15-744: Computer Networking

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15-744: Computer Networking

L-14 Network Topology

Sensor Networks

• Structural generators

• Power laws

• HOT graphs

• Graph generators

• Assigned reading

• On Power-Law Relationships of the Internet

Topology

• A First Principles Approach to Understanding

the Internet’s Router-level Topology

2

Outline

• Motivation/Background

• Power Laws

• Optimization Models

• Graph Generation

3

Why study topology?

• Correctness of network protocols typically

independent of topology

• Performance of networks critically

dependent on topology

• e.g., convergence of route information

• Internet impossible to replicate

• Modeling of topology needed to generate

test topologies

4

2

Internet topologies

AT&T

SPRINT MCI

AT&T

MCI SPRINT

Router level Autonomous System (AS) level

5

More on topologies..

• Router level topologies reflect physical connectivity

between nodes

• Inferred from tools like traceroute or well known public measurement projects like Mercator and Skitter

• AS graph reflects a peering relationship between two

providers/clients

• Inferred from inter-domain routers that run BGP and publlic

projects like Oregon Route Views

• Inferring both is difficult, and often inaccurate

6

Hub-and-Spoke Topology

• Single hub node

• Common in enterprise networks

• Main location and satellite sites

• Simple design and trivial routing

• Problems

• Single point of failure

• Bandwidth limitations

• High delay between sites

• Costs to backhaul to hub

7

Simple Alternatives to Hub-and-Spoke

• Dual hub-and-spoke • Higher reliability

• Higher cost

• Good building block

• Levels of hierarchy

• Reduce backhaul cost

• Aggregate the bandwidth

• Shorter site-to-site delay …

8

3

Abilene Internet2 Backbone

9

SOX

SFGP/

AMPATH

U. Florida

U. So. Florida

Miss State

GigaPoP

WiscREN

SURFNet

Rutgers U.

MANLAN

Northern

Crossroads

Mid-Atlantic

Crossroads

Drexel U.

U. Delaware

PSC

NCNI/MCNC

MAGPI

UMD NGIX

DARPA

BossNet

GEANT

Seattle

Sunnyvale

Los Angeles

Houston

Denver

Kansas

City Indian-

apolis

Atlanta

Wash

D.C.

Chicago

New York

OARNET

Northern Lights

Indiana GigaPoP

Merit U. Louisville

NYSERNet

U. Memphis

Great Plains

OneNet Arizona St.

U. Arizona

Qwest Labs

UNM

Oregon

GigaPoP

Front Range

GigaPoP

Texas Tech

Tulane U.

North Texas

GigaPoP

Texas

GigaPoP

LaNet

UT Austin

CENIC

UniNet

WIDE

AMES NGIX

Pacific

Northwest

GigaPoP U. Hawaii

Pacific

Wave

ESnet

TransPAC/APAN

Iowa St.

Florida A&M UT-SW

Med Ctr.

NCSA

MREN

SINet

WPI

StarLight

Intermountain

GigaPoP

Abilene Backbone

Physical Connectivity (as of December 16, 2003)

0.1-0.5 Gbps

0.5-1.0 Gbps

1.0-5.0 Gbps

5.0-10.0 Gbps

10

Points-of-Presence (PoPs)

• Inter-PoP links

• Long distances

• High bandwidth

• Intra-PoP links

• Short cables between

racks or floors

• Aggregated bandwidth

• Links to other

networks

• Wide range of media

and bandwidth

Intra-PoP

Other networks

Inter-PoP

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Deciding Where to Locate Nodes and Links

• Placing Points-of-Presence (PoPs)

• Large population of potential customers

• Other providers or exchange points

• Cost and availability of real-estate

• Mostly in major metropolitan areas

• Placing links between PoPs

• Already fiber in the ground

• Needed to limit propagation delay

• Needed to handle the traffic load

12

4

Trends in Topology Modeling

Observation

• Long-range links are expensive

• Real networks are not random,

but have obvious hierarchy

• Internet topologies exhibit

power law degree distributions

(Faloutsos et al., 1999)

• Physical networks have hard technological (and economic)

constraints.

Modeling Approach

• Random graph (Waxman88)

• Structural models (GT-ITM

Calvert/Zegura, 1996)

• Degree-based models replicate

power-law degree sequences

• Optimization-driven models topologies consistent with design

tradeoffs of network engineers

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Waxman model (Waxman 1988)

• Router level model

• Nodes placed at random

in 2-d space with

dimension L

• Probability of edge (u,v):

• ae^{-d/(bL)}, where d is

Euclidean distance (u,v), a

and b are constants

• Models locality

14

v

u d(u,v)

Real world topologies

• Real networks exhibit

• Hierarchical structure

• Specialized nodes (transit, stub..)

• Connectivity requirements

• Redundancy

• Characteristics incorporated into the

Georgia Tech Internetwork Topology Models

(GT-ITM) simulator (E. Zegura, K.Calvert

and M.J. Donahoo, 1995)

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Transit-stub model (Zegura 1997)

• Router level model

• Transit domains

• placed in 2-d space

• populated with routers

• connected to each other

• Stub domains

• placed in 2-d space

• populated with routers

• connected to transit

domains

• Models hierarchy

16

5

So…are we done?

• No!

• In 1999, Faloutsos, Faloutsos and

Faloutsos published a paper, demonstrating

power law relationships in Internet graphs

• Specifically, the node degree distribution

exhibited power laws

That Changed Everything…..

