Collective navigation of complex networks:Participatory greedy routing

Kaj Kolja Kleineberg | kkleineberg@ethz.ch @KoljaKleineberg | koljakleineberg.wordpress.com

“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)

We are actually quite good at this

mapwe can build a

of the system

networkRoad

networkAirtravel

spaceEuclidean

Maps:

Maps:

spaceHyperbolic

[Network Science, Barabasi]

networkRoad

networkAirtravel

spaceEuclidean

Maps:

Maps:

spaceHyperbolic

[PRE 82, 036106]

[Figures: Network Science, Barabasi]

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)

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)

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.

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 )

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 )

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 )

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 )

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 )

Greedy routing requires

active participationfrom agents.

Greedy routing requires

active participationfrom agents.

Greedy routing requires

active participationfrom agents.

What if they

don't?

Game theory:

Sending messagehas a cost

Succesul deliverycreates value

Agents may defect Value is shared

Individuals obtain a payoff if message is deliveredbut forwarding has a cost

Cooperator

Defector

Message is sent

Message is lost

Succ

ess

Failu

re

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.

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.

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.

Bistability: the system is either highly functionalor performance breaks down completely

b: Value generated by successful deliveryC0: Initial density of cooperators

System self-organizes into local clusters of cooperatorsprior to the emergence of global cooperation

Distributing the initial cooperators into local clustersfavors significantly the emergence of cooperation

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

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

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

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

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

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

Reference:

»Collective navigation of complex networks: Participatory greedyrouting«arXiv:1611.04395 (2016)K-K. Kleineberg & Dirk Helbing

Thanks to:

Dirk Helbing

Kaj Kolja Kleineberg:

• kkleineberg@ethz.ch

• @KoljaKleineberg

• koljakleineberg.wordpress.com

Reference:

»Collective navigation of complex networks: Participatory greedyrouting«arXiv:1611.04395 (2016)K-K. Kleineberg & Dirk Helbing

Thanks to:

Dirk Helbing

Kaj Kolja Kleineberg:

• kkleineberg@ethz.ch

• @KoljaKleineberg← Slides

• koljakleineberg.wordpress.com

Reference:

»Collective navigation of complex networks: Participatory greedyrouting«arXiv:1611.04395 (2016)K-K. Kleineberg & Dirk Helbing

Thanks to:

Dirk Helbing

Kaj Kolja Kleineberg:

• kkleineberg@ethz.ch

• @KoljaKleineberg← Slides

• koljakleineberg.wordpress.com← Slides