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- 1. Towards controlling evolutionary dynamics through network geometry: some very first steps Kaj Kolja Kleineberg | kkleineberg@ethz.ch @KoljaKleineberg | koljakleineberg.wordpress.com
- 2. The puzzle of cooperation
- 3. Tragedy of the commons
- 4. Humans tend to imitate more successful individuals rather than behaving fully rational -2 -1 0 1 2 Payoff difference 0,0 0,2 0,4 0,6 0,8 1,0 Probabilitytochangestrategy Cooperation to defection Defection to cooperation Traulsen et al., PNAS 107 (7) 2962-2966, (2010)
- 5. Individuals collect a payoff from playing with their neighbors and update their strategy by imitation
- 6. Result as a function of the payoff parameters for fully mixed populations ˙c = c(1 − c) tanh [⟨k⟩ (c(1 − T) + S(1 − c))]
- 7. Result as a function of the payoff parameters for fully mixed populations ˙c = c(1 − c) tanh [⟨k⟩ (c(1 − T) + S(1 − c))]
- 8. Games are played on networks of contacts: »Structured populations«
- 9. Cooperation in social dilemmas on structured populations: two important papers Spatial selection: Cooperators survive in lattices by forming clusters in Euclidean space, because groups of cooperators are “shielded” from surrounding defectors [Nowak, Nature 1992] Credits: Nowak Scale-free networks: Heterogeneity promotes cooperation in the prisoner’s dilemma [Santos el al, PRL, 2005]
- 10. Common properties of real networks: existence of underlying geometry 1. Heterogeneous: p(k) ∝ k−γ (often scale-free, i.e 2 < γ < 3) 2. Small world: d ∝ ln N 3. Highly clustered (i.e. many triangles compared to random networks)
- 11. Common properties of real networks: existence of underlying geometry 1. Heterogeneous: p(k) ∝ k−γ (often scale-free, i.e 2 < γ < 3) 2. Small world: d ∝ ln N 3. Highly clustered (i.e. many triangles compared to random networks)
- 12. Hidden metric spaces
- 13. Hidden metric spaces underlying real complex networks provide a fundamental explanation of their observed topologies Nature Physics 5, 74–80 (2008)
- 14. Hidden metric spaces underlying real complex networks provide a fundamental explanation of their observed topologies One can infer the coordinates of nodes embedded in metric spaces by inverting models [PRE 92, 022807].
- 15. Hyperbolic geometry emerges from Newtonian model and similarity × popularity optimization in growing networks S1 p(κ) ∝ κ−γ
- 16. Hyperbolic geometry emerges from Newtonian model and similarity × popularity optimization in growing networks S1 p(κ) ∝ κ−γ r = 1 1+ [ d(θ,θ′) µκκ′ ]1/T PRL 100, 078701
- 17. Hyperbolic geometry emerges from Newtonian model and similarity × popularity optimization in growing networks S1 H2 p(κ) ∝ κ−γ ri = R − 2 ln κi κmin r = 1 1+ [ d(θ,θ′) µκκ′ ]1/T PRL 100, 078701
- 18. Hyperbolic geometry emerges from Newtonian model and similarity × popularity optimization in growing networks S1 H2 p(κ) ∝ κ−γ ρ(r) ∝ e 1 2 (γ−1)(r−R) r = 1 1+ [ d(θ,θ′) µκκ′ ]1/T PRL 100, 078701
- 19. Hyperbolic geometry emerges from Newtonian model and similarity × popularity optimization in growing networks S1 H2 p(κ) ∝ κ−γ ρ(r) ∝ e 1 2 (γ−1)(r−R) r = 1 1+ [ d(θ,θ′) µκκ′ ]1/T p(xij) = 1 1+e xij−R 2T PRL 100, 078701 PRE 82, 036106
- 20. Hyperbolic geometry emerges from Newtonian model and similarity × popularity optimization in growing networks S1 H2 growing p(κ) ∝ κ−γ ρ(r) ∝ e 1 2 (γ−1)(r−R) t = 1, 2, 3 . . . r = 1 1+ [ d(θ,θ′) µκκ′ ]1/T p(xij) = 1 1+e xij−R 2T mins∈[1...t−1] s · ∆θst PRL 100, 078701 PRE 82, 036106 Nature 489, 537–540
- 21. 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
- 22. 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
- 23. 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
- 24. 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
- 25. Coordinate inference via maximum likelihood estimation Goal: to infer ri, θi for every node i = 1...N in a given (real) network Approach: Maximum a posteriori probability estimation (MAP) L(αij|{ri, θi}) = ∏ j<i p(xij)αij (1 − p(xij)(1−αij) Inferring ri: ri ≈ R − 2 ln ki Inferring θi: Numerical maximization of the Likelihood L(αij|{ri, θi}) See F. Papadopoulos et al., PRE 92, 022807 (2015)
- 26. Metric spaces underlying scale-free and clustered networks A: Low temperature (high mean local clustering, ¯c). B: High temperature (low ¯c).
