Workshop on Random Graphs and their applications: доклад Сергея дороговцева

1. Basic epidemiology
of 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

2. 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
(arXiv:1207.0448)
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3. The big data problem
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4. 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 the
past two years.
Google invested $1.6bn on building data centres just in
the three months from April to June.
Technology firms to spend $140bn on building new data
centres this year.
(arXiv:1207.0448)
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5. Progress: 1997
Google’s servers in 1997
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6. Progress: 2001
Google’s servers in 2001
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7. Progress: now
Google’s server room in council Bluffs, Iowa.
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8. A central cooling plant in Google’s data centre in Georgia.
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9. Estimating their share in the world’s
economy
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10. 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 the
past two years.
Google invested $1.6bn on building data centres just in
the three months from April to June.
Technology firms to spend $140bn on building new data
centres this year.
(arXiv:1207.0448)
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11. 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 spent
more on patent wars than on innovations.
(arXiv:1207.0448)
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12. Mapping the global social network
Vertices - 94 entity types.
Edges - 164 relationship types.
- a huge coloured graph (edges and vertices are of
different kinds).
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13. Interdependent networks:
Networks of networks
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14. (arXiv:1207.0448)
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15. Remaining giant cluster:
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16. The SIS model without approximations
On complex nets, the SIS model = contact process.
Deroulers & Monasson (2003), the complete graph of 100 nodes
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17. The SIS on an individual graph,
(but not on a statistical ensemble!)
The evolution equation neglecting correlations between
infective and susceptible nodes:
dρi (t)
= −ρi (t) + λ[1 − ρi (t)]
dt
N
Aij ρj (t).
j=1
The steady state:
ρi =
(arXiv:1207.0448)
λ j Aij ρj
.
1 + λ j Aij ρj
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18. The SIS on an annealed network
The evolution equation neglecting correlations between
infective and susceptible nodes:
dρk (t)
= −ρk (t)+λ[1−ρk (t)]
dt
kP(k)ρk (t)/
k
qP(q).
q
The steady state:
ρk =
(arXiv:1207.0448)
kλ k kP(k)ρk (t)/ q qP(q)
.
1 + kλ k kP(k)ρk (t)/ q qP(q)
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19. ρi =
c(Λ)fi (Λ).
Λ
N
c(Λ) = λ
Λ c(Λ )
i=1
Λ
fi (Λ)fi (Λ )
1+λ
Λ Λc(Λ)fi (Λ)
.
If τ ≡ λΛ1 −1 1, then
N
ρ≡
ρi /N ≈ α1 τ,
i=1
N
α1 = [
fi 3 (Λ1 )].
fi (Λ1 )]/[N
i=1
(arXiv:1207.0448)
N
i=1
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20. Localized and delocalized eigenvectors:
N 2
Normalization:
i fi (Λ) = 1
The inverse participation ratio:
N
fi 4 (Λ).
IPR(Λ) ≡
i=1
N→∞
A delocalized state: IPR(Λ) → 0.
N→∞
A localized state: IPR(Λ) → const.
It depends on a particular network realization.
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21. √
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 vertices
in contrast to “localization” within k-cores (a finite
fraction of vertices), see Kitsak (2010), Castellano and
Pastor-Satorras (2012).
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22. Unweighted networks:
The first iteration (g(0) =1):
(1)
Λ1
=
1
q2 N
qi Aij qj = ΛMF
i,j
q σ2r
+
,
q2
IPRMF = q 4 /[N q 2 2 ] ∼ O(1/N)
where r is the Pearson coefficient, ΛMF ≡ q 2 / q .
So assortative degree–degree correlations increase Λ1 and
decrease λc .
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23. 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.
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24. 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.
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25. A Bethe lattice with a hub:
Find Λ1 if B = k − 1 is the branching and then B
substitute with B .
The localization of the principal eigenvector at a hub
with qmax occurs if
Λ1 = qmax / qmax − B ≥ Λd >≈ ΛMF
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26. 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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27. Conclusion
If the principal eigenvector is localized,
then below the endemic epidemic threshold λc ,
the disease localized on a finite number of vertices
slowly decays with time.
The decay time diverges at the epidemic threshold.
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28. Lectures on
Complex Networks
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