Many networks are used to transfer information or goods, in other words, they are navigated. The larger the network, the more difficult it is to navigate efficiently. Indeed, information routing in the Internet faces serious scalability problems due to its rapid growth, recently accelerated by the rise of the Internet of Things. Large networks like the Internet can be navigated efficiently if nodes, or agents, actively forward information based on hidden maps underlying these systems. However, in reality most agents will deny to forward messages, which has a cost, and navigation is impossible. Can we design appropriate incentives that lead to participation and global navigability? Here, we present an evolutionary game where agents share the value generated by successful delivery of information or goods. We show that global navigability can emerge, but its complete breakdown is possible as well. Furthermore, we show that the system tends to self-organize into local clusters of agents who participate in the navigation. This organizational principle can be exploited to favor the emergence of global navigability in the system.

1. Collective navigation of complex networks:
Participatory greedy routing
Kaj Kolja Kleineberg | kkleineberg@ethz.ch
@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)

3. We are actually quite good at this

4. map
we can build a
of the system

5. networkRoad
networkAirtravel
space
Euclidean
Maps:
Maps:
space
Hyperbolic
[Network Science, Barabasi]

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)

15. Greedy routing requires
active participation
from agents.

16. Greedy routing requires
active participation
from agents.

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:
• kkleineberg@ethz.ch
• @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:
• kkleineberg@ethz.ch
• @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:
• kkleineberg@ethz.ch
• @KoljaKleineberg ← Slides
• koljakleineberg.wordpress.com ← Slides