This document proposes mapping the Internet's autonomous systems (ASes) to a hyperbolic geometric space to enable greedy routing. It finds that mapping ASes with angular coordinates equal to their geographic positions and radial coordinates based on degree preserves 99.99% of shortest paths. The model places higher-degree ASes farther from the origin to match the Internet's topology. Greedy routing in this mapped hyperbolic space remains efficient even when links are removed.

1. Sustaining Internet with
Hyperbolic Mapping
- Marian Boguna, Fragkiskos
Papadopoulos & Dmitri Krioukov

2. 29/8/13 Hyperbolic Mapping 2
Overview
• Hyperbolic Map –basic philosophy
• Routing Scheme – Greedy Forwarding
• Results-stretch, %shortest path, RT
• Model to map current Internet on hyperbolic
Space

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Hyperbolic Map-Basic Philosophy
• Coordinate system in geometric space!
• Assign AS coordinates in some geometric space.
• Use the space to forward information packets in the
right directions towards their destinations.
• Hyperbolic Map
• Angular coordinate- as per geography
• Radial coordinate – as a function of the AS degree,
making the space hyperbolic
• Only information that ASs must maintain is the
coordinates of their neighbours

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5. Greedy routing (Kleinberg)
• Greedy forwarding implements routing in the right
direction: upon reading the destination address in
the packet, the current packet holder forwards the
packet to its neighbour closest to the destination
in the space
• Uses local neighborhood information only.

7. Advantages- Greedy Routing
• Greedy routing in complex networks, like the real
AS Internet, embedded in hyperbolic spaces, is
always successful and always follows shortest
paths
• Even if some links are removed, emulating
topology dynamics, greedy routing finds
remaining paths if they exist.
• No recomputation of node coordinates required.

8. Results
• Percentage of successful greedy paths 99.99%
• Avg Stretch - 1.1
• Routing Information at AS  AS Degree
• approximately the same traffic load on nodes as
shortest-path forwarding
• Percentage of successful greedy paths after removal of
X% of links or nodes
• X=10% 99%
• X=30% 95%

9. The Model:Einsteinian
• The main property of hyperbolic geometry is the
exponential expansion of space expansion of
space
• Angular node density
is uniform, but in radial
the number of nodes
grows exponentially
as we move away from
the origin.
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10. Einsteinian Model
By hyperbolic law of cosines,
Length(ab) is –
coshxab = coshra coshrb-sinhrasinhrbcosab;
ab – is the angle between Oa and Ob.
• rO = 0 in formula gives, Oa = ra and Ob = rb.
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11. Methods
•
10/5/06 Lecture #12: Inter-Domain Routing 11

12. Methods Cont..
•
10/5/06 Lecture #12: Inter-Domain Routing 12

13. Methods Cont..
•
10/5/06 Lecture #12: Inter-Domain Routing 13

16. Internet Topology
• 23572 Ass, 58416 AS links
• avg AS degree=4.92, and max degree=2778
• avg clustering=0.61
• hyperbolic disc radius=27
• power law exponent=2.1.
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17. Mapping AS to countries
• From CAIDA AS ranking project.
• Uses 2 methods –
• 1) IP based: splits the IP address space
advertised by an AS into small blocks, and then
maps each block to a country.
• 2) IP and WHOIS based: reports the country
where the AS headquarters are located according
to the WHOIS database, for AS’s located in many
countries.
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18. Geographic Routing
• AS angular coordinates equal to their geographic
coordinates, the radial coordinate is obtained by
Einsteinian model according to the relationship
between node degrees and radial positions.
• Then greedy forwarding in the three-dimensional
hyperbolic space is performed for Routing.
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19. Traffic & Congestion considerations
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20. success ratio& average stretch
On removal of a given fraction of AS nodes
20

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Thank You

22. Metropolis-Hastings
• Compute current likelihood Lc
• Select a random node
• Move it to a new random angular coordinate
• Compute new likelihood Ln
• If Ln > Lc, accept the move
• If not, accept it with probability Ln / Lc
• Repeat
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23. Sensitivity to missing links
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24. Mapping the real Internet
using statistical inference methods
• Measure the Internet topology properties
• Map them to model parameters
• Place nodes at hyperbolic coordinates (r,)
• ’s are uniformly distributed on [0,2]
• Apply the Metropolis-Hastings algorithm to find ’s
maximizing the likelihood that Internet is produced
by the model