Collective navigation of complex networks: Participatory greedy routing
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The document discusses participatory greedy routing in the navigation of complex networks, emphasizing the ability to forward messages based on local knowledge and the associated costs and benefits. It outlines a system where agents can defect or cooperate based on the success of message deliveries and the dynamics of imitation among individuals. Additionally, it highlights the importance of self-organization into clusters of cooperators, which can optimize the initial requirement for cooperative agents in the network.
![networkRoad
networkAirtravel
space
Euclidean
Maps:
Maps:
space
Hyperbolic
[Network Science, Barabasi]](https://image.slidesharecdn.com/navgamenovid-170322171116/85/Collective-navigation-of-complex-networks-Participatory-greedy-routing-5-320.jpg)
![networkRoad
networkAirtravel
space
Euclidean
Maps:
Maps:
space
Hyperbolic
[PRE 82, 036106]
[Figures: Network Science, Barabasi]](https://image.slidesharecdn.com/navgamenovid-170322171116/85/Collective-navigation-of-complex-networks-Participatory-greedy-routing-6-320.jpg)
![Maps of scale-free clustered networks
are hyperbolic
“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)](https://image.slidesharecdn.com/navgamenovid-170322171116/85/Collective-navigation-of-complex-networks-Participatory-greedy-routing-7-320.jpg)
![Maps of scale-free clustered networks
are hyperbolic
“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)](https://image.slidesharecdn.com/navgamenovid-170322171116/85/Collective-navigation-of-complex-networks-Participatory-greedy-routing-8-320.jpg)
![Maps of scale-free clustered networks
are hyperbolic
“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.](https://image.slidesharecdn.com/navgamenovid-170322171116/85/Collective-navigation-of-complex-networks-Participatory-greedy-routing-9-320.jpg)
![Inferred maps can be used to navigate the network
relying only on local information (greedy routing)
[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)](https://image.slidesharecdn.com/navgamenovid-170322171116/85/Collective-navigation-of-complex-networks-Participatory-greedy-routing-10-320.jpg)
![Inferred maps can be used to navigate the network
relying only on local information (greedy routing)
[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)](https://image.slidesharecdn.com/navgamenovid-170322171116/85/Collective-navigation-of-complex-networks-Participatory-greedy-routing-11-320.jpg)
![Inferred maps can be used to navigate the network
relying only on local information (greedy routing)
[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)](https://image.slidesharecdn.com/navgamenovid-170322171116/85/Collective-navigation-of-complex-networks-Participatory-greedy-routing-12-320.jpg)
![Inferred maps can be used to navigate the network
relying only on local information (greedy routing)
[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)](https://image.slidesharecdn.com/navgamenovid-170322171116/85/Collective-navigation-of-complex-networks-Participatory-greedy-routing-13-320.jpg)
![Inferred maps can be used to navigate the network
relying only on local information (greedy routing)
[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)](https://image.slidesharecdn.com/navgamenovid-170322171116/85/Collective-navigation-of-complex-networks-Participatory-greedy-routing-14-320.jpg)
Collective navigation of complex networks: Participatory greedy routing
- 1. Collective navigation of complex networks: Participatory greedy routing Kaj Kolja Kleineberg | [email protected] @KoljaKleineberg | koljakleineberg.wordpress.com
- 2. “I read somewhere that on this planet is separated by only six other people. separation. Between us and everybody else on this planet. The president of the United States. A gondolier in Venice. Fill in the names. . . . Six degrees of separation between me and everyone else on this planet. everybody Six degrees of But to find the the right six people..." John Guare, Six Degrees of Separation (1990)
- 4. map we can build a of the system
- 6. networkRoad networkAirtravel space Euclidean Maps: Maps: space Hyperbolic [PRE 82, 036106] [Figures: Network Science, Barabasi]
- 7. Maps of scale-free clustered networks are hyperbolic “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)
- 8. Maps of scale-free clustered networks are hyperbolic “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)
- 9. Maps of scale-free clustered networks are hyperbolic “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.
