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