1. 1. Spatial patterns in evolutionary games on scale-free networks and multiplexes Kaj Kolja Kleineberg | kkleineberg@ethz.ch @KoljaKleineberg | koljakleineberg.wordpress.com
2. 2. Evolutionary games on structured populations: It's complicated!
3. 3. Evolutionary games on structured populations: It's complicated! Does heterogeneity always favor cooperation? Spatial effects in scale-free, clustered networks?
4. 4. Real complex networks are scale-free and clustered Clustering implies an underlying geometry
5. 5. Scale-free clustered networks can be embedded into hyperbolic space Hyperbolic geometry of complex networks [PRE 82, 036106] Distribute: (r) e 1 2 (1)r Connect: p(xij) = 1 1 + e xijR 2T xij = cosh1 (cosh ri cosh rj sinh ri sinh rj cos ij)
6. 6. Scale-free clustered networks can be embedded into hyperbolic space Hyperbolic geometry of complex networks [PRE 82, 036106] Distribute: (r) e 1 2 (1)r Connect: p(xij) = 1 1 + e xijR 2T xij = cosh1 (cosh ri cosh rj sinh ri sinh rj cos ij)
7. 7. Scale-free clustered networks can be embedded into hyperbolic space Hyperbolic geometry of complex networks [PRE 82, 036106] Distribute: (r) e 1 2 (1)r Connect: p(xij) = 1 1 + e xijR 2T xij = cosh1 (cosh ri cosh rj sinh ri sinh rj cos ij) Real networks can be embedded into hyperbolic space by inverting the model.
8. 8. Hyperbolic maps of complex networks: Poincar disk Nature Communications 1, 62 (2010) Polar coordinates: ri : Popularity (degree) i : Similarity Distance: xij = cosh1 (cosh ri cosh rj sinh ri sinh rj cos ij) Connection probability: p(xij) = 1 1 + e xijR 2T
9. 9. Hyperbolic maps of complex networks: Poincar disk Internet IPv6 topology Polar coordinates: ri : Popularity (degree) i : Similarity Distance: xij = cosh1 (cosh ri cosh rj sinh ri sinh rj cos ij) Connection probability: p(xij) = 1 1 + e xijR 2T
10. 10. Hyperbolic maps of complex networks: Poincar disk Internet IPv6 topology Polar coordinates: ri : Popularity (degree) i : Similarity Distance: xij = cosh1 (cosh ri cosh rj sinh ri sinh rj cos ij) Connection probability: p(xij) = 1 1 + e xijR 2T
11. 11. Hyperbolic maps of complex networks: Poincar disk Internet IPv6 topology Polar coordinates: ri : Popularity (degree) i : Similarity Distance: xij = cosh1 (cosh ri cosh rj sinh ri sinh rj cos ij) Connection probability: p(xij) = 1 1 + e xijR 2T
12. 12. Temperature parameters related clustering and the strength of the metric space A: Low temperature (high mean local clustering, c). B: High temperature (low c).
13. 13. Individuals collect a payoff form playing with their neighbors and update their strategy by imitation
14. 14. Self-organization into metric clusters allows cooperators to survive in social dilemmas A B DC E F HG A B C t Prisoners dilemma, T = 1.2, S = 0.2
15. 15. We can use the initial conditions as a proxy of the effectiveness of different structures Lack of analytical solution Random initial conditions may not reveal all possible solutions (no ergodicity)
16. 16. We can use the initial conditions as a proxy of the effectiveness of different structures Lack of analytical solution Random initial conditions may not reveal all possible solutions (no ergodicity) Random Hubs Connected cluster Metric cluster FullgraphCooperatorsubgraph
17. 17. Metric clusters can be better in sustaining cooperation than hubs and heterogeneity can even hinder cooperation /connected cluster Prisoner's Dilemma, T=1.5, S=-0.5
18. 18. Metric clusters can be better in sustaining cooperation than hubs and heterogeneity can even hinder cooperation /connected cluster Prisoner's Dilemma, T=1.5, S=-0.5 Heterogeneity does not always favorbut can even hindercooperation in social dilemmas.
