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Towards controlling evolutionary dynamics through network geometry: some very first steps

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Towards controlling evolutionary dynamics through network geometry: some very first steps

  1. 1. Towards controlling evolutionary dynamics through network geometry: some very first steps Kaj Kolja Kleineberg | kkleineberg@ethz.ch @KoljaKleineberg | koljakleineberg.wordpress.com
  2. 2. The puzzle of cooperation
  3. 3. Tragedy of the commons
  4. 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. 5. Individuals collect a payoff from playing with their neighbors and update their strategy by imitation
  6. 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. 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. 8. Games are played on networks of contacts: »Structured populations«
  9. 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. 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. 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. 12. Hidden metric spaces
  13. 13. Hidden metric spaces underlying real complex networks provide a fundamental explanation of their observed topologies Nature Physics 5, 74–80 (2008)
  14. 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. 15. Hyperbolic geometry emerges from Newtonian model and similarity × popularity optimization in growing networks S1 p(κ) ∝ κ−γ
  16. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 26. Metric spaces underlying scale-free and clustered networks A: Low temperature (high mean local clustering, ¯c). B: High temperature (low ¯c).
  27. 27. Evolutionary games
  28. 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. 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. 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. 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. 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. 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. 34. Abundance of intercluster links explains how cooperators can be shielded from defectors Intercluster links Connected cluster Metric cluster
  35. 35. Navigation
  36. 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. 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. 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. 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. 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. 41. Individuals can choose to forward messages or deny participation in the process Cooperator Defector Message is sent Message is lost SuccessFailure
  42. 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. 43. System self-organizes into local clusters of cooperators prior to the emergence of global cooperation
  44. 44. Distributing the initial cooperators into local clusters favors significantly the emergence of cooperation
  45. 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. 46. Take home
  47. 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. 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. 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. 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. 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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