the internet and the hyperbolic plane -...
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The internet and the hyperbolic plane
image by Sandy Ressler nist.gov
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http://www.caida.org
The internet infrastructure is composed of connections
between autonomous systems (AS).
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The number of ASs increases by approx 2400/year.
To route information between locations, the ASs must discover the best path to each possible destination.
The constantly changing structure of the internet leads to immense and quickly growing routing costs: the existing internet routing architecture may not last the decade.
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Roads without an atlas
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We need a map!
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Goal: find a path metric space X so that the internet graph can be drawn in X and so that the best path in the graph is approximately a geodesic in X.
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What are the essential properties of the AS graph?
The graph is scale-free
P(k) = # nodes w/ degree k
# nodes
degree = 8
≈ 1
kγγ ≥ 2with
This strongly corresponds to robustness to failure.
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The hyperbolic plane can be used to create scale-free networks.
Randomly distribute N nodes in a hyperbolic disc of radius R.
Angles are uniformly distributed in [0, 2π].
Hyperbolic radii are distributed in [0, R]with PDF sinh(r)
cosh(R)− 1
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The hyperbolic plane can be used to create scale-free networks.
Join two vertices by an edge if they are within hyperbolic distance R
Krioukov, et al.
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Area of intersection decreases exponentially with r.
k̄(r) ≈ e−r/2
Inverting we find:
r̄(k) ≈ −2 ln(k)
ρ(r) ≈ er−R
ρ(r̄(k))|r̄�(k)| ≈ 2
k3
Recalling:
We have:
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Goal: find a path metric space X so that the internet graph can be drawn in X and so that the best path in the graph is approximately a geodesic in X.
In fact, it is possible to place the nodes of any scale-free network into the hyperbolic plane so that shortest paths track geodesics.
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Doing this for the AS network
Boguña, et al
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Doing this for the AS network
Produces a routing algorithm which is successful 97% of the time.
Reflects geo-political boundaries
Does not cause abnormal traffic congestion
Is unaffected by random removals from the graph.
Has limited performance degradation over time.
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Sources:
Krioukov, et al. Hyperbolic Geometry of Complex NetworksarXiv: 1006.5169
Boguña, et al. Sustaining the Internet with hyperbolic mappingNature Communications September 7, 2010.