Emergence of scaling in random networksAlbert-Laszlo Barabasi, Reka Albert
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Barabasi, Albert-Laszlo, and Reka Albert. ”Emergence ofscaling in random networks.” science 286.5439 (1999):509-512.
Number of citations: 29239 (Google scholar)
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Complex networks: vertices - edges
Genetic network: Proteins & genes - chemical interactionsNervous system: Nerve cells - axonsWorld Wide Web: HTML pages - links
First model: Random graph theory by Erdos and Renyi
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Power law
Self-organize into a scale-free stateP(k) = k−γ
Two features cause this scaling
GrowthPreferential attachment
Not present in existing network models
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Collaboration of movie actors (A)
γ = 2.3
World Wide Web (B)
γ = 2.1
Electrical power grid of the western US (C)
γ = 4
Citation patterns
γ = 3
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Erdos-Renyi (ER) model (random graph)
Watts-Strogatz (WS) model (small-world)
Probability of highly connected nodes decreases exponentiallyNo highly connected nodes
In contrast with power law
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Growth
ER and WS models are static: fixed number of nodes
However...
Actors network: new actors joinWWW: new webpages are createdCitation network: New publication are done
Networks expand!
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Preferential Attachment
The probability of an edge existing between any two nodes isthe same
However:
New actor more likely to be cast with an established actorNew webpage more likely to link to a popular webpageNew paper more likely to cite a well-known paper
The connections formed by a vertex are not arbitrary!
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The Model
Growth + preferential attachment = scale-free
Growth: each new vertex connects to m already existingvertices
Preferential attachment: π(ki ) = ki∑j kj
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The Model
Growth + preferential attachment = scale-free
Growth: each new vertex connects to m already existingvertices
Preferential attachment: π(ki ) = ki∑j kj
At time t:
m0 + t verticesmt edgesγ = 2.9± 0.1
P(k) independent of t
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Verification
No preferential attachment
P(k) = exp(−βk)→ no power-law
No growth
No stationary P(k)→ no power-law
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Preferential attachment: rich-get-richer
After enough time:
P(k) =2m2
k3
γ = 3, independent of m
Restricted to a single γ value
To get different γ values:
Use non-linear preferential attachmentChange a fraction of links to directed linksAdd edges between existing vertices
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Growth & preferential attachment: Common mechanisms inreal life networks
Even genetic networks may have these mechanisms
Can be able to explain social and economic systems
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Questions
Are the two features, growth & preferential attachment,realistic for the networks in real life? Does growth work onlyas explained?
Does the change in networks happen only through nodes?
Is the model complete? Can it be improved in any way? Whatwould be done?
Are there any phenomenon appearing in networks disregardedin this model?
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