Clarkson University

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Reuven Cohen
Tomer Kalisky
Alex Rozenfeld
Eugene Stanley Lidia Braunstein Sameet Sreenivasan
Boston Universtiy
Gerry Paul




Clarkson University

2

References






Paul et al Europhys. J. B (in
press) (cond-mat/0404331)




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Percolation theory
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Internet Network
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Networks in Physics
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Cohen, Havlin, Phys. Rev. Lett. 90, 58701(2003)

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Distance in Scale Free Networks
(Bollobas, Riordan, 2002)
(Bollobas, 1985) (Newman, 2001)
Small World
Cohen, Havlin Phys. Rev. Lett. 90,
58701(2003) Cohen, Havlin and ben-Avraham, in
Handbook of Graphs and Networks Eds.
Bornholdt and Shuster (Willy-VCH, NY, 2002) Chap.
4 Confirmed also by Dorogovtsev et al (2002),
Chung and Lu (2002)
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Critical Threshold Scale Free
General result
robust
Poor immunization
Random
Acquaintance
vulnerable
Intentional
Efficient immunization
Efficient Immunization Strategies Acquaintance
Immunization
Cohen et al Phys. Rev. Lett. 91 , 168701 (2003)
Not only critical thresholds but also critical
exponents are different ! THE UNIVERSALITY CLASS
DEPENDS ON THE WAY CRITICALITY REACHED
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Critical Exponents
Using the properties of power series (generating
functions) near a singular point (Abelian
methods), the behavior near the critical point
can be studied. (Diff. Eq. Molloy Reed (1998)
Gen. Func. Newman Callaway PRL(2000),
PRE(2001)) For random breakdown the behavior near
criticality in scale-free networks is different
than for random graphs or from mean field
percolation. For intentional attack-same as
mean-field.
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Optimal Distance - Disorder
.
.
lmin 2(ACB) lopt 3(ADEB)
Path from A to B
.
.
.
weight price, quality, time..
minimal optimal
path Weak disorder (WD) all contribute
to the sum (narrow distribution) Strong disorder
(SD) a single term dominates the sum (broad
distribution) SD example Broadcasting video
over the Internet. A transmission at
constant high rate is needed. The
narrowest band width link in the path
between transmitter and receiver controls the
rate.
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Random Graph (Erdos-Renyi)
Scale Free (Barabasi-Albert)
Small World (Watts-Strogatz)
Z 4
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Optimal path strong disorder Random Graphs and
Watts Strogatz Networks
N total number of nodes
Analytically and Numerically
LARGE WORLD!!
Mapping to shortest path at critical percolation
Compared to the diameter or average shortest path

(small world)
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Scale Free ER Optimal Path
Strong Disorder
Diameter shortest path
LARGE WORLD!!
SMALL WORLD!!
Weak Disorder
For
Braunstein et al Phys. Rev. Lett. 91, 247901
(2003)
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Optimal Networks
Simultaneous waves of targeted and random
attacks
- Fraction of targeted
Bimodal fraction of (1-r) having k links and
r having links
1
- Fraction of random
r 0.0010.15 from left to right
Optimal Bimodal
Condition for connectivity
P(k) changes
P(k) changes also due to targeted attack
Bimodal
Bi-modal
For
is the only parameter
- critical threshold
SF
SF
r0.001
Paul et al. Europhys. J. B 38, 187 (2004),
(cond-mat/0404331)
0.01
Tanizawa et al. Optimization of Network
Robustness to Waves of Targeted and Random
Attacks (Cond-mat/0406567)
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Optimal Bimodal
Specific example
Given N100, , and
i.e, 10 hubs of
degree using
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• Conclusions and Applications
• Generalized percolation , ?gt4
  Erdos-Renyi, ?lt4 novel topology novel
  physics.  
• Distance in scale free networks ?lt3 dloglogN -
  ultra small world, ?gt3 dlogN.       
• Optimal distance strong disorder ER, WS and
  SF (?gt4)
• scale free
• Scale Free networks (2lt?lt3) are robust to random
  breakdown. 
•  Scale Free networks are vulnerable to targeted
  attack on the highly connected nodes. 
•  Efficient immunization is possible without
  knowledge of topology, using Acquaintance
  Immunization.
•  The critical exponents for scale-free directed
  and non-directed networks are different than
  those in exponential networks different
  universality class!  
• Large networks can have their connectivity
  distribution optimized for maximum robustness to
  random breakdown and/or intentional attack. 

Large World

Large World Small World