Topologies of Complex NetworksFunctions vs. Structures

Lun LiAdvisor: John C. Doyle

Co-advisor: Steven H. LowCollaborators: David Alderson (NPS)

Walter Willinger (AT&T-labs Research)

• Incredibly large size• Dramatic growth, rapid and ongoing evolution• Multitude of measurement data (some are

biased and incomplete)

Challenges

• Structures always affect functions• Evaluate the performances of new regulations

that run on top of the structure.• A better design of complex networks

Importance

Trends

• Identify unifying large-scale properties– Power-law degree distribution

• Build universal models to match properties– Scale-free Networks

• Derive “Emergent properties” from models– “Achilles’ heel”, self-similar, generic…

Trends

• Identify unifying large-scale properties– Power-law degree distribution

• Build universal models to match properties– Scale-free Networks

• Derive “Emergent properties” from models– “Achilles’ heel”, Self-similar, generic

“New Science of Networks!”

Power Laws

Call D={d1, d2, …, dn } degree sequence of graph

Let di denote the degree of node i

ckdk where c>0 and α>0

k: rank of a degree

α: power-law tail index, 0<α≤2 in complex networks

10000

1000

100

10

1

Source: Faloutsos et al. (1999)

Degree

Ran

k

Power Laws

10000

1000

100

10

1

Source: Faloutsos et al. (1999)Most nodes have few connections

Call D={d1, d2, …, dn } degree sequence of graph

Let di denote the degree of node i A few nodes have lots of connections

ckdk where c>0 and α>0

)log()log()log( cdk k

Degree

Ran

k

Power-law and High Variability• For a sequence D,

– CV characterizes the variability of a degree sequence– Regular Graph (di=c), CV(D) = 0– ER random graph (Poisson), CV(D) = c– Some other random graphs (Exponential), CV(D) = c

• If D is Power-law (n∞), <2 , CV(D) = ∞• High variability significantly deviates from classical graphs• Power-law is discovered in many complex networks• Lead to pursue of universal theories to explain it.

Scale-Free (SF) Models

• Preferential Attachment (PA)Barabasi & Albert (1999)

– Growth by sequentially adding new nodes– New nodes connect preferentially to nodes

having more connections

Scale-Free (SF) Networks

• Reproduce power-law degree sequence• Generated by random process (PA, GRG,…)• Highly connected central “hubs”, which are

crucial to the system, “hold network together”– Achilles’ heel: fragile to specific attack

• Self-similar and fractal, Small-world properties… • SF networks have been suggested as

representative models of complex networks

PA GRG

However…

• Scale-free network theories are incomplete and in need of corrective actions. – Power laws are “more normal than Normal”…– Power laws are popular but not universal…– And not a “signature” of specific mechanisms

• Focus on network functions and structures

Functions Internet Router-level topology

s-metricStructures

Our Approach for Internet Topology• Consider the explicit design of the Internet

– Annotated network graphs (bandwidth)– Network Functions

• Carry expected traffic demand

– Constraints• Technological constraints• Economic limitations

– Heuristic optimized tradeoffs (HOT)• Maximize network function subject to constraints

Our Approach for Internet Topology• Consider the explicit design of the Internet

– Annotated network graphs (bandwidth)– Network Functions

• Carry expected traffic demand

– Constraints• Technological constraints• Economic limitations

– Heuristic optimized tradeoffs (HOT)

10K

100K

1M

10M

100M

1G

10G

100G

1000G

1

10 100 1000 10000

Total Router Degree (physical connections)

To

tal

Ro

ute

r B

and

wid

th (

bit

s/se

c)

Shared media at network edge (LAN, DSL, Cable, Wireless, Dial-up)

Corebackbone

High-end gateways

Older/cheapertechnology

Abstracted Technologically Feasible Region

Flow conservation in routers:Routers can either have a few

high-bandwidth connections, or many low bandwidth

connections.

Individual router models specialize in different bandwidth-degree combinations and therefore tend to used in different

regions of the network.

Rank (number of users)

Con

nect

ion

Spee

d (M

bps)

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 (2003)

High cost of links drives traffic

aggregation at network edge

Hosts

Edges

Core

Heuristically Optimal Topology

High degree nodes are at the edges.

Sparse, mesh-like core of fast, low-degree routers.

Relatively uniform low connectivity within core: carry high

bandwidth

high variability in connectivity at

edge:aggregate end

users

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

Optimization-based models• Core: Mesh-like, low

degree • Edge: High degree• From engineering design• Tradeoffs in constraints • Match the real Internet

SF models• Core: Hub-like, high

degree • Edge: Low degree• From random process• Ignore engineering details• Match aggregate

statistics

SF HOT

How to reconcile these two perspectives?

SF

PLRG/GRG

HOT

Abilene-inspired Sub-optimal

What are the key differences among these graphs?

