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ScaleScale--free networksfree networks
Péter KómárPéter Kómár
Statistical physics seminarStatistical physics seminar
07/10/200807/10/2008
22
Elements of graph Elements of graph theory I.theory I.
A graph consists:A graph consists: vertices vertices edgesedges
Edges can be:Edges can be: directed/undirecteddirected/undirectedweighted/non-weightedweighted/non-weighted self loopsself loopsmultiple edgesmultiple edges
Non-regularNon-regulargraphgraph
33
Elements of graph Elements of graph theory II.theory II.
Degree of a vertex:Degree of a vertex: the number of edges the number of edges
going in and/or outgoing in and/or out Diameter of a graph:Diameter of a graph: distance between the distance between the
farthest verticesfarthest vertices Density of a graph:Density of a graph: sparsesparse densedense
44
Networks around us I.Networks around us I.
InternetInternet:: routersrouters cablescables
WWW:WWW:HTML pagesHTML pages hyperlinkshyperlinks
Social networks:Social networks: peoplepeople social relationshipsocial relationship
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Networks around us II.Networks around us II.
Transportation systems:Transportation systems: stations / routesstations / routes routes / stationsroutes / stations
Nervous system:Nervous system: neuronsneurons axons and dendritesaxons and dendrites
Biochemical pathways:Biochemical pathways: chemical substanceschemical substances reactionsreactions
66
Real networksReal networks
Properties:Properties: Self-organized structureSelf-organized structure Evolution in time Evolution in time
(growing and varying)(growing and varying) Large number of verticesLarge number of verticesModerate densityModerate densityRelatively small diameter Relatively small diameter
(Small World phenomenon)(Small World phenomenon)Highly centralized subnetworksHighly centralized subnetworks
77
Random networksRandom networks
Measuring real networks:Measuring real networks:Relevant state-parametersRelevant state-parameters Evolution in timeEvolution in time
Creating models:Creating models:Analytical formulasAnalytical formulasGrowing phenomenonGrowing phenomenon
Checking: Checking: ‘‘Raising’ random networksRaising’ random networksMeasuring Measuring
88
Scale-free propertyScale-free property
1999. A1999. A..-L. Barab-L. Barabási, R. Albertási, R. Albertmeasured the vertex degree measured the vertex degree
distributiondistribution→ power-law tail:→ power-law tail:
movie actors:movie actors:www:www:US power grid:US power grid:
kkP
1.03.2actors 1.01.2www
4power
A.-L. Barabási, R. Albert (1999) ‘Emergence of Scaling in Random Networks’, Science Vol. 286
actors
www
99
Small diameterSmall diameter
2000. A2000. A..-L. Barab-L. Barabási, R. Albertási, R. Albert measured the diameter of a HTML measured the diameter of a HTML
graphgraph 325 729 documents, 1 469 680 links325 729 documents, 1 469 680 links found logarithmic found logarithmic
dependence:dependence:
‘‘small world’small world’
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
Nlog06.235.0
1010
ErdErdős-Rényi ős-Rényi graph (ER)graph (ER)
Construction:Construction:NN vertices vertices probability of each edge: probability of each edge: ppERER
Properties:Properties: ppERER ≥ 1/N → ≥ 1/N →
→ Asympt. connected → Asympt. connected degree distribution:degree distribution:
Poisson (short tail)Poisson (short tail) not centralizednot centralized small diametersmall diameter
A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187
(1960. (1960. PP.. Erd Erdős, A. Rényi)ős, A. Rényi)
N=104pER = 6∙10-
4
10-
31.5∙10-
3
1111
ER graph exampleER graph example
1212
Small World graph (WS)Small World graph (WS)
Construction:Construction:NN vertices in sequence vertices in sequence 11stst and 2 and 2ndnd neighbor edges neighbor edges rewiring probability: rewiring probability: ppWSWS
Properties:Properties: ppWS WS = 0 → clustered,= 0 → clustered, 0 <0 < p pWS WS < 0.01 → clustered< 0.01 → clustered
→ small-world propery → small-world propery ppWS WS = 1 → not clustered,= 1 → not clustered,
A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187
(1998. D.J. Watts, S.H. Strogatz)(1998. D.J. Watts, S.H. Strogatz)
N
Nln
1313
WS graph exampleWS graph example
1414
ER graph - WS graphER graph - WS graph
WS ER
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BarabBarabáási-Albert graph si-Albert graph (BA)(BA)
New aspects:New aspects:Continuous growingContinuous growing Preferential attachmentPreferential attachment
Construction:Construction:mm00 initial vertices initial vertices in every step: in every step:
+1 vertex with +1 vertex with mm edges edges PP(edge to vertex (edge to vertex ii) ~ degree of ) ~ degree of ii
A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187
m0 = 3m = 2
1616
BarabBarabáási-Albert graph si-Albert graph II.II.
Properties:Properties: Power-law distribution Power-law distribution
of degrees:of degrees:
Stationary scale-free stateStationary scale-free stateVery high clusteringVery high clustering Small diameterSmall diameter
A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187
kkP
1.09.2
1 = m0 = m
3
57
N = 300 000
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BA graph exampleBA graph example
1818
ER graph – BA graphER graph – BA graph
1919
Mean-field Mean-field approximation I.approximation I.
Time dependence of Time dependence of kkii (continuous):(continuous):
solution:solution:
ii kmdt
dk
jj
i
k
km
t
k
mt
km ii
22
21
ii t
tmtk
probability of anprobability of anedge to iedge to ithth vertex vertex
time of time of occurrenceoccurrence of the i of the ithth vertex vertex
A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187
kkii(t(t))
ttttii
~ t ~ t 1/21/2
2020
Mean-field Mean-field approximation II.approximation II.
Distribution of degrees:Distribution of degrees:
Distribution of Distribution of tti i ::
Probability density:Probability density:
ktkPkP i
21
ii t
tmtk
2
2
k
tmtP i
tCtm
tP i
0
1
2
2
1...k
tmtPkP i tmk
tm
02
2
1
3
0
2 12
ktm
tmkP
dk
dkp
3
A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187
2121
Without preferential Without preferential attachmentattachment
Uniform growth:Uniform growth:
Exponential degreeExponential degree distribution: distribution:
10
tm
m
dt
dki
1
1ln1
0
0
ii tm
tmmtk
m
kkp exp...
A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187
pp((kk))
kk
scale-freescale-free
exponentialexponential
2222
Without growthWithout growth
Construction:Construction:Constant # of verticesConstant # of vertices+ new edges with + new edges with
preferential attachmentpreferential attachment Properties:Properties:At early stages At early stages
→ power-law scaling → power-law scalingAfter After tt ≈ ≈ NN2 2 steps steps
→ dense graph → dense graph
A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187
t = Nt = N
5N5N
40N40N
N=10 000N=10 000
2323
ConclusionConclusion
Power-law = Growth + Pref. Attach.Power-law = Growth + Pref. Attach. VarietiesVarietiesNon-linear attachment probability:Non-linear attachment probability:
→ affects the power-law scaling→ affects the power-law scaling Parallel adding of new edges → Parallel adding of new edges → Continuously adding edges (eg. actors)Continuously adding edges (eg. actors)
→ may result complete graph→ may result complete graphContinuous reconnecting (preferentially)Continuous reconnecting (preferentially)
→ may result ripened state→ may result ripened state
kk
2ln/3ln
A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187
2424
Network research todayNetwork research today
A.-L. Barabási, R. Albert, ‘Statistical Mechanics of Complex Networks’, arXiv:cond-mat/0106096 v1 6 Jun 2001
CentralityCentrality
Adjacency Adjacency matrixmatrix
Spectral Spectral densitydensity
Attack toleranceAttack tolerance
Thank you for the Thank you for the attention!attention!
2626
2727
ER – WS – BAER – WS – BA