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.
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