Understanding Router-level Topology: Principles, Models, and Validation
David AldersonCalifornia Institute of Technology
ISMA Workshop on Internet Topology May 10, 2006
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 2
AcknowledgmentsPrimary Coauthors• John Doyle (Caltech)• Walter Willinger (AT&T Labs-Research)• Lun Li (Caltech)
Contributions• Reiko Tanaka (RIKEN, Japan)• Matt Roughan (U. Adelaide, Australia)• Steven Low (Caltech)• Ramesh Govindan (USC)• Neil Spring (U. Maryland)• Stanislav Shalunov (Abilene)• Heather Sherman (CENIC)
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 3
On modeling
“All models are wrong, but some models are useful.”
- G. P. E. Box
“When exactitude is elusive, it is better to be approximately right than certifiably wrong.”
- B. B. Mandelbrot
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 4
the Internet as an inspiration for the development of elegant mathematical models of
networks
wanting to say “something meaningful” about the Internet (something about which decision
makers are concerned)
vs
what is the MESSAGE? what “MATTERS”? and TO WHOM?
who has RESPONSIBILTY for the message?
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 5
The application of graph theory and statistics to the study of Internet topology without the details of system architecture
and engineering can lead to incorrect (and possibly misleading) conclusions.
Let’s consider the router-level Internet
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 6
The Router-Level Internet
mycomputer
router router
webserver
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 7
The Internet is a LAYERED Network
HTTP
TCPIP
LINK
mycomputer
router router
webserver
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 8
The Internet is a LAYERED Network
HTTP
TCPIP
LINK
mycomputer
router router
webserver
packetpacketpacketpacketpacketpacket
The perception of the Internet as a simple, user-friendly, and robust
system is enabled by FEEDBACK and other CONTROLS that operate both
WITHIN LAYERS and ACROSS LAYERS.
These ARCHITECTURAL DETAILS (protocols, interfaces, etc.) are MOST ESSENTIAL to
the nature of the Internet.
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 9
Internet structure can be viewedas a solution to a DESIGN problem
• physical constraints on components
– distance/delay, capacity
• functional constraints on the system as a whole
– “X-ities”: functionality, maintainability, adaptability, evolvability, etc.
design approach: modularity• simplify the problem by breaking it up
• but still with provable properties as if it were an integrated whole
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 10
Internet Architecture: Dual Decomposition
HTTP
TCPIP
LINK
mycomputer
router router
webserver
Ver
tical
dec
ompo
sitio
nP
roto
col S
tack Benefits: • Each layer can evolve
independently• Substitutes, complementsRequirements:1. Each layer follows the rules2. Every other layer does “good
enough” with its implementation
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 11
networks and their properties are different at each layer
IP
TRANSMISSION
TCP
virtual
physical static
dynamicAPPLICATION
Router-level connectivity
IP-level connectivity
Autonomous System (AS) graph
Web graphEmail graphP2P graphand many others …
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 12
Internet Architecture: Dual Decomposition
HTTP
TCPIP
LINK
mycomputer
router router
webserver
Horizontal decompositionEach level is decentralized and asynchronous
Benefit: Individual components can fail (provided that they “fail off”) without disrupting the network.
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 13
The Router-Level Internet
mycomputer
router router
webserver
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 14
Bigger Picture: Internet Architecture
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 15
Bigger Picture: Internet Architecture
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 16
Bigger Picture: Internet Architecture
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 17
Autonomous System (AS) Graphs = Business Relationships
AS 1 AS 3
AS 4AS 2
Nodes = ASesLinks = peeringrelationships
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 18
AS graphs obscure topology!
The AS graphmay look like this. Reality may be closer to this…
Courtesy Tim Griffin
see talks by HyunseokChang and others on
Thursday AM for more on AS topology modeling
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 19
MESSAGE #1: specify WHICH aspect of Internet topology
• There is no “generic” Internet topology• Router-level, IP-level, AS-level, application-level, …• Details of each make a big difference
PITFALL: Lack of specificity causes confusion– Albert, Jeong, and Barabasi (2000) study robustness
properties of the Internet by equating AS-level topology with router-level topology⇒Knocking out nodes in the AS graph??
– Berger, Borgs, Chayes, and Saberi (2005) study the spread of viruses on the Internet by equating the Web graph with the router-level topology.⇒Virus propagation on the Web graph??
