Scale-free networks
Péter Kómár
Statistical physics seminar
07/10/2008
Elements of graph theory I.
A graph consists:
 vertices
 edges
Edges can be:
 directed/undirected
 weighted/non-weighted
 self loops
Non-regular
graph
 multiple edges
2
Elements of graph theory II.
Degree of a vertex:
 the number of edges
going in and/or out
Diameter of a graph:
 distance between the
farthest vertices
Density of a graph:
 sparse
 dense
3
Networks around us I.
Internet:
 routers
 cables
WWW:
 HTML pages
 hyperlinks
Social networks:
 people
 social relationship
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Networks around us II.
Transportation systems:
 stations / routes
 routes / stations
Nervous system:
 neurons
 axons and dendrites
Biochemical pathways:
 chemical substances
 reactions
5
Real networks
Properties:
 Self-organized structure
 Evolution in time
(growing and varying)
 Large number of vertices
 Moderate density
 Relatively small diameter
(Small World phenomenon)
 Highly centralized subnetworks
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Random networks
Measuring real networks:
 Relevant state-parameters
 Evolution in time
Creating models:
 Analytical formulas
 Growing phenomenon
Checking:
 ‘Raising’ random networks
 Measuring
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Scale-free property
1999. A.-L. Barabási, R. Albert
 measured the vertex degree
distribution



P
k

k
→ power-law tail:
 movie actors:  actors  2.3  0.1
 www  2.1  0.1
 www:
 US power grid:  power  4
actors
www
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A.-L. Barabási, R. Albert (1999) ‘Emergence of Scaling in Random Networks’, Science Vol. 286
Small diameter
2000. A.-L. Barabási, R. Albert
 measured the diameter of a HTML graph
 325 729 documents, 1 469 680 links
 found logarithmic
dependence:
  0.35  2.06 log N 
 ‘small world’
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A.-L. Barabási, R. Albert, H. Jeong (2000)
‘Scale-free characteristics of random networks: the topology of the wold-wide web’, Physica A, Vol. 281 p. 69-77
Erdős-Rényi graph (ER)
(1960. P. Erdős, A. Rényi)
Construction:
 N vertices
 probability of each edge: pER
Properties:
 pER ≥ 1/N →
→ Asympt. connected
 degree distribution:
Poisson (short tail)
 not centralized
 small diameter
pER =
6∙10-4
N=104
10-3
1.5∙10-3
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A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187
ER graph example
11
Small World graph (WS)
(1998. D.J. Watts, S.H. Strogatz)
Construction:
 N vertices in sequence
 1st and 2nd neighbor edges
 rewiring probability: pWS
Properties:
 pWS = 0 → clustered,   N
 0 < pWS < 0.01 → clustered
→ small-world propery
 pWS = 1 → not clustered,   ln N
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A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187
WS graph example
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ER graph - WS graph
WS
ER
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Barabási-Albert graph (BA)
New aspects:
 Continuous growing
 Preferential attachment
m0 = 3
m =2
Construction:
 m0 initial vertices
 in every step:
+1 vertex with m edges
 P(edge to vertex i) ~ degree of i
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A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187
Barabási-Albert graph II.
Properties:
 Power-law distribution
of degrees: Pk   k 
  2.9  0.1
 Stationary scale-free state
 Very high clustering
 Small diameter
5
7
3
1 = m0 = m
N = 300 000
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A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187
BA graph example
17
ER graph – BA graph
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Mean-field approximation I.
Time dependence of ki (continuous):
ki
ki
dki
ki
 m

 m   ki   m 
2mt 2t
dt
kj
probability of an
edge to ith vertex
 solution:
t
ki t   m
 ti



1
j
ki(t)
~ t 1/2
2
time of occurrence
of the ith vertex
ti
t
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A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187
Mean-field approximation II.
Distribution of degrees:
1

m2t  
 t  2
Pk   Pki t   k   P ti  2   ki t   m  
ti  


k


 
1
 Distribution of ti : Pti  
 C t 
m0  t

m 2t
m 2t 
Pk   ...  1  P ti  2   1  2
k m0  t 
k 

 Probability density:
2
d
2m t 1
pk  
Pk  
dk
m0  t k 3
 3
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A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187
Without preferential attachment
Uniform growth:

dki
m
m0  t  1 


 ki t   m1  ln
dt m0  t  1
m0  ti  1 

Exponential degree
distribution:
 k
pk   ...  exp   
 m
p(k)
exponential
scale-free
k
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A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187
Without growth
Construction:
 Constant # of vertices
 + new edges with
preferential attachment
t=N
N=10 000
5N
40N
Properties:
 At early stages
→ power-law scaling
 After t ≈ N2 steps
→ dense graph
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A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187
Conclusion
Power-law = Growth + Pref. Attach.
Varieties




k

k
 Non-linear attachment probability:
→ affects the power-law scaling
 Parallel adding of new edges →   ln 3 / ln 2
 Continuously adding edges (eg. actors)
→ may result complete graph
 Continuous reconnecting (preferentially)
→ may result ripened state
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A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187
Network research today
Centrality
Adjacency matrix
Spectral density
Attack tolerance
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A.-L. Barabási, R. Albert, ‘Statistical Mechanics of Complex Networks’, arXiv:cond-mat/0106096 v1 6 Jun 2001
Thank you for the attention!
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ER – WS – BA
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