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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