Topology Modeling: First-Principles Approach
Aditya Akella
Supplemental Slides03/30/2007
• Evaluate performance of protocols• Protect Internet• Resource provisioning• Understand large scale networks
Why Topology Modeling
Challenges
• Large Size• Real topologies are not publicly available• Incredible variability in many aspects
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
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
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
power-law generator
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
Preferential Attachment Expected Degree Sequence
Li et al.• Consider the explicit design of the Internet
– Annotated network graphs (capacity, bandwidth)– Technological and economic limitations– Network performance
• 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
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
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 Feasibilitycore technologies
edge technologies
older/cheaper technologies
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
Heuristically Optimal Topology
Hosts
Edges
Cores
Mesh-like core of fast, low degree routers
High degree nodes are at the edges.
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
Metrics for Comparison: Network Performance
Given 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
PA PLRG/GRGHOT
Structure Determines Performance
P(g) = 1.19 x 1010 P(g) = 1.64 x 1010 P(g) = 1.13 x 1012
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?
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