Scale Free Networks

Robin Coope

April 4 2003

Abert-László Barabási, Linked (Perseus, Cambridge, 2002).

Réka Albert and AL Barabási,Statistical Mechanics of Complex Networks, Rev. Mod. Phys 74 (1) 2002

Réka Albert and AL Barabási, Topology of Evolving Networks:Local Events and Universality, Phys. Rev. Lett. 85 (24) 2000

Motivation

• Many networks, (www links, biochemical & social networks) show P(k) ~ k- scale free behaviour.

• Classical theories predict P(k) ~ exp(-k).• Something must be done!

Properties of Networks• Small World Property

kN

lrand ln

ln

• Clustering – “Grade Seven Factor”

1

2

ii

ii kk

EC

• Degree – Distribution of # of links

Random Graphs (Erdõs-Rényi )

!)(

kekP

k

k

kN

lrand ln

ln

N

kpCrand

kNk ppk

NN

11

1

Predictions of Random Graphs

Path Length vs. Theory Clustering vs. Theory

What About Scale Free Random Graphs?

• Restrict distributions to P(k) ~ k-

• Still doesn’t make good predictions

• Conclusion: Network connections are not random!

Average Path Length

Measured Network Values

Measured Network Values

Comparison

Evolution of a SF Network

7

7

3

2 2

2

2

25

2

4

Charleton Heston > 150 links

Nancy Kerrigan ~ 1 link

Assumptions for Scale Free Model

• Networks are open – they add and lose nodes, and nodes can be rewired.

• Older nodes get more new links.• More popular nodes get more new links• Result: no characteristic nodes – Scale Free• Both growth and rewiring required.

jj

i

i

i

k

kpm

pm

t

k

)1(

1

)(

1. Addition of m new links with prob. p

jj

i

i

i

k

kqm

qm

t

k

)1(

1

)(

2. Rewiring of m links with prob. q

jj

i

i

i

k

kmqp

t

k

)1(

1)1(

)(

3. Add a new node with prob. (1-p-q)

Continuum Theory

pq

p

10

10

Avoid isolated links

jj

i

i

i

k

km

mqp

t

k

)1(

1)(

)(

Combined Equation

mmtqtkj

j 2)1()(tqpmt o )1()(

Time Dependency of system size and # of links

Initial Condition for connectivity of a node added at time ti: mtk ii

1),,(1),,()(),,(

1

mqpA

t

tmmqpAtk

mqpB

ii

11

12,,

qp

qmqpmqpA

m

qpqmmqpB

112,,

Solution

YOU MANIACS! YOU BLEW IT UP! DAMN

YOU! GOD DAMN YOU ALL TO HELL!!

Finding P(k)

tmqpAk

mqpAmtPkkP

mqpB

ii

),,(

1,,

1,,

11

12,,

qp

qmqpmqpA

Can get analytic solution for P(k) if:

1

1,,

1,,0

),,(

mqpB

mqpAk

mqpAm

Finding P(k)

tm

t

mqpAk

mqpAmktkP

mqpB

i

0

),,(

1,,

1,,1

mqpB

mqpB

mqpAkmqpB

mqpAmtm

kP

,,1

..

0

1,,,,

1,,1

tm

tP ii

0

1 k

ktkPkP i

)(

Finally…….

mqpmqpkkP ,,,,

1,,,,

1,,,,

mqpBmqp

mqpAmqp

where

And for fixed p,m:

mmppqq 21/1,1minmax

Regimes

m

0m

maxqq

maxqq

mm

slope21

As q -> qmax, distribution gets exponential.

Simulation Results

Experimental Results

07.3,68.31

,1,937.0

mp

93.7% new linksfor current actors 6.3% new actors

Implications – Attack Tolerance

• Robust. For <3, removing nodes does not break network into islands.

• Very resistant to random attacks, but attacks targeting key nodes are more dangerous.

Ma x

Clu

s te r

Siz

e P

ath

Leng

th

Implications

• Infections will find connected nodes.

• Cascading node failures a problem

• Treatment with novel strategies like targeting nodes for treatment - AIDS

• Protein hubs critical for cells 60-70%

• Biological complexity: # states ~2# of genes

Conclusion

• Real world networks show both power law and exponential behaviour.

• A model based on a growing network with preferential attachment of new links can describe both regimes.

• Scale free networks have important implications for numerous systems.