Collective navigation of complex networks: Participatory greedy routing
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Transcript of Collective navigation of complex networks: Participatory greedy routing
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Collective navigation of complex networks:Participatory greedy routing
Kaj Kolja Kleineberg | [email protected] @KoljaKleineberg | koljakleineberg.wordpress.com
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“I read somewhere thaton 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 thethe right six people ..."
John Guare, Six Degrees of Separation (1990)
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We are actually quite good at this
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mapwe can build a
of the system
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networkRoad
networkAirtravel
spaceEuclidean
Maps:
Maps:
spaceHyperbolic
[Network Science, Barabasi]
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networkRoad
networkAirtravel
spaceEuclidean
Maps:
Maps:
spaceHyperbolic
[PRE 82, 036106]
[Figures: Network Science, Barabasi]
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Maps of scale-free clustered networksare hyperbolic
“Hyperbolic geometry of complex networks” [PRE 82, 036106]
Distribute:
ρ(r) ∝ e12(γ−1)r
Connect:
p(xij) =1
1 + exij−R
2T
xij = cosh−1(cosh ri cosh rj
− sinh ri sinh rj cos∆θij)
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Maps of scale-free clustered networksare hyperbolic
“Hyperbolic geometry of complex networks” [PRE 82, 036106]
Distribute:
ρ(r) ∝ e12(γ−1)r
Connect:
p(xij) =1
1 + exij−R
2T
xij = cosh−1(cosh ri cosh rj
− sinh ri sinh rj cos∆θij)
![Page 9: Collective navigation of complex networks: Participatory greedy routing](https://reader031.fdocuments.in/reader031/viewer/2022022001/58eff7251a28ab625d8b45d9/html5/thumbnails/9.jpg)
Maps of scale-free clustered networksare hyperbolic
“Hyperbolic geometry of complex networks” [PRE 82, 036106]
Distribute:
ρ(r) ∝ e12(γ−1)r
Connect:
p(xij) =1
1 + exij−R
2T
xij = cosh−1(cosh ri cosh rj
− sinh ri sinh rj cos∆θij)
Real networks can be embedded into hyperbolicspace by inverting the model.
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Inferred maps can be used to navigate the networkrelying only on local information (greedy routing)
[Credits: Marian Boguna]
Forward messageto contact closest totarget in metric space
Delivery failsif message runs into aloop (define success
rate P )
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Inferred maps can be used to navigate the networkrelying only on local information (greedy routing)
[Credits: Marian Boguna]
Forward messageto contact closest totarget in metric space
Delivery failsif message runs into aloop (define success
rate P )
![Page 12: Collective navigation of complex networks: Participatory greedy routing](https://reader031.fdocuments.in/reader031/viewer/2022022001/58eff7251a28ab625d8b45d9/html5/thumbnails/12.jpg)
Inferred maps can be used to navigate the networkrelying only on local information (greedy routing)
[Credits: Marian Boguna]
Forward messageto contact closest totarget in metric space
Delivery failsif message runs into aloop (define success
rate P )
![Page 13: Collective navigation of complex networks: Participatory greedy routing](https://reader031.fdocuments.in/reader031/viewer/2022022001/58eff7251a28ab625d8b45d9/html5/thumbnails/13.jpg)
Inferred maps can be used to navigate the networkrelying only on local information (greedy routing)
[Credits: Marian Boguna]
Forward messageto contact closest totarget in metric space
Delivery failsif message runs into aloop (define success
rate P )
![Page 14: Collective navigation of complex networks: Participatory greedy routing](https://reader031.fdocuments.in/reader031/viewer/2022022001/58eff7251a28ab625d8b45d9/html5/thumbnails/14.jpg)
Inferred maps can be used to navigate the networkrelying only on local information (greedy routing)
[Credits: Marian Boguna]
Forward messageto contact closest totarget in metric space
Delivery failsif message runs into aloop (define success
rate P )
![Page 15: Collective navigation of complex networks: Participatory greedy routing](https://reader031.fdocuments.in/reader031/viewer/2022022001/58eff7251a28ab625d8b45d9/html5/thumbnails/15.jpg)
Greedy routing requires
active participationfrom agents.
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Greedy routing requires
active participationfrom agents.
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Greedy routing requires
active participationfrom agents.
What if they
don't?
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Game theory:
Sending messagehas a cost
Succesul deliverycreates value
Agents may defect Value is shared
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Individuals obtain a payoff if message is deliveredbut forwarding has a cost
Cooperator
Defector
Message is sent
Message is lost
Succ
ess
Failu
re
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Individuals imitate the behaviorof more successful contacts
AfterN message sending events, individuals can update theirstrategies according to imitation dynamics:
i copies strategy of randomlyselected neighbor j withprobability
pi←j =1
1 + e−(pj−pi)/K
pi,j denotes collected payoffs
After each update step, we reset the payoffs.
