Spatial patterns in evolutionary games on scale-free networks and multiplexes
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Spatial patterns in evolutionary games on scale-free networks and multiplexes Kaj Kolja Kleineberg | [email protected]@KoljaKleineberg | koljakleineberg.wordpress.com
Spatial patterns in evolutionary games on scale-free networks and multiplexes
1. Spatial patterns in evolutionary games on scale-free
networks and multiplexes Kaj Kolja Kleineberg | [email protected]
@KoljaKleineberg | koljakleineberg.wordpress.com
2. Evolutionary games on structured populations: It's
complicated!
3. Evolutionary games on structured populations: It's
complicated! Does heterogeneity always favor cooperation? Spatial
effects in scale-free, clustered networks?
4. Real complex networks are scale-free and clustered
Clustering implies an underlying geometry
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. 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. 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. 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. 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. 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. 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. 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. Individuals collect a payoff form playing with their
neighbors and update their strategy by imitation
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. 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. 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. 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. 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. Metric clusters or hubs can be more efficient in sustaining
cooperation depending on network topology
20. Abundance of intercluster links explains why and when
metric clusters are successful Intercluster links Connected cluster
Metric cluster
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. Metric clusters as initial conditions might even be more
realistic than random ones Nature Communications 1, 62 (2010)
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. 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. Formation of metric clusters in collective intelligence
with minority incentives Model from PNAS 114, 20:50775082
26. Human interactions take place in different domains:
Multiplex networks
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. 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. Self-organization into clusters of cooperators only occurs
if angular correlations are present
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. 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. Reference: Metric clusters in evolutionary games on
scale-free networks arXiv:1704.00952 K-K. Kleineberg Kaj Kolja
Kleineberg: [email protected] @KoljaKleineberg
koljakleineberg.wordpress.com
33. Reference: Metric clusters in evolutionary games on
scale-free networks arXiv:1704.00952 K-K. Kleineberg Kaj Kolja
Kleineberg: [email protected] @KoljaKleineberg Slides
koljakleineberg.wordpress.com
34. Reference: Metric clusters in evolutionary games on
scale-free networks arXiv:1704.00952 K-K. Kleineberg Kaj Kolja
Kleineberg: [email protected] @KoljaKleineberg Slides
koljakleineberg.wordpress.com Data & Model