15-744: Computer Networking L-14 Network Topology

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

Internet topologies

AT&T

SPRINTMCI

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

Abilene Internet2 Backbone

9

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

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

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

13

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)

15

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

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…..

17

Outline


• Power Laws



18

Power laws in AS level topology

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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

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

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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

22

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

23

2

11

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 / ∑ kj

24

0.5

0.5 0.25

0.5 0.25

new nodeexisting node

may contain multi-edges, self-loops

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

25

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

26

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

27

Outline


• Power Laws



28

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

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100

101

102

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

30

Cisco 12416 GSR, circa 2002high 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 Feasibility

core technologies

edge technologies

older/cheaper technologies

Source: Cisco Product Catalog, June 2002 31

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.

33

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)

kBxts

BBx

ijrkjikij

ji jijiij

,..

maxmax

:,

, ,

Step 3: Compute max flow

Bi

Bj

xij

Step 2: Compute traffic demand

jiij BBx

34

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?

j

connectedji

iddgL ,

)(Define the metric (di = degree of node i)

35

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

36

PA PLRG/GRGHOT

Structure Determines Performance

P(g) = 1.19 x 1010 P(g) = 1.64 x 1010 P(g) = 1.13 x 1012

37

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 generated by random processes!

38

Outline


• Power Laws



39

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)

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

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

50

Graph Reproduction

51

Other Useful Tools

• Rocketfuel• Biases?

• RouteViews

52

53

• Faloutsos3 (Sigcomm’99)

• frequency vs. degree

Power Laws

topology from BGP tables of 18 routers54



Power Laws




Power Laws


• Faloutsos• frequency vs.

degree• empirical ccdf

P(d>x) ~ x-

Power Laws

57

Power Laws



• empirical ccdf P(d>x) ~ x-

α ≈1.15

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Documents

15-744: Computer Networking L-14 Network Topology