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Individuals imitate the behaviorof more successful contacts
AfterN message sending events, individuals can update theirstrategies according to imitation dynamics:
i copies strategy of randomlyselected neighbor j withprobability
pi←j =1
1 + e−(pj−pi)/K
pi,j denotes collected payoffs
After each update step, we reset the payoffs.
![Page 22: Collective navigation of complex networks: Participatory greedy routing](https://reader031.fdocuments.in/reader031/viewer/2022022001/58eff7251a28ab625d8b45d9/html5/thumbnails/22.jpg)
Individuals imitate the behaviorof more successful contacts
AfterN message sending events, individuals can update theirstrategies according to imitation dynamics:
i copies strategy of randomlyselected neighbor j withprobability
pi←j =1
1 + e−(pj−pi)/K
pi,j denotes collected payoffs
After each update step, we reset the payoffs.
![Page 23: Collective navigation of complex networks: Participatory greedy routing](https://reader031.fdocuments.in/reader031/viewer/2022022001/58eff7251a28ab625d8b45d9/html5/thumbnails/23.jpg)
Bistability: the system is either highly functionalor performance breaks down completely
b: Value generated by successful deliveryC0: Initial density of cooperators
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System self-organizes into local clusters of cooperatorsprior to the emergence of global cooperation
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Distributing the initial cooperators into local clustersfavors significantly the emergence of cooperation
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Heterogeneity favors cooperation in the systemin addition to initial localization
Rand.Clust.
5 10 15 20 25 30 350.1
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
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Collective navigation of complex networks:Participatory greedy routing
Results:
- Greedy routing: Forwarding of messages with localknowledge 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 underlyingmetric space)
- This can be exploited to lower necessary number of initialcooperators (localization)
Outlook:
- Reputation system
- Adaptive networks
![Page 28: Collective navigation of complex networks: Participatory greedy routing](https://reader031.fdocuments.in/reader031/viewer/2022022001/58eff7251a28ab625d8b45d9/html5/thumbnails/28.jpg)
Collective navigation of complex networks:Participatory greedy routing
Results:
- Greedy routing: Forwarding of messages with localknowledge 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 underlyingmetric space)
- This can be exploited to lower necessary number of initialcooperators (localization)
Outlook:
- Reputation system
- Adaptive networks
![Page 29: Collective navigation of complex networks: Participatory greedy routing](https://reader031.fdocuments.in/reader031/viewer/2022022001/58eff7251a28ab625d8b45d9/html5/thumbnails/29.jpg)
Collective navigation of complex networks:Participatory greedy routing
Results:
- Greedy routing: Forwarding of messages with localknowledge 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 underlyingmetric space)
- This can be exploited to lower necessary number of initialcooperators (localization)
Outlook:
- Reputation system
- Adaptive networks
![Page 30: Collective navigation of complex networks: Participatory greedy routing](https://reader031.fdocuments.in/reader031/viewer/2022022001/58eff7251a28ab625d8b45d9/html5/thumbnails/30.jpg)
Collective navigation of complex networks:Participatory greedy routing
Results:
- Greedy routing: Forwarding of messages with localknowledge 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 underlyingmetric space)
- This can be exploited to lower necessary number of initialcooperators (localization)
Outlook:
- Reputation system
- Adaptive networks
![Page 31: Collective navigation of complex networks: Participatory greedy routing](https://reader031.fdocuments.in/reader031/viewer/2022022001/58eff7251a28ab625d8b45d9/html5/thumbnails/31.jpg)
Collective navigation of complex networks:Participatory greedy routing
Results:
- Greedy routing: Forwarding of messages with localknowledge 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 underlyingmetric space)
- This can be exploited to lower necessary number of initialcooperators (localization)
Outlook:
- Reputation system
- Adaptive networks
![Page 32: Collective navigation of complex networks: Participatory greedy routing](https://reader031.fdocuments.in/reader031/viewer/2022022001/58eff7251a28ab625d8b45d9/html5/thumbnails/32.jpg)
Reference:
»Collective navigation of complex networks: Participatory greedyrouting«arXiv:1611.04395 (2016)K-K. Kleineberg & Dirk Helbing
Thanks to:
Dirk Helbing
Kaj Kolja Kleineberg:
• @KoljaKleineberg
• koljakleineberg.wordpress.com
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Reference:
»Collective navigation of complex networks: Participatory greedyrouting«arXiv:1611.04395 (2016)K-K. Kleineberg & Dirk Helbing
Thanks to:
Dirk Helbing
Kaj Kolja Kleineberg:
• @KoljaKleineberg← Slides
• koljakleineberg.wordpress.com
![Page 34: Collective navigation of complex networks: Participatory greedy routing](https://reader031.fdocuments.in/reader031/viewer/2022022001/58eff7251a28ab625d8b45d9/html5/thumbnails/34.jpg)
Reference:
»Collective navigation of complex networks: Participatory greedyrouting«arXiv:1611.04395 (2016)K-K. Kleineberg & Dirk Helbing
Thanks to:
Dirk Helbing
Kaj Kolja Kleineberg:
• @KoljaKleineberg← Slides
• koljakleineberg.wordpress.com← Slides