Clarkson University

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




Clarkson University

2

References








Sreenivasan et al Phys. Rev. E Submitted
(2004)

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Percolation theory
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Airlines
New Type of Networks
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Networks in Physics
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Internet Network
Faloutsos et. al., SIGCOMM 99
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Metabolic network
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More Examples
Email
Trust networks Guardiola et al (2002) Email
networks Ebel etal PRE (2002)
Trust
Trust
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Cohen, Havlin, Phys. Rev. Lett. 90, 58701(2003)

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Erdös-Rényi model (1960)
Pál Erdös (1913-1996)
- Democratic - Random
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BA model
Scale-free model
(1) GROWTH
At every timestep we
add a new node with m edges (connected to the
nodes already present in the system). (2)
PREFERENTIAL ATTACHMENT
The probability ? that a new node will be
connected to node i depends on the connectivity
ki of that node
A.-L.Barabási, R. Albert, Science 286, 509 (1999)
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Shortest Paths 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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We find that not only critical thresholds but
also critical exponents are different ! THE
UNIVERSALITY CLASS DEPENDS ON THE WAY CRITICALITY
REACHED
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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)
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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 weak disorder Random Graphs and
Watts Strogatz Networks
Typical short range neighborhood
Crossover from large to small world
For
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Scale Free Optimal Path Weak disorder
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Optimal path strong disorder Random Graphs and
Watts Strogatz Networks
N total number of nodes
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Scale Free Optimal Path
Strong Disorder
LARGE WORLD!!
SMALL WORLD!!
Weak Disorder
Diameter shortest path
Braunstein, Buldyrev, Cohen, Havlin, Stanley,
Phys. Rev. Lett. 91, 247901 (2003)
Cond-mat/0305051
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Theoretical Approach Strong Disorder
(i) Distribute random numbers 0ltult1 on the links
of the network.

with
(ii) Strong disorder represented by
(iii) The largest
in each path between two nodes dominates the sum.

(iv) The lowest are on the percolation
cluster where
(v) The optimal path must therefore be on the
percolation cluster at criticality.
(vi) Percolation on random networks is like
percolation in or

(vii) Since loops can be neglected the optimal
path can be identified with the shortest path.
where
Mass of infinite cluster
(see also Erdos-Renyi, 1960)
Thus,
From percolation
Thus,
for ER, WS and SF with

change due to novel topology
For SF with
and
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Sreenivasan et al Phys. Rev. E Submitted
(2004)For details see POSTER
Transition from weak to strong disorder
For a given disorder strength
for
for
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• Conclusions and Applications
•   Distance in scale free networks ?lt3 dloglogN
  - ultra small world, ?gt3 dlogN.       
• Optimal distance strong disorder Random
  Graphs and WS
• scale free
• Transition between weak and strong disorder
• Scale Free networks (2lt?lt3) are robust to random
  breakdown. 
•  Scale Free networks are vulnerable to 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
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• Conclusions and Applications
• Distance in scale free networks ?lt3 dloglogN
  - ultra small world, ?gt3 dlogN.       
• Optimal distance strong disorder Random
  Graphs and WS
• scale free
• Scale Free networks (2lt?lt3) are robust to random
  breakdown. 
•  Scale Free networks are vulnerable to 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! 
•  Method for embedding Scale Free Networks
  embedded in Euclidean space - dmin lt 1
• Large networks can have their connectivity
  distribution optimized for maximum resilience to
  random breakdown and/or intentional attack. 

Large World

Large World Small World
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• Conclusions and Applications
• Generalized random graphs
  , ?gt4 Erdos-Renyi, ?lt4 novel topology.
• Distance in scale free networks dloglogN -
  ultra small world      
• Scale free networks (2lt?lt3) are resilient to
  random breakdown. 
•  Scale free networks are sensitive to attacks on
  the most highly connected nodes. 
•  Efficient immunization is possible without
  knowledge of topology, using Acquaintance
  Immunization. 
• The optimal distance makes random graphs large
  worlds while scale-free networks are still small
  worlds! 
•   Large networks can have their connectivity
  distribution optimized for maximum resilience to
  random breakdown and/or intentional attack. 

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Scale Free Optimal Path Weak disorder
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