19. 19. Metric clusters or hubs can be more efficient in sustaining cooperation depending on network topology
20. 20. Abundance of intercluster links explains why and when metric clusters are successful Intercluster links Connected cluster Metric cluster
21. 21. Abundance of intercluster links explains why and when metric clusters are successful Intercluster links Connected cluster Metric cluster Metric clusters shield cooperators from surrounding defectors similar to spatial selection.
22. 22. Metric clusters as initial conditions might even be more realistic than random ones Nature Communications 1, 62 (2010)
23. 23. Formation of metric clusters in the dynamical navigation game Cooperator Defector Message is sent Message is lost SuccessFailure Source Target Sci. Rep. 7, 2897 (2017)
24. 24. Formation of metric clusters in the dynamical navigation game Cooperator Defector Message is sent Message is lost SuccessFailure Source Target Sci. Rep. 7, 2897 (2017)
25. 25. Formation of metric clusters in collective intelligence with minority incentives Model from PNAS 114, 20:50775082
26. 26. Human interactions take place in different domains: Multiplex networks
27. 27. Radial and angular coordinates are correlated between different layers in many real multiplexes Arx12 Arx42 Arx41 Arx28 Phys12 Arx52 Arx15 Arx26 Internet Arx34 CE23 Phys13 Phys23 Sac13 Sac35 Sac23 Sac12 Dro12 CE13 Sac14 Sac24 Brain Rattus CE12 Sac34 AirTrain 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 Angular correlations (g) Radialcorrelations() Model: - Tune correlations independently from constituent layer topologies - Similarity (angular) correlations: g [0, 1] - Degree (radial) correlations: [0, 1] [Nature Physics 12, 10761081 (2016)]
28. 28. Geometric correlations can lead to the formation of coherent patterns among different layers GN ON +T+S C D Layer 1: Evolutionary games Stag Hunt, Prisoners Dilemma & imitation dynamics Layer 2: Social influence Voter model & bias towards cooperation Coupling: at each timestep, with probability (1 ) perform respective dynamics in each layer nodes copy their state from one layer to the other
29. 29. Self-organization into clusters of cooperators only occurs if angular correlations are present
30. 30. Overlapping clusters of cooperators also happen in the mutual prisoner's dilemma 2 1 1 2 1 2 1 2 a) b) c) d) Both layers play prisoners dilemma with the same coupling as before.
31. 31. Summary: metric clusters in evolutionary games on scale-free networks - Cooperation can be sustained in metric clusters in scale-free networks - Metric clusters shield cooperators from surrounding defectors (similar to spatial selection) - Survival of metric clusters is favored if: - The network is less heterogeneous - The network has a higher clustering coefficient (lower temperature, stronger metric structure) - The clusters (networks) are larger - If started with metric clusters, heterogeneity can even hinder cooperation - We find similar clusters for different games and on correlated multiplexes
32. 32. Reference: Metric clusters in evolutionary games on scale-free networks arXiv:1704.00952 K-K. Kleineberg Kaj Kolja Kleineberg: kkleineberg@ethz.ch @KoljaKleineberg koljakleineberg.wordpress.com
33. 33. Reference: Metric clusters in evolutionary games on scale-free networks arXiv:1704.00952 K-K. Kleineberg Kaj Kolja Kleineberg: kkleineberg@ethz.ch @KoljaKleineberg Slides koljakleineberg.wordpress.com
34. 34. Reference: Metric clusters in evolutionary games on scale-free networks arXiv:1704.00952 K-K. Kleineberg Kaj Kolja Kleineberg: kkleineberg@ethz.ch @KoljaKleineberg Slides koljakleineberg.wordpress.com Data & Model