- 27. Evolutionary games
- 28. 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
- 29. 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
- 30. 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 Random Hubs Connected cluster Metric cluster FullgraphCooperatorsubgraph
- 31. 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
- 32. 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.
- 33. Metric clusters or hubs can be more efficient in sustaining cooperation depending on network topology M = C D C 1 S D T 0 (1)
- 34. Abundance of intercluster links explains how cooperators can be shielded from defectors Intercluster links Connected cluster Metric cluster
- 35. Navigation
- 36. Greedy routing in network using hyperbolic space allows efficient navigation relying only on local knowledge [Credits: Marian Boguna] Forward message to contact closest to target in metric space Delivery fails if message runs into a loop (define success rate P) Boguñá et al., Nat. Com. 1, 62 (2010)
- 37. Greedy routing in network using hyperbolic space allows efficient navigation relying only on local knowledge [Credits: Marian Boguna] Forward message to contact closest to target in metric space Delivery fails if message runs into a loop (define success rate P) Boguñá et al., Nat. Com. 1, 62 (2010)
- 38. Greedy routing in network using hyperbolic space allows efficient navigation relying only on local knowledge [Credits: Marian Boguna] Forward message to contact closest to target in metric space Delivery fails if message runs into a loop (define success rate P) Boguñá et al., Nat. Com. 1, 62 (2010)
- 39. Greedy routing in network using hyperbolic space allows efficient navigation relying only on local knowledge [Credits: Marian Boguna] Forward message to contact closest to target in metric space Delivery fails if message runs into a loop (define success rate P) Boguñá et al., Nat. Com. 1, 62 (2010)
- 40. Greedy routing in network using hyperbolic space allows efficient navigation relying only on local knowledge [Credits: Marian Boguna] Forward message to contact closest to target in metric space Delivery fails if message runs into a loop (define success rate P) Boguñá et al., Nat. Com. 1, 62 (2010)
- 41. Individuals can choose to forward messages or deny participation in the process Cooperator Defector Message is sent Message is lost SuccessFailure
- 42. Bistability: the system is either highly functional or performance breaks down completely b: Value generated by successful delivery C0: Initial density of cooperators
- 43. System self-organizes into local clusters of cooperators prior to the emergence of global cooperation
- 44. Distributing the initial cooperators into local clusters favors significantly the emergence of cooperation
- 45. Mitigation of required initial cooperation density is observed for different levels of heterogeneity Rand. Clust. 5 10 15 20 25 30 35 0.1 0.3 0.5 0.7 0.9 b C0Threshold γ = 3.1 γ = 2.9 γ = 2.7 γ = 2.5 γ = 2.3 γ = 2.1 Different values of power-law exponent γ
- 46. Take home
- 47. Control of evolutionary dynamics via network geometry: the impact of initial conditions Summary: - Network geometry allows to detect the self-organized formation of cooperating clusters - The final state can be controlled to some extend by how the initial cooperators are placed in space - Example: Pairwise games and navigation game
- 48. Control of evolutionary dynamics via network geometry: the impact of initial conditions Summary: - Network geometry allows to detect the self-organized formation of cooperating clusters - The final state can be controlled to some extend by how the initial cooperators are placed in space - Example: Pairwise games and navigation game Future: - Can we actively control evolutionary dynamics by using underlying geometry? - Can the geometry help us to determine where to strategically place control agents? - Can we develop a more general theory of geometric network control?
- 49. References: »Metric clusters in evolutionary games on scale-free networks«, Nature Communications 8, 1888 (2017), K-K. Kleineberg »Collective navigation of complex networks: Participatory greedy routing«, Scientific Reports 7, 2897 (2017), K-K. Kleineberg, D. Helbing Kaj Kolja Kleineberg: • kkleineberg@ethz.ch • @KoljaKleineberg • koljakleineberg.wordpress.com
- 50. References: »Metric clusters in evolutionary games on scale-free networks«, Nature Communications 8, 1888 (2017), K-K. Kleineberg »Collective navigation of complex networks: Participatory greedy routing«, Scientific Reports 7, 2897 (2017), K-K. Kleineberg, D. Helbing Kaj Kolja Kleineberg: • kkleineberg@ethz.ch • @KoljaKleineberg ← Slides • koljakleineberg.wordpress.com
- 51. References: »Metric clusters in evolutionary games on scale-free networks«, Nature Communications 8, 1888 (2017), K-K. Kleineberg »Collective navigation of complex networks: Participatory greedy routing«, Scientific Reports 7, 2897 (2017), K-K. Kleineberg, D. Helbing Kaj Kolja Kleineberg: • kkleineberg@ethz.ch • @KoljaKleineberg ← Slides • koljakleineberg.wordpress.com ← Data & Model

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