•Functions

•Structures

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

dw

idth

(M

bp

s)

kBxts

BBxgPerf

ijrkjikij

ji jijiij

,..

maxmax)(

:,

, ,

Step 3: Compute max flow

Bi

Bj

xij

Step 2: Compute traffic demand

jiij BBx

SF HOT

Perf(g) = 1.19 x 1010

(bps)Perf(g) = 1.13 x 1012 (bps)

Engineering-based models• Core: Mesh-like, low

degree • Edge: High degree• From explicit design• Tradeoffs in constraints • High throughput• High router utilization• No Achilles’ Heel

Degree-based models• Core: Hub-like, high

degree • Edge: Low degree• From random process• Ignore engineering details• Low throughput• Low router utilization• Achilles’ Heel

SF HOT

Engineering-based models• Core: Mesh-like, low

degree • Edge: High degree• From explicit design• Tradeoffs in constraints • High throughput• High router utilization• No Achilles’ Heel

Degree-based models• Core: Hub-like, high

degree • Edge: Low degree• From random process• Ignore engineering details• Low throughput• Low router utilization• Achilles’ Heel

PA HOT

A Structural Approach

• s-metric

– Structural metric, depending only on the connectivity of a given graph not on the generation mechanism

– Not for a specific network

• High s(g) is achieved by connecting high degree nodes to each other

• Measures how “hub-like” the network core is

jji

iddgs

),(

)(

s and Joint Degree Distribution

• Joint Degree Distribution (JDD): p(k,k’) correlation between the degrees k, k’ of connected nodes– Degree distribution is a first order statistic– JDD is a second order statistic

• For a graph having degree sequence D, s is the aggregation of JDD

– Corollary: If two graphs have the same JDD, define a metric as the aggregation of the third order correlation.

Dkk

kkpkks',

)',('2

1

P(g) Perfomance (bps)

SFHOT

0 0.2 0.4 0.6 0.8 1

1010

1011

1012

S(g)

minmax

min)()(

ss

sgsgS

s-metric

• Structural metric, depending only on the connectivity of a given graph not on the generation mechanism

• Define the extent to which the graph is scale-free

• Differentiate graphs with the same highly variable degree sequence, among them:– smax graph is the one with the highest s value– smin graph is the one with the lowest s value

• smax – smin defines the graph diversity of a given degree sequence in the simple and connected graph space.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

smax

s-va

lue

smin

Variability of a degree sequence

Graph diversity

cv

Variability vs. graph diversity of a degree sequence

A tree with 100 nodes, 99 links

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

smax (SF is closed to smax)

s-va

lue

smin (HOT is closed to smin)

Variability vs. graph diversity of a degree sequence

A tree with 100 nodes, 99 links

Variability of a degree sequence

Graph diversity

cv

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

s-va

lue

chain

star

Variability of a degree sequence

Graph diversity

Low variability graphs

cv

Variability vs. graph diversity of a degree sequence

A tree with 100 nodes, 99 links

s and Assortativity r(g)

• For a given graph, assortativity is:

– r>0, assortative, high degree nodes connect to high degree nodes

– r<0, dissortative, high degree nodes connect to low degree nodes

– A popular metric to measure the degree correlation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

CV(D)

Almost all the simple, connected trees with high variable connectivity have negative assortativity

Ass

ort

ativ

ity

rmax

rmin

Variability vs. assortivity of a degree sequence

A tree with 100 nodes, 99 links

r(SF) = -0.42

r(HOT) = -0.46

Assortativity r(g)

• For a given graph, assortativity is:

Assortativity r(g)

• For a given graph, assortativity is:

• Normalization termsmax of unconstrained graphs:all the nodes connect themselves

Assortativity r(g)

• For a given graph, assortativity is:

• Normalization term

• Centering term

smax of unconstrained graph

Center of unconstrained graph

Assortativity r(g)

• For a given graph, assortativity is:

• Normalization term

• Centering term

• Background set is the unconstrained graph!

smax of unconstrained graph

Center of unconstrained graph

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

s-metric assortativity

smax

smin

rmax

rmin

CV(D) CV(D)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

s/smax assortativity

smin/smax

rmax

rmin

•Assortativity is a metric directed borrowed from classic graph theory

•It works well for the low variability case

•Extremely misleading for the high variability complex network

CV(D) CV(D)

smax (SF) and graph metrics

With s, we can quantitatively characterize the properties claimed in SF literature.

• Node Centrality– In smax graph, node centrality increases with degree

• Small-world phenomena– smax has lowest average shortest path

• Self similarity– smax graph remains smax by trimming, coarse graining,

highest connect motif• Generic

– smax graph is most likely to appear by GRG

Conclusions

FunctionsStructures

• The Internet• Functions vs.

Constraints• HOT vs. SF

•s-metric can highlight the difference (HOT vs SF)•s-metric measures graph diversity•s-metric has a rich connection to self-similarity, assortativity