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 20
Unfortunately, direct inspection of Internet topology is generally NOT possible
• Economic incentive for ISPs to obscure network structure• Recent trend
– Empirical measurement studies– Generative models
• Obstacles– Mismatch between what we want to measure and can
measure– Imperfect measurements– What macro/microscopic statistics characterize a
topology?– How to determine what matters?
Remainder of talk: focus on router-level topology
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 21
considerable progress in measuring router-level topology…
• traceroute tool – Discovers compliant (i.e., IP) routers along path
between selected network host computers
• Large-scale traceroute experiments– Pansiot and Grad (router-level map from around
1995)– Cheswick and Burch (mapping project 1997--)– Mercator (router-level maps from around 1999 by R.
Govindan and H. Tangmunarunkit)– Skitter (ongoing mapping project by CAIDA folks)– Rocketfuel (state-of-the-art router-level maps of
individual ISPs by UW folks)
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 22
http://research.lumeta.com/ches/map/
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 23
http://www.isi.edu/scan/mercator/mercator.html
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 24
http://www.caida.org/tools/measurement/skitter/
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 25http://www.cs.washington.edu/research/networking/rocketfuel/bb
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 26
…but considerable drawbacks to existing approaches• traceroute-based measurements are ambiguous
– traceroute is strictly about IP-level connectivity– traceroute cannot distinguish between high connectivity
nodes that are for real and that are fake and due to underlying Layer 2 (e.g., Ethernet, ATM) or Layer 2.5 technologies (e.g., MPLS)
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 27
http://www.caida.org/tools/measurement/skitter/
www.savvis.netmanaged IP and hosting companyfounded 1995offering “private IP with ATM at core”
Possible that this “node” is
an entire network! (not just a router)
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 28
…but considerable drawbacks to existing approaches• traceroute-based measurements are ambiguous
– traceroute is strictly about IP-level connectivity– traceroute cannot distinguish between high connectivity
nodes that are for real and that are fake and due to underlying Layer 2 (e.g., Ethernet, ATM) or Layer 2.5 technologies (e.g., MPLS)
• traceroute-based measurements are inaccurate– Requires some guesswork in deciding which IP
addresses/interface cards refer to the same router (“alias resolution” problem)
• traceroute-based measurements are incomplete/biased– IP-level connectivity is more easily/accurately inferred
the closer the routers are to the traceroute source(s)– Node degree distribution is inferred to be of the power-
law type even when the actual distribution is not see talk by Aaron Clauset et al. on Thu AM for more on this…
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 29
MESSAGE #2: Idiosyncracies of network measurements require careful interpretation
• Each technique is typically specific to network of interest (e.g., traceroute for IP-level, BGP tables for AS-level)
• Even best-of-breed measurement data is ambiguous, inaccurate, and incomplete
PITFALL: Taking (someone else’s) data at face value may provide a false basis for results– example: use of MERCATOR data to support claims of
power-law degree distribution for router-level Internet⇒Are routers with >1000 connections plausible??
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 30
100 101 102 103 104100
101
102
103
104
105
106
100 101 102 10310
0
101
102
103
104
105
100 101 102 103100
101
102
103
104
105
106
0 50 100 150 200 250 300 350100
101
102
103
104
105
106
Noncumulative Size-Frequency Binned Size-Frequency
Size-Rank (log-log scale) Size-Rank (log-linear scale)
raw MERCATOR data a common reporting technique
without 2 largest nodes
without 2 largest nodes
exponentialin tail…
plausibledata?
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 31
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 modelsreplicate power-law degree sequences (e.g. scale-free networks, 1999-2004)
It is difficult to know what “matters”when it comes to representing router-level
topology
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 32
Node Degree (d)
Source: Faloutsos et al. (1999)R
ank:
R(d
) = P
(D>d
) x #
node
s
100 101 10210 0
10 1
10 2
10 3
10 4
Node Degree (d)
Nod
e R
ank
Power Laws and Internet Topology
Node Degree: d = # connections
Router-Level Graph Autonomous System (AS) Graph
• A random variable X is said to follow a power law with index α > 0 if
• Led to active research in degree-based network models
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 33
Degree-Based Network Models• Basic Idea: traditional random graphs [Erdös & Renyí, 59] do
not produce power laws, so develop new models that explicitly attempt to match the observed (power law) distribution in node degree
• Preferential Attachment
– Incremental growth + new nodes attach to high-degree nodes
– “Rich get richer”—power laws in asymptotic limit– Scale-free networks [Barabási & Albert, 99]– Generators: Inet, GPL, AB, BA, BRITE, CMU power-law
generator
• Expected Degree Sequence
– Based on random graph models that skew probability distribution to produce power laws in expectation
– Power law random graph (PLRG) [Aiello et al., 00]– Generalized random graph (GRG) [Chung & Lu, 03]
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 34
“Error tolerance”= Loss of random
node has little effect
“Attack vulnerability”= Targeted loss of
hub fragments network
“Scale-free” networks and the
“Achilles’ heel” of the InternetReference: R. Albert, H. Jeong,
and A.-L. Barabási. Attack and error tolerance of complex networks. Nature 406, 378-382, 2000.
