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A First-Principles Approachto Understanding the Internet’s
Router-level Topology
Lun LiDavid Alderson
Walter Willinger
John C. Doyle
2004 ACM SIGCOMM
Portland, OR
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• Evaluate performance of protocols• Protect Internet• Resource provisioning• Understand large scale networks
Why Topology
Challenges
• Large Size• Real topologies are not publicly available• Incredible variability in many aspects
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Trends in Topology Modeling
Observation Modeling Approach
• Real networks are not random, but have obvious hierarchy.
• Structural models (GT-ITM Calvert/Zegura, 1996)
• Long-range links are expensive • Random graph models (Waxman, 1988)
• Internet topologies exhibit power law degree distributions (Faloutsos et al., 1999)
• Degree-based models replicate power-law degree sequences
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A few nodes have lots of connections
Ran
k R(d)
Degree d
Source: Faloutsos et al. (1999)Power Laws and Internet Topology
• Router-level graph & Autonomous System (AS) graph• Led to active research in degree-based network models
Most nodes have few connections
R(d
) =
P (
D>
d) x
#no
des
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Degree-Based Models of Topology
• Preferential Attachment– Growth by sequentially adding new nodes– New nodes connect preferentially to nodes having
more connections– Examples: Inet, GPL, AB, BA, BRITE, CMU powe
r-law generator
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Degree-Based Models of Topology
• Expected Degree Sequence– Based on random graph models that skew
probability distribution to produce power laws in expectation
– Examples: Power Law Random Graph (PLRG), Generalized Random Graph (GRG)
• Preferential Attachment– Growth by sequentially adding new nodes– New nodes connect preferentially to nodes having
more connections– Examples: Inet, GPL, AB, BA, BRITE, CMU powe
r-law generator
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Features of Degree-Based Models
• Degree sequence follows a power law (by construction)• High-degree nodes correspond to highly connected
central “hubs”, which are crucial to the system• Achilles’ heel: robust to random failure, fragile to specific
attack • “scale-free” in complex networks
Preferential Attachment Expected Degree Sequence
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Our Approach• Consider the explicit design of the Internet
– Annotated network graphs (capacity, bandwidth)– Technological and economic limitations– Network performance– Heuristic optimized tradeoffs (HOT)
• Seek a theory for Internet topology that is explanatory and not merely descriptive.– Explain high variability in network connectivity– Ability to match large scale statistics (e.g. power
laws) is only secondary evidence
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Trends in Topology ModelingObservation Modeling Approach
• Real networks are not random, but have obvious hierarchy.
• Structural models (GT-ITM Calvert/Zegura, 1996)
• Long-range links are expensive • Random graph models (Waxman, 1988)
• Internet topologies exhibit power law degree distributions (Faloutsos et al., 1999)
• Degree-based models replicate power-law degree sequences
• Physical networks have hard technological (and economic) constraints.
• Optimization-driven models topologies consistent with design tradeoffs of network engineers
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100
101
102
Degree
10-1
100
101
102
103
Ban
dwid
th (
Gbp
s)
15 x 10 GE
15 x 3 x 1 GE
15 x 4 x OC12
15 x 8 FE
Technology constraint
Total Bandwidth
Bandwidth per Degree
Router Technology ConstraintCisco 12416 GSR, circa 2002
high BW low degree high
degree low BW
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0.01
0.1
1
10
100
1000
10000
100000
1000000
1 10 100 1000 10000degree
To
tal R
ou
ter
BW
(M
bp
s)
cisco 12416
cisco 12410
cisco 12406
cisco 12404
cisco 7500
cisco 7200
linksys 4-port router
uBR7246 cmts(cable)
cisco 6260 dslam(DSL)
cisco AS5850(dialup)
approximateaggregate
feasible region
Aggregate Router Feasibility
Source: Cisco Product Catalog, June 2002
core technologies
edge technologies
older/cheaper technologies
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Rank (number of users)
Con
nect
ion
Spe
ed (
Mbp
s)
1e-1
1e-2
1
1e1
1e2
1e3
1e4
1e21 1e4 1e6 1e8
Dial-up~56Kbps
BroadbandCable/DSL~500Kbps
Ethernet10-100Mbps
Ethernet1-10Gbps
most users have low speed
connections
a few users have very high speed
connections
high performancecomputing
academic and corporate
residential and small business
Variability in End-User Bandwidths
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Heuristically Optimal Topology
Hosts
Edges
Cores
Mesh-like core of fast, low degree routers
High degree nodes are at the edges.
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SOX
SFGP/AMPATH
U. Florida
U. So. Florida
Miss StateGigaPoP
WiscREN
SURFNet
Rutgers U.
MANLAN
NorthernCrossroads
Mid-AtlanticCrossroads
Drexel U.
U. Delaware
PSC
NCNI/MCNC
MAGPI
UMD NGIX
DARPABossNet
GEANT
Seattle
Sunnyvale
Los Angeles
Houston
Denver
KansasCity
Indian-apolis
Atlanta
Wash D.C.
Chicago
New York
OARNET
Northern LightsIndiana GigaPoP
MeritU. Louisville
NYSERNet
U. Memphis
Great Plains
OneNetArizona St.
U. Arizona
Qwest Labs
UNM
OregonGigaPoP
Front RangeGigaPoP
Texas Tech
Tulane U.
