The document discusses evolutionary games on scale-free networks and multiplexes. It finds that cooperation can be sustained in metric clusters that form on scale-free networks. These metric clusters shield cooperators from surrounding defectors similar to spatial selection. The survival of metric clusters is favored when the network is less heterogeneous, has a higher clustering coefficient, and the clusters are larger. Similar clusters are also found for different games played on correlated multiplex networks.

1. Spatial patterns in evolutionary games on
scale-free networks and multiplexes
Kaj Kolja Kleineberg | kkleineberg@ethz.ch
@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:
ρ(r) ∝ e
1
2
(γ−1)r
Connect:
p(xij) =
1
1 + e
xij−R
2T
xij = cosh−1
(cosh ri cosh rj
− sinh ri sinh rj cos ∆θij)

6. Scale-free clustered networks
can be embedded into hyperbolic space
“Hyperbolic geometry of complex networks” [PRE 82, 036106]
Distribute:
ρ(r) ∝ e
1
2
(γ−1)r
Connect:
p(xij) =
1
1 + e
xij−R
2T
xij = cosh−1
(cosh ri cosh rj
− sinh ri sinh rj cos ∆θij)

7. Scale-free clustered networks
can be embedded into hyperbolic space
“Hyperbolic geometry of complex networks” [PRE 82, 036106]
Distribute:
ρ(r) ∝ e
1
2
(γ−1)r
Connect:
p(xij) =
1
1 + e
xij−R
2T
xij = cosh−1
(cosh ri cosh rj
− sinh ri sinh rj cos ∆θij)
Real networks can be embedded into hyperbolic
space by inverting the model.

8. Hyperbolic maps of complex networks:
Poincaré disk
Nature Communications 1, 62 (2010)
Polar coordinates:
ri : Popularity (degree)
θi : Similarity
Distance:
xij = cosh−1
(cosh ri cosh rj
− sinh ri sinh rj cos ∆θij)
Connection probability:
p(xij) =
1
1 + e
xij−R
2T

9. Hyperbolic maps of complex networks:
Poincaré disk
Internet IPv6 topology
Polar coordinates:
ri : Popularity (degree)
θi : Similarity
Distance:
xij = cosh−1
(cosh ri cosh rj
− sinh ri sinh rj cos ∆θij)
Connection probability:
p(xij) =
1
1 + e
xij−R
2T

10. Hyperbolic maps of complex networks:
Poincaré disk
Internet IPv6 topology
Polar coordinates:
ri : Popularity (degree)
θi : Similarity
Distance:
xij = cosh−1
(cosh ri cosh rj
− sinh ri sinh rj cos ∆θij)
Connection probability:
p(xij) =
1
1 + e
xij−R
2T

11. Hyperbolic maps of complex networks:
Poincaré disk
Internet IPv6 topology
Polar coordinates:
ri : Popularity (degree)
θi : Similarity
Distance:
xij = cosh−1
(cosh ri cosh rj
− sinh ri sinh rj cos ∆θij)
Connection probability:
p(xij) =
1
1 + e
xij−R
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

15. Self-organization into metric clusters
allows cooperators to survive in social dilemmas
A B DC
E F HG
A B C
t
Prisoner’s dilemma, T = 1.2, S = −0.2

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)

17. 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

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

19. 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 favor—but can even
hinder—cooperation in social dilemmas.

20. Metric clusters or hubs can be more efficient
in sustaining cooperation depending on network topology

21. Abundance of intercluster links explains
why and when metric clusters are successful
Intercluster links
Connected cluster Metric cluster

22. 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.

23. Metric clusters as initial conditions
might even be more realistic than random ones
Nature Communications 1, 62 (2010)

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 the dynamical navigation game
Cooperator
Defector
Message is sent
Message is lost
SuccessFailure
Source
Target
Sci. Rep. 7, 2897 (2017)

26. Formation of metric clusters
in collective intelligence with minority incentives
Model from PNAS 114, 20:5077–5082

27. Human interactions take place in different domains:
Multiplex networks

28. 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(ν)
Model:
- Tune correlations
independently from
constituent layer
topologies
- Similarity (angular)
correlations: g ∈ [0, 1]
- Degree (radial)
correlations: ν ∈ [0, 1]
[Nature Physics 12, 1076–1081
(2016)]

29. Geometric correlations can lead to the formation
of coherent patterns among different layers
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

30. Self-organization into clusters of cooperators
only occurs if angular correlations are present

31. 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 prisoner’s dilemma with the same coupling as before.

32. 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

33. 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

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

35. 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