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Outline


• Power Laws



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Power laws in AS level topology

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• Faloutsos3 (Sigcomm’99)

• frequency vs. degree

Power Laws

topology from BGP tables of 18 routers 20

6



Power Laws




Power Laws


• Faloutsos

• frequency vs.

degree

• empirical ccdf

P(d>x) ~ x-a

Power Laws

23

Power Laws


• frequency vs.

degree

• empirical ccdf

P(d>x) ~ x-a

1.15

24

7

GT-ITM abandoned..

• GT-ITM did not give power law degree

graphs

• New topology generators and explanation

for power law degrees were sought

• Focus of generators to match degree

distribution of observed graph

25

Inet (Jin 2000)

• Generate degree sequence

• Build spanning tree over nodes

with degree larger than 1,

using preferential connectivity

• randomly select node u not in

tree

• join u to existing node v with

probability d(v)/ d(w)

• Connect degree 1 nodes using

preferential connectivity

• Add remaining edges using

preferential connectivity 26

Power law random graph (PLRG)

• Operations • assign degrees to nodes drawn from power law distribution

• create kv copies of node v; kv degree of v.

• randomly match nodes in pool

• aggregate edges

may be disconnected, contain multiple edges, self-loops

• contains unique giant component for right choice of parameters

27

2

1 1

Barabasi model: fixed exponent

• incremental growth

• initially, m0 nodes

• step: add new node i with m edges

• linear preferential attachment

• connect to node i with probability

(ki) = ki / kj

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0.5

0.5 0.25

0.5 0.25

new node existing node

may contain multi-edges, self-loops

8

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

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Preferential Attachment Expected Degree Sequence

Does Internet graph have these properties?

• No…(There is no Memphis!)

• Emphasis on degree distribution - structure

ignored

• Real Internet very structured

• Evolution of graph is highly constrained

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Problem With Power Law

• ... but they're descriptive models!

• No correct physical explanation, need an

understanding of:

• the driving force behind deployment

• the driving force behind growth

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Outline


• Power Laws



32

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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

33 10

010

110

2

Degree10

-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 Constraint

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Cisco 12416 GSR, circa 2002 high BW low degree

high degree low BW

approximate

aggregate

feasible region

Aggregate Router Feasibility

core technologies

edge technologies

older/cheaper

technologies

Source: Cisco Product Catalog, June 2002 35 Rank (number of users)

Co

nn

ecti

on

Sp

eed

(M

bp

s)

1e-1

1e-2

1

1e1

1e2

1e3

1e4

1e2 1 1e4 1e6 1e8

Dial-up

~56Kbps

Broadband

Cable/DSL

~500Kbps

Ethernet

10-100Mbps

Ethernet

1-10Gbps

most users

have low speed

connections

a few users have very

high speed

connections

high

performance

computing academic

and corporate

residential and

small business

Variability in End-User Bandwidths

36

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Heuristically Optimal Topology

Mesh-like core of fast, low degree routers

High degree nodes

are at the edges.

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Comparison Metric: 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)

Step 3: Compute max flow

Bi

Bj

xij

Step 2: Compute traffic demand

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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

• 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?

Define the metric (di = degree of

node i)

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Lmaxl(g) = 1P(g) = 1.08 x 1010

P(g) Perfomance (bps)

PA PLRG/GRG HOT Abilene-inspired Sub-optimal

0 0.2 0.4 0.6 0.8 1

10 10

10 11

10 12

l(g) = Relative Likelihood 40

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PA PLRG/GRG HOT

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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Summary Network Topology

• Faloutsos3 [SIGCOMM99] on Internet topology • Observed many “power laws” in the Internet structure

• Router level connections, AS-level connections, neighborhood sizes

• Power law observation refuted later, Lakhina [INFOCOM00]

• Inspired many degree-based topology generators • Compared properties of generated graphs with those of measured graphs

to validate generator

• What is wrong with these topologies? Li et al [SIGCOMM04]

• Many graphs with similar distribution have different properties

• Random graph generation models don’t have network-intrinsic meaning

• Should look at fundamental trade-offs to understand topology • Technology constraints and economic trade-offs

• Graphs arising out of such generation better explain topology and its properties, but are unlikely to be generted by random processes!

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Outline


• Power Laws



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Graph Generation

• Many important topology metrics

• Spectrum

• Distance distribution

• Degree distribution

• Clustering…

• No way to reproduce most of the important

metrics

• No guarantee there will not be any other/

new metric found important

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dK-series approach

• Look at inter-dependencies among topology

characteristics

• See if by reproducing most basic, simple,

but not necessarily practically relevant

characteristics, we can also reproduce

(capture) all other characteristics, including

practically important

• Try to find the one(s) defining all others

0K

Average degree <k>

1K

Degree distribution P(k)

2K

Joint degree distribution P(k1,k2)

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3K

“Joint edge degree” distribution P(k1,k2,k3)

3K, more exactly

4K Definition of dK-distributions

dK-distributions are degree correlations

within simple connected graphs of size d

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Nice properties of properties Pd

• Constructability: we can construct graphs

having properties Pd (dK-graphs)

• Inclusion: if a graph has property Pd, then

it also has all properties Pi, with i < d (dK-

graphs are also iK-graphs)

• Convergence: the set of graphs having

property Pn consists only of one element, G

itself (dK-graphs converge to G)

Rewiring

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15-744: Computer Networking

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