North TexasGigaPoP
TexasGigaPoP
LaNet
UT Austin
CENIC
UniNet
WIDE
AMES NGIX
PacificNorthwestGigaPoP
U. Hawaii
PacificWave
ESnet
TransPAC/APAN
Iowa St.
Florida A&MUT-SWMed Ctr.
NCSA
MREN
SINet
WPI
StarLight
IntermountainGigaPoP
Abilene BackbonePhysical Connectivity(as of December 16, 2003)
0.1-0.5 Gbps0.5-1.0 Gbps1.0-5.0 Gbps5.0-10.0 Gbps
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Cisco 750X
Cisco 12008
Cisco 12410
OC-3 (155 Mb/s)OC-12 (622 Mb/s)GE (1 Gb/s)OC-48 (2.5 Gb/s)10GE (10 Gb/s)
CENIC Backbone (as of January 2004)
dc1
dc2
dc3
hpr
dc1
dc3
hpr
dc2
dc1
dc1 dc2
hpr
hpr
SACOAK
SVL
LAX
SDG
SLOdc1
FRGdc1
FREdc1
BAKdc1
TUSdc1
SOLdc1
CORdc1
hprdc1
dc2
dc3
hpr
AbileneLos Angeles
AbileneSunnyvale
Corporation for Education Network Initiatives in California (CENIC)
1
10
100
1000
10000
100000
1 10 100degree
tota
l BW
(Mbp
s)12410
12008
750X
12410 Feasible Region
12008 Feasible Region
750X Feasible Region
CENIC Router Configurations, Jan. 2004
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Two Different Perspectives
• Degree-based perspective– Match aggregate statistics– Suggest high-degree central hubs
• First principles perspective– Technology and economic constraints – Performance– Suggest fast, low-degree core routers– Consistent with physical design of real networks
How to reconcile these two perspectives?
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Same Degree Distribution
PA
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PLRG
Same Degree Distribution
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HOT
Same Degree Distribution
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PA PLRG/GRG
HOT Abilene-inspired Sub-optimal
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Network PerformanceGiven realistic technology constraints on routers, how well is the
network able to carry traffic?
Step 1: Constrain to be feasible
Abstracted Technologically Feasible Region
1
10
100
1000
10000
100000
1000000
10 100 1000
degree
Ban
dw
idth
(M
bp
s)
kBxts
BBx
ijrkjikij
ji jijiij
,..
maxmax
:,
, ,
Step 3: Compute max flow
Bi
Bj
xij
Step 2: Compute traffic demand
jiij BBx
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PA PLRG/GRGHOT
Structure Determines Performance
P(g) = 1.19 x 1010 P(g) = 1.64 x 1010 P(g) = 1.13 x 1012
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Likelihood-Related Metric
• Easily computed for any graph• Depends on the structure of the graph, not the
generation mechanism• Measures how “hub-like” the network core is
j
connectedji
iddgL ,
)(Define the metric (di = degree of node i)
For graphs resulting from probabilistic construction (e.g. PLRG/GRG),
LogLikelihood (LLH) L(g)
Interpretation: How likely is a particular graph (having given node degree distribution) to be constructed?
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Lmax
l(g) = 1P(g) = 1.08 x 1010
P(g) Perfomance (bps)
PA PLRG/GRGHOT Abilene-inspired Sub-optimal
0 0.2 0.4 0.6 0.8 1
1010
1011
1012
l(g) = Relative Likelihood
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Lmax
l(g) = 1P(g) = 1.08 x 1010
P(g) Perfomance (bps)
PA PLRG/GRGHOT Abilene-inspired Sub-optimal
0 0.2 0.4 0.6 0.8 1
1010
1011
1012
l(g) = Relative Likelihood
???
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Points to Emphasize
• Same Degree distribution can have different core structures
• Same Core structure can have different degree distributions
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100 101 102100
101
102
Node Degree
No
de
Ran
k
100 101 102100
101
102
Node Degree
No
de
Ran
k
100 101 102100
101
102
No
de
Ran
k
Node Degree
Abilene-inspired core
Uniform high BW users
Low variability deg dist.
Abilene-inspired core
High variability at edges
Power-law deg distribution
Abilene-inspired core
Uniform low BW users
low variability deg dist.
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Conclusions1. Probabilistic degree-based generation
mechanisms are likely to produce networks that have poor performance and are very unlikely to generate realistic networks.
2. Models of router-level topology should consider inherent technological and economic tradeoffs in network design, and they should consider issues such as performance over simple connectivity.
3. Realistic router-level topology generators will require additional work to incorporate other key features (e.g. geography, population density) into the framework
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Future Work
• Validation– rely on cooperative ISPs to validate router technology
constraints– combine with heuristics and supplementary
measurements– develop annotated prototype ISP topology generator
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A Tier 1 Router Bandwidth-Degree (Courtesy: Matt Roughan)
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Future Work
• Validation– rely on cooperative ISPs to validate router technology
constraints– combine with heuristics and supplementary measure
ments– develop annotated prototype ISP topology generator– Contact: David Alderson <[email protected]>
• HOT-inspired framework for protocol stack– consider layering as optimization decomposition– study properties of `natural' decompositions– Contact: Lun Li <[email protected]>