node degree101
101
102
100
node
rank
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 35
The literature on Scale-Free Networks claims broad implications for the Internet and other
networksPower laws in network connectivity…⇔ Are necessary and sufficient for “scale-free structure”⇔ Imply critically connected “hubs”⇒ Create an Achilles’ heel vulnerability⇒ Yield a zero epidemic threshold for contagion
⇒Are evidence of fundamental self-organization in networks
⇒This self-organization is believed by some to be a universal feature of technological, biological, social and business networks
⇒Efforts to protect complex networks should focus on the most highly-connected components
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 36node degree
101
101
102
100
node
rank
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 37
• Low degree core• Result of design• High performance and
robustness
• High degree central “hubs”
• From random construction • Poor performance and
robustness
MESSAGE #3: networks with the same statistical features can be OPPOSITES in terms
of engineering
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 38
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 modelsreplicate power-law degree sequences (scale-free networks, 1999-2004)
• Degree-based models are fundamentally inconsistent with engineering reality
• Optimization-driven modelsyield topologies consistent with design tradeoffs of network engineers (SIGCOMM’04)
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 39
HTTP
TCPIP
LINK
mycomputer
router router
webserver
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 40
• Who builds real router-level topologies?
• How do technology and cost influence deployment?
• How does one evaluate a “good” design?
• What drives their structure?
• What about power laws?
LINK
some form of an (implicit) OPTIMIZATION problem, although actual “design” may be decentralized and heuristic
the “decision makers” are individual ISPs
network PERFORMANCE can be measured in terms of tra
they provide CONSTRAINTS on what the ISP can do
a mere consequence of the inputs to the optimization prob
Our Perspective
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 41
Cisco 12000 Series Routers
80 Gbps41/812404
120 Gbps61/412406
200 Gbps101/212410
320 Gbps16Full12416
Switching CapacitySlotsRack sizeChassis
• Modular in design, creating flexibility in configuration.• Router capacity is constrained by the number and speed of line
cards inserted in each slot.
Source: www.cisco.com
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 4210 0 10 1 10 2Degree
10-1
100
101
102
103
Ban
dwid
th (G
bps)
15 x 1-port 10 GE
15 x 3-port 1 GE
15 x 4-port OC12
15 x 8-port FE
Technology constraint
Total Bandwidth
Router Technology ConstraintCisco 12416 GSR, circa 2002
high BW low degree
high degree low BW
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 43
bandwidth
degree1 16
10Gb
155Mb
256log/log
625Mb
2.5Gb
Technically feasible
160Gb
Technological advance
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 44
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+00 1.E+01 1.E+02 1.E+03 1.E+04Router Degree
Tota
l BW
Cisco 12416 GSR (c.2002)Cisco 7600 WAN Agg.Aggregated DSLBroadband CableAggregated Dialup
Abstracted Technologically Feasible Region
router models specialize by “role”
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 45
HostsEdges
Core
Heuristically Optimal Topology
High degree nodes are at the edges.
Sparse, mesh-like core of fast, low-degree routers.
High cost of links drives
traffic aggregation at network edge
Relatively uniform connectivity within
core.
Possibly high variability in
connectivity at edge.
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 46
SOX
U. Florida
U. So. Florida
Miss StateGigaPoP
WiscREN
SURFNet
MANLAN
NorthernCrossroads
Mid-AtlanticCrossroads
Drexel U.
NCNI/MCNC
MAGPI
UMD NGIX
Seattle
Sunnyvale
Los Angeles
Houston
DenverKansas
City Indian-apolis
Atlanta
Wash D.C.