- 10. Inferred maps can be used to navigate the network relying only on local information (greedy routing) [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)
- 11. Inferred maps can be used to navigate the network relying only on local information (greedy routing) [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)
- 12. Inferred maps can be used to navigate the network relying only on local information (greedy routing) [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)
- 13. Inferred maps can be used to navigate the network relying only on local information (greedy routing) [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)
- 14. Inferred maps can be used to navigate the network relying only on local information (greedy routing) [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)
- 17. Greedy routing requires active participation from agents. What if they don't?
- 18. Game theory: Sending message has a cost Succesul delivery creates value Agents may defect Value is shared
- 19. Individuals obtain a payoff if message is delivered but forwarding has a cost Cooperator Defector Message is sent Message is lost SuccessFailure
- 20. Individuals imitate the behavior of more successful contacts After N message sending events, individuals can update their strategies according to imitation dynamics:
- 21. Individuals imitate the behavior of more successful contacts After N message sending events, individuals can update their strategies according to imitation dynamics: i copies strategy of randomly selected neighbor j with probability pi←j = 1 1 + e−(pj−pi)/K pi,j denotes collected payoffs
- 22. Individuals imitate the behavior of more successful contacts After N message sending events, individuals can update their strategies according to imitation dynamics: i copies strategy of randomly selected neighbor j with probability pi←j = 1 1 + e−(pj−pi)/K pi,j denotes collected payoffs After each update step, we reset the payoffs.
- 23. Bistability: the system is either highly functional or performance breaks down completely b: Value generated by successful delivery C0: Initial density of cooperators
- 24. System self-organizes into local clusters of cooperators prior to the emergence of global cooperation
- 25. Distributing the initial cooperators into local clusters favors significantly the emergence of cooperation
- 26. Heterogeneity favors cooperation in the system in addition to initial localization 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
- 27. Collective navigation of complex networks: Participatory greedy routing Results: - Greedy routing: Forwarding of messages with local knowledge based on underlying metric spaces
- 28. Collective navigation of complex networks: Participatory greedy routing Results: - Greedy routing: Forwarding of messages with local knowledge based on underlying metric spaces - Participatory greedy routing: Sending messages has a cost, but successful deliveries create value (agents can defect)
- 29. Collective navigation of complex networks: Participatory greedy routing Results: - Greedy routing: Forwarding of messages with local knowledge based on underlying metric spaces - Participatory greedy routing: Sending messages has a cost, but successful deliveries create value (agents can defect) - Self-organization into local clusters (visible in underlying metric space)
- 30. Collective navigation of complex networks: Participatory greedy routing Results: - Greedy routing: Forwarding of messages with local knowledge based on underlying metric spaces - Participatory greedy routing: Sending messages has a cost, but successful deliveries create value (agents can defect) - Self-organization into local clusters (visible in underlying metric space) - This can be exploited to lower necessary number of initial cooperators (localization)
- 31. Collective navigation of complex networks: Participatory greedy routing Results: - Greedy routing: Forwarding of messages with local knowledge based on underlying metric spaces - Participatory greedy routing: Sending messages has a cost, but successful deliveries create value (agents can defect) - Self-organization into local clusters (visible in underlying metric space) - This can be exploited to lower necessary number of initial cooperators (localization) Outlook: - Reputation system - Adaptive networks
- 32. Reference: »Collective navigation of complex networks: Participatory greedy routing« arXiv:1611.04395 (2016) K-K. Kleineberg & Dirk Helbing Thanks to: Dirk Helbing Kaj Kolja Kleineberg: • [email protected] • @KoljaKleineberg • koljakleineberg.wordpress.com
- 33. Reference: »Collective navigation of complex networks: Participatory greedy routing« arXiv:1611.04395 (2016) K-K. Kleineberg & Dirk Helbing Thanks to: Dirk Helbing Kaj Kolja Kleineberg: • [email protected] • @KoljaKleineberg ← Slides • koljakleineberg.wordpress.com
- 34. Reference: »Collective navigation of complex networks: Participatory greedy routing« arXiv:1611.04395 (2016) K-K. Kleineberg & Dirk Helbing Thanks to: Dirk Helbing Kaj Kolja Kleineberg: • [email protected] • @KoljaKleineberg ← Slides • koljakleineberg.wordpress.com ← Slides