Basic epidemiologyof complex networks

Sergey Dorogovtsev

University of Aveiro and Ioffe Institute

Avalanche collapse in interdependent networks,

Baxter, S.D., Goltsev, and Mendes,

Phys. Rev. Lett. 109, 248701 (2012);

arXiv:1207.0448

Solid state physics vs science of networks:

Researcher: “I have a dream” —

(object – new material) =⇒ device or technol.process

⇓ ⇑understanding → · · · · · · · · →

⇑ ⇓(object – new network) =⇒ algorithm

Google PageRank

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The big data problem

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Lords of the world: intellect & information

The big data problem.

Privacy is actually reduced to the big data problem.

90% of the data in the world has been generated in thepast two years.

Google invested $1.6bn on building data centres just inthe three months from April to June.

Technology firms to spend $140bn on building new datacentres this year.

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Progress: 1997

Google’s servers in 1997(arXiv:1207.0448) 5 / 28

Progress: 2001

Google’s servers in 2001(arXiv:1207.0448) 6 / 28

Progress: now

Google’s server room in council Bluffs, Iowa.(arXiv:1207.0448) 7 / 28

A central cooling plant in Google’s data centre in Georgia.

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Estimating their share in the world’seconomy

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Lords of the world: intellect & information

The big data problem.

Privacy is actually reduced to the big data problem.

90% of the data in the world has been generated in thepast two years.

Google invested $1.6bn on building data centres just inthe three months from April to June.

Technology firms to spend $140bn on building new datacentres this year.

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Information:

Automated processing of big data.

Software is more expensive than hardware.

Expensive writing the code.

2012 is the first year when both Apple and Google spentmore on patent wars than on innovations.

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Mapping the global social network

Vertices - 94 entity types.

Edges - 164 relationship types.

- a huge coloured graph (edges and vertices are ofdifferent kinds).

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Interdependent networks:

Networks of networks

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Remaining giant cluster:

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The SIS model without approximationsOn complex nets, the SIS model 6= contact process.

Deroulers & Monasson (2003), the complete graph of 100 nodes(arXiv:1207.0448) 16 / 28

The SIS on an individual graph,(but not on a statistical ensemble!)

The evolution equation neglecting correlations betweeninfective and susceptible nodes:

dρi (t)

dt= −ρi (t) + λ[1− ρi (t)]

N∑j=1

Aijρj(t).

The steady state:

ρi =λ

∑j Aijρj

1 + λ∑

j Aijρj.

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The SIS on an annealed network

The evolution equation neglecting correlations betweeninfective and susceptible nodes:

dρk(t)

dt= −ρk(t)+λ[1−ρk(t)]

∑k

kP(k)ρk(t)/∑

q

qP(q).

The steady state:

ρk =kλ

∑k kP(k)ρk(t)/

∑q qP(q)

1 + kλ∑

k kP(k)ρk(t)/∑

q qP(q).

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ρi =∑

Λ

c(Λ)fi (Λ).

c(Λ) = λ∑

Λ′

Λ′c(Λ′)N∑

i=1

fi (Λ)fi (Λ′)

1 + λ∑

Λ̃ Λ̃c(Λ̃)fi (Λ̃).

If τ ≡ λΛ1−1�1, then

ρ ≡N∑

i=1

ρi/N ≈ α1τ,

α1 = [N∑

i=1

fi (Λ1)]/[NN∑

i=1

f 3i (Λ1)].

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Localized and delocalized eigenvectors:

Normalization:∑N

i f 2i (Λ) = 1

The inverse participation ratio:

IPR(Λ) ≡N∑

i=1

f 4i (Λ).

A delocalized state: IPR(Λ)N→∞→ 0.

A localized state: IPR(Λ)N→∞→ const.

It depends on a particular network realization.

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A delocalized principal eigenvector: fi (Λ) = O(1/√

N),so

α1 = O(1)

A localized principal eigenvector:

α1 = O(1/N)

— the disease is localized on a finite number of verticesin contrast to “localization” within k-cores (a finitefraction of vertices), see Kitsak (2010), Castellano andPastor-Satorras (2012).

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Unweighted networks:

The first iteration (g(0)=1):

Λ(1)1 =

1

〈q2〉N∑

i ,j

qiAijqj = ΛMF +〈q〉σ2r

〈q2〉,

IPRMF = 〈q4〉/[N〈q2〉2] ∼ O(1/N)

where r is the Pearson coefficient, ΛMF≡〈q2〉/〈q〉.So assortative degree–degree correlations increase Λ1 anddecrease λc .

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Weighted and unweighted real-world nets:

(a) astro-phys (upper) and cond-mat-2005 (lower) weighted

networks. (b) Karate-club network (unweighted). The lower curve

accounts for only the eigenstate Λ1. The most upper curve is the

“exact” ρ.(arXiv:1207.0448) 23 / 28

An uncorrelated scale-free net:

A scale-free network of 105 vertices generated by the static model

with γ = 4, 〈q〉 = 10. (b) Zoom of the prevalence at λ near

λc = 1/Λ1. I, II eigenvectors are localized, III is delocalized.(arXiv:1207.0448) 24 / 28

A Bethe lattice with a hub:

Find Λ1 if B = k − 1 is the branching and then Bsubstitute with 〈B〉.The localization of the principal eigenvector at a hubwith qmax occurs if

Λ1 = qmax/√

qmax − B ≥ Λd >≈ ΛMF

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Uncorrelated scale-free networks:

The principal eigenvector is localized if γ > 5/2

The principal eigenvector is delocalized if 2 < γ ≤ 5/2(but notice loops!)

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Conclusion

If the principal eigenvector is localized,then below the endemic epidemic threshold λc ,

the disease localized on a finite number of verticesslowly decays with time.

The decay time diverges at the epidemic threshold.

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Lectures on

Complex Networks

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