Chicago
New York
OARNET
Northern Lights Indiana GigaPoP
MeritU. LouisvilleNYSERNet
U. Memphis
Great Plains
OneNet
U. Arizona
Qwest Labs
CHECS-NETOregon
GigaPoP
Front RangeGigaPoP
Texas Tech
Tulane U.
TexasGigaPoP
LaNetUT Austin
CENIC
UniNet
NISN
PacificNorthwestGigaPoP
U. Hawaii
PacificWave
TransPAC/APAN
Iowa St.
Florida A&MUT-SWMed Ctr.
SINetWPI
Star-Light
IntermountainGigaPoP
Abilene BackbonePhysical Connectivity
(as of August 2004)
0.1-0.5 Gbps0.5-1.0 Gbps1.0-5.0 Gbps5.0-10.0 Gbps
DREN
Jackson St.
NRENUSGS
U. So. Miss.
PSC
DARPABossNet
SFGP/AMPATH
Arizona St.
ESnet
GEANT
North TexasGigaPoP
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 47
Cisco 750XCisco 12008Cisco 12410
dc1
dc2
dc3
hpr
dc1dc3
hpr
dc2
dc1
dc1 dc2
hpr
hpr
SACOAK
SVL
LAX
SDG
SLOdc1
FRGdc1
FREdc1
BAKdc1
TUSdc1
SOLdc1
CORdc1
hprdc1
dc2
dc3
hpr
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)
AbileneLos Angeles
AbileneSunnyvale
The Corporation for Education Network Initiatives in California (CENIC) acts as ISP for the state's colleges and universitieshttp://www.cenic.org
Like Abilene, its backbone is a sparsely-connected mesh, with relatively low connectivity and minimal redundancy.• no high-degree hubs?• no Achilles’ heel?
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 48
Router Deployment: Abilene and CENIC
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+00 1.E+01 1.E+02 1.E+03Router Degree
Tota
l BW
(Mbp
s)
Abilene T640sJuniper T640 Feasible RegionCENIC 12410sCisco 12410 Feasible RegionCENIC 12008sCisco 12008 Feasible RegionCENIC 750XsCisco 750X Feasible Region
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 49
AT&T Router Deployment (c.2003)
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+00 1.E+01 1.E+02 1.E+03 1.E+04Router Degree
Tota
l Ban
dwid
th
ISP 'Core' RoutersCore Router Config RegionISP 'High Speed Access' RoutersISP 'Low Speed Access' RoutersAccess Router Config Region
core routers
“low speed”access routers
“high speed”access routers
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 50
HostsEdges
Core
Heuristically Optimal Topology
High degree nodes are at the edges.
Sparse, mesh-like core of fast, low-degree routers.
High cost of links drives
traffic aggregation at network edge
Relatively uniform connectivity within
core.
Possibly high variability in
connectivity at edge.
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 51
A Closer Look Router-Level Measurement Data• Rocketfuel Project: Higher fidelity maps of individual ISPs• Shows that core routers do not follow a power law distribution
100 101 102100
101
102
103
104
Node Degree
Nod
e R
ank
100 101 102 103100
101
102
103
104
Node DegreeN
ode
Ran
k
Node Degree Distribution for AS 7018
Source: Rocketfuel
Pansiot and Grad datain Faloutsos (1999)
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 52
4
Source: Rocketfuel
all routersaccess routersbackbone routers
Node Degree Distribution for AS 7018
100 101 102 103
Node Degree100
101
102
103
104
Nod
e R
ank
A Closer Look Router-Level Measurement Data
100 101 102100
101
102
103
104
Node Degree
Nod
e R
ank
Again, traceroute measurement data requires careful scrutiny
• Rocketfuel Project: Higher fidelity maps of individual ISPs• Shows that core routers do not follow a power law distribution
Nonetheless, power laws in aggregate connectivity are plausible.
High variabilityis toward the network edge.
Pansiot and Grad datain Faloutsos (1999)
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 53
Rocketfuel: Interpretation, Validation, Augmentation
• application of “first principles” (e.g. router technology)• alias resolution: discovery of duplicate nodes• new graph annotation methods
AS 7018 9261 total nodes640 core nodes156 duplicates (24%)484 unique core nodes
AS 1239 7043 total nodes673 core nodes215 duplicates (32%)458 unique core nodes
AS 7018: Austin, TX
ISP Points of Presence (POPs) have highly organized structure
(simplicity, hierarchy, redundancy)
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 54node degree
101
101
102
100
node
rank
Preferential Attachment
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 55
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
DenverKansas
City 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
TexasGigaPo
P
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
0.1-0.5 Gbps0.5-1.0 Gbps1.0-5.0 Gbps5.0-10.0 Gbps
Abilene-inspired
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 56
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
dwid
th (M
bps)
kBxts
BBx
ijrkjikij
ji jijiij
∀≤
=
∑
∑ ∑
∈
,..
maxmax
:,
, ,α
α
Step 3: Compute max flow
Bi
Bj
xij
Step 2: Compute traffic demand
jiij BBx ∝
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 57
PAHeuristicallyOptimized
Structure Determines Performance
P(g) = 1.19 x 1010P(g) = 1.13 x 1012
102
101
100
10-1
10-2 10
0101
Degree
102
101
100
10-1
10-2 10
0101
Degree
P(g) = 3.95 x 1011
102
101
Ach
ieve
d B
W (G
bps)
100
10-1
10-2 10
0101
Degree
Abilene-inspired
1
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 58
Structural Metric
• Introduced in SIGCOMM’04• 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
iddgs ∑=,
)(Define the metric (di = degree of node i)
max
)()(s
gsgS =
We can renormalize so that 0 ≤ S(g) ≤ 1
where smax has the largest value of s(g) among all graphs g having the same degree distribution.
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 59
Properties of the S(g) metric
• Captures all of the information in degree correlationstatistics [Dorogovtsev and Mendes, 2003]
• Closely related to notion of assortativity [Newman, 2002]• Graphs with high-S(g) have certain self-similar
properties as defined by notions of rewiring, coarse graining, trimming
• For graphs resulting from probabilistic construction (e.g. PLRG/GRG), LogLikelihood (LLH) ∝ S(g)
Relevant Interpretation: How likely is a particular graph (having given degree sequence) to arise at random?
For details:L. Li, D. Alderson, J.C. Doyle, W. Willinger. Toward a Theory of Scale-Free Networks: Definition, Properties, and Implications. Internet Mathematics, In Press (2006).
see also the talk by Lun Li on Thurs P
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 60
PA PLRG
Heuristically Optimal Abilene-inspired Sub-optimal
Networks with the Same Degree Sequence
102
101
Ran
k
100
100 101 Degree
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 61
PA PLRG/GRGHOT Abilene-inspired Sub-optimal
smaxS(g) = 1P(g) = 1.08 x 1010
0.2 0.4 0.6 0.8 1 Relative Likelihood
1010
1011
1012
Per
fom
ance
(bps
)
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 62
PA PLRG/GRGHOT Abilene-inspired Sub-optimal
smaxS(g) = 1P(g) = 1.08 x 1010
???
0.2 0.4 0.6 0.8 1 Relative Likelihood
1010
1011
1012
Per
fom
ance
(bps
)
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 63
PA
HOT
Worst case = low-degree core router
Worst case = highdegree central hub
Response to router attack
real Tier-1 ISPs typically only ~10% loaded
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 64
MESSAGE #4: importance of model validation
• Descriptive modeling that replicates statistical features is no more than an exercise in “data fitting”
• Emphasis on “closing the loop” (using complementary measurements and domain expertise)
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 65
• The “nodes” and “links” are physical things that have hard constraints (technology).
• ISPs are constrained in what they can afford to build, operate, and maintain (economics).
• Decisions of ISPs are driven by objectives (performance) and reflect tradeoffs between what is feasible and what is desirable (heuristic optimization)
• Many important questions (robustness) only make sense in the context of the broader system (protocol stack)
PITFALL: Emphasis on power laws– “Full of sound and fury, signifying nothing?”
(Strogatz)– Power laws as “more normal than Normal” (ask
Summary: What “Matters” For Router-Level Topologies?
D. Alderson, Caltech 2006 ISMA Workshop on Internet Topology 66
http://hot.caltech.edu/topology.html• L. Li, D. Alderson, J.C. Doyle, W. Willinger. Toward a Theory of Scale-Free
Networks: Definition, Properties, and Implications. Internet Math. In Press (2006).
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• D. Alderson, W. Willinger, L. Li, and J. Doyle. The Role of Optimization in Understanding and Modeling Internet Topology. In Telecommunications Planning: Innovations in Pricing, Network Design and Management. S. Raghavan and G. Anandlingham, eds. Springer, 2005.
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