, clustering leads to patterns Game dynamics Everything so far + more complex dynamics + multiplex topology Multiple layers Heterogeneity can favor cooperation Very rich behavior, sometime cooperation is favored, sometimes not Am. Math. Soc. 40, 479-519 Nature 359, 826–829 PRL 95, 098104 Nat. Comm. 8, 1888 New J. Phys. 19 073017 New J. Phys. 20 053030 Sci. Rep. 2, 620 Physics Reports 687, 1-51 Static networks
- 5. Evolutionary games on networks: from simple to complex topologies Topology Scale-free & clusteredNone Fixed neighbors, dynamical correlations Static networks Complex networks Multiplex networks No static neighbors, no dynamical correlations Replicator equation (or other dynamics) MechanismInsight Grid/"simple" networks "Spatial selection" "Network reciprocity" Hubs play special role, clustering leads to patterns Game dynamics Everything so far + more complex dynamics + multiplex topology Multiple layers Heterogeneity can favor cooperation Very rich behavior, sometime cooperation is favored, sometimes not Am. Math. Soc. 40, 479-519 Nature 359, 826–829 PRL 95, 098104 Nat. Comm. 8, 1888 New J. Phys. 19 073017 New J. Phys. 20 053030 Sci. Rep. 2, 620 Physics Reports 687, 1-51 Complex networks
- 6. Evolutionary games on networks: from simple to complex topologies Topology Scale-free & clusteredNone Fixed neighbors, dynamical correlations Static networks Complex networks Multiplex networks No static neighbors, no dynamical correlations Replicator equation (or other dynamics) MechanismInsight Grid/"simple" networks "Spatial selection" "Network reciprocity" Hubs play special role, clustering leads to patterns Game dynamics Everything so far + more complex dynamics + multiplex topology Multiple layers Heterogeneity can favor cooperation Very rich behavior, sometime cooperation is favored, sometimes not Am. Math. Soc. 40, 479-519 Nature 359, 826–829 PRL 95, 098104 Nat. Comm. 8, 1888 New J. Phys. 19 073017 New J. Phys. 20 053030 Sci. Rep. 2, 620 Physics Reports 687, 1-51 Multiplex networks
- 8. Multiplex: nodes are simultaneously present in different network layers Several networking layers
- 9. Multiplex: nodes are simultaneously present in different network layers Several networking layers Same nodes exist in different layers
- 10. Multiplex: nodes are simultaneously present in different network layers Several networking layers Same nodes exist in different layers One-to-one mapping between nodes in different layers
- 11. Human interactions take place in different domains that can be abstracted as different layers of networks - Human interactions take place in many domains. - Which ones are significant? - Layers have different meaning, i.e. different dynamics
- 12. Human interactions take place in different domains that can be abstracted as different layers of networks - Human interactions take place in many domains. - Which ones are significant? - Layers have different meaning, i.e. different dynamics - Layer 1: Evolutionary games Stag Hunt, Prisoner’s Dilemma & imitation dynamics - Layer 2: Social influence Voter model & bias towards cooperation
- 14. Individuals play strategical games with their neighbors and tend to imitate more successful players
- 15. Social influence
- 16. Human interactions take place in different domains that can be abstracted as different layers of networks - Individuals have opinions “cooperate” or “defect”, which can be interpreted as proclamations of intend in the game - Bias voter model, bias β - Individuals copy opinion of a randomly chosen neighbor with probability β if the opinion is “cooperate” and with 1 − β if it is “defect”
- 18. Putting it all together: interplay between social influence and evolutionary game dynamics γ β GN ON +T+S C D Layer 1: Evolutionary games Stag Hunt, Prisoner’s Dilemma & imitation dynamics Layer 2: Social influence Voter model & bias towards cooperation Coupling: at each timestep, with probability (1 − γ) perform respective dynamics in each layer γ nodes copy their state from one layer to the other
- 19. Meanfield equations: interplay between social influence and evolutionary game dynamics Meanfield solution: ∂tcI =(1 − γ)cI(1 − cI) tanh [⟨k⟩ (cI(1 − T) + S(1 − cI))] + γ(cII − cI) ∂tcII =(1 − γ)(2β − 1)cII(1 − cII) + γ(cI − cII) (1) cI : Density of cooperators in the game layer cII : Density of individuals with the “cooperate” attitude in the opinion layer γ: Coupling strength Note that Pi←j = 1 2 (1 − tanh [πi − πj]) is just another way of writing the Fermi Dirac probability
- 20. Opinion dynamics with pro-cooperation bias can transform prisoner's dilemma into a snowdrift game γ = 0.2, β = 0.7, and ⟨k⟩ = 6.
- 21. Multiplex topology with complex individual layers and two types of correlations Correlated Uncorrelated Real mul�plexes Model from [1] [1] Nat. Phys. 12, 1076–1081 (2016) Individual layer topologies: tune and clustering Mul�plex: tune similarity correla�ons and degree correla�ons Implemena�on at: koljakleineberg.wordpress.com/materials/
- 22. Layer topology and multiplexity can increase or decrease cooperation a) b) c) d) e) f) g) h) Erdős Rényi layers no correlations Scale-free & clustered no correlations Scale-free & clustered with correlations a) b) c) e) f) g)
- 24. Pattern formation of evolutionary games on heterogeneous networks - Multiplex model used here is based on latent metric spaces - Cooperators can form clusters in the latent space similar to “spatial selection” [Nat. Com. 8, 1888 (2017)] - Especially similarity correlations “align” the metric spaces of different layers in the multiplex
- 25. Pattern formation of evolutionary games on heterogeneous networks - Multiplex model used here is based on latent metric spaces - Cooperators can form clusters in the latent space similar to “spatial selection” [Nat. Com. 8, 1888 (2017)] - Especially similarity correlations “align” the metric spaces of different layers in the multiplex Emergence of overlapping cooperating clusters in both layers becomes visible in the metric space.
- 26. Self-organization into clusters of cooperators only occurs if angular correlations are present
- 27. Take home
- 28. Interplay between games and influence reveals new region and shows the importance of multiplex topology - Opinion dynamics with pro-cooperation bias can transform prisoner’s dilemma into a snowdrift game
- 29. Interplay between games and influence reveals new region and shows the importance of multiplex topology - Opinion dynamics with pro-cooperation bias can transform prisoner’s dilemma into a snowdrift game - We found a new mixed state (bistability of snowdrift and harmony solution)
- 30. Interplay between games and influence reveals new region and shows the importance of multiplex topology - Opinion dynamics with pro-cooperation bias can transform prisoner’s dilemma into a snowdrift game - We found a new mixed state (bistability of snowdrift and harmony solution) - Multiplex topology (i.e. correlations) can have an even bigger impact than individual layer topologies → it is important to consider such correlations
- 31. Interplay between games and influence reveals new region and shows the importance of multiplex topology - Opinion dynamics with pro-cooperation bias can transform prisoner’s dilemma into a snowdrift game - We found a new mixed state (bistability of snowdrift and harmony solution) - Multiplex topology (i.e. correlations) can have an even bigger impact than individual layer topologies → it is important to consider such correlations - Formation of mutual patterns across layers (if “similarity” correlations exist)
- 32. Reference: Interplay between social influence and competitive strategical games in multiplex networks Roberta Amato, Albert Díaz-Guilera Kaj-Kolja Kleineberg Scientific Reports 7, 7087 (2017) Contact: kkleineberg@ethz.ch @KoljaKleineberg Multiplex model: koljakleineberg.wordpress.com/materials/ Nature Physics 12, 1076–1081 (2016) Pattern formation: Nat. Com. 8, 1888 (2017)