Greedy Forwarding in
Dynamic Scale-Free Networks
Embedded in Hyperbolic Metric Spaces
Dmitri Krioukov
CAIDA/UCSD
Joint work with
F. Papadopoulos, M. Boguñá, A. Vahdat
1
Outline
Motivation
Model of scale-free networks embedded
in hyperbolic metric spaces
Greedy forwarding in the model
Conclusion
2
Motivation
Routing overhead is a serious scaling limitation in
many networks (Internet, wireless, overlay/P2P
networks, etc.)
Search for infinitely scalable routing without any
overhead
Do not propagate any information about changing
topology
Route without any global topology knowledge,
using only local information
How is it possible?
3
Greedy geometric forwarding as
routing using only local information
Network topology is embedded in a
geometric space
To reach a destination, each node forwards
the packet to the neighbor that is closest to
the destination in the space
4
Hidden space visualized
5
Desired properties of greedy
forwarding, and related metrics
Property 1: Greedy routes should never get stuck at
local minima, nodes that do not have any neighbor
closer to the destination than themselves

Success ratio, percentage of successful greedy paths
reaching their destinations, should be close to 1
Property 2: Greedy paths should be close to
shortest paths

Stretch, ratio of the lengths of greedy to shortest paths,
should be also close to 1
Property 3: Even if topology changes, success ratio
and stretch should stay close to 1 without any
recomputation (e.g., without nodes changing their
positions in the space)
6
Problem formulation (high-level)
Find a combination of network topology and underlying
geometric space which would satisfy these desired
properties
Any suggestions?
Nature offers some: many dynamic networks in nature and
society do route information without any topology
knowledge (brain, regulatory, social networks, etc.)
All these complex networks have power-law degree
distributions (scale-free) and strong clustering (many
triangular subgraphs)
Let’s focus on these topologies (which, luckily, also
characterize the Internet and P2P networks)
But what about the underlying space?
7
Conjecture: space is hyperbolic
Nodes in real complex networks can often
be classified hierarchically
Hierarchies are tree-like structures
Hyperbolic geometry is the geometry of
tree-like structures

Formally: trees embed almost isometrically in
hyperbolic spaces, not in Euclidean ones
8
Main hyperbolic property: the
exponential expansion of space
Circle length and disc area grow with their radius
R as
~ eR
They are exactly
2 sinh R
2 (cosh R  1)
The numbers of nodes in a tree at or within R hops
from the root grow as
~ bR
where b is the tree branching factor
The metric structures of hyperbolic spaces and
trees are essentially the same
9
Problem formulation (low-level)
Verify the conjecture: check if hyperbolic
geometry, in the simplest possible settings,
can naturally give rise to scale-free, strongly
clustered topologies
Check if greedy forwarding satisfies the
desired properties in the resulting embedding
10
Outline
Motivation
Model of scale-free networks embedded
in hyperbolic metric spaces
Greedy forwarding in the model
Conclusion
11
The model
12
The model (cont.)
13
14
Average node degree at distance r
from the disc center
15
Node degree distribution
16
Model vs. AS Internet
17
Growing networks
The model can be adjusted for networks
growing in hyperbolic spaces
All results stay the same
18
Outline
Motivation
Model of scale-free networks embedded
in hyperbolic metric spaces
Greedy forwarding in the model
Conclusion
19
Two greedy forwarding algorithms
Original Greedy Forwarding (OGF): select
closest neighbor to destination, drop the
packet if no one closer than current hop
Modified Greedy Forwarding (MGF): select
closest neighbor to destination, drop the
packet if a node sees it twice
20
Property 1:
success ratio
21
Property 2:
average and maximum stretch
22
Property 3: Robustness of greedy
forwarding w.r.t. network dynamics
Scenario 1: Randomly remove a percentage
of links and compute the new success ratio
Scenario 2: Remove a link and compute the
percentage of paths that were going through
it and are still successful (that is, the
percentage of paths that found a by-pass)
23
Percentage of successful paths
(dynamic networks, scenario 1)
24
Percentage of successful paths
(dynamic networks, scenario 2)
25
Shortest paths in scale-free graphs
and hyperbolic spaces
26
Outline
Motivation
Model of scale-free networks embedded
in hyperbolic metric spaces
Greedy forwarding in the model
Conclusion
27
Conclusion (low-level)
Hyperbolic geometry naturally explains the two
main topological characteristics of complex
networks
scale-free degree distributions
 strong clustering

Greedy forwarding in complex networks
embedded in hyperbolic spaces is exceptionally
efficient
28
Conclusion (mid-level)
Complex network topologies are naturally congruent
with hyperbolic geometries

Greedy paths follow shortest paths that approximately follow
geodesics in the hyperbolic space

Both topology and geometry are tree-like
This congruency is robust w.r.t. topology dynamics

There are many link/node-disjoint shortest paths between the
same source and destination that satisfy the above property


Strong clustering (many by-passes) boosts up the path diversity
If some of shortest paths are damaged by link failures, many
others remain available, and greedy routing still finds them
29
Conclusion (high-level)
To efficiently route without topology
knowledge, the topology should be both
hierarchical (tree-like) and have high path
diversity (not like a tree)
Complex networks do borrow the best out of
these two seemingly mutually-exclusive worlds
Hidden hyperbolic geometry naturally explains
how this balance is achieved
30
Implications
Greedy forwarding mechanisms in these settings
may offer virtually infinitely scalable information
dissemination (routing) strategies for
communication networks
Zero communication costs (no routing updates!)
 Constant routing table sizes (coordinates in the space)
 No stretch (all paths are shortest, stretch=1)

31
Applications
Internet routing (hard): need to reverse the problem and
find an embedding for a given Internet topology first
Overlay networks:


with underlay (easier; examples: existing P2P): have freedom
of constructing a name space and its embedding according to
the model, so that all the desired properties are satisfied
without underlay (harder; examples: CCN, pocket switching):
need to make sure that the underlay network topology and its
dynamics are congruent with the overlay name space and its
dynamics
32