Scale-free networks Péter Kómár Statistical physics seminar 07/10/2008.

ScaleScale--free networksfree networks

Péter KómárPéter Kómár

Statistical physics seminarStatistical physics seminar

07/10/200807/10/2008

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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

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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

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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

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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

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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

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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

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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

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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

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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,


(1998. D.J. Watts, S.H. Strogatz)(1998. D.J. Watts, S.H. Strogatz)

N

Nln

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WS graph exampleWS graph example

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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


m0 = 3m = 2

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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


kkP

1.09.2

1 = m0 = m

3

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N = 300 000

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BA graph exampleBA graph example

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ER graph – BA graphER graph – BA graph

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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

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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


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


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...


pp((kk))

kk

scale-freescale-free

exponentialexponential

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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


t = Nt = N

5N5N

40N40N

N=10 000N=10 000

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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


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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!

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ER – WS – BAER – WS – BA

Scale-free networks Péter Kómár Statistical physics seminar 07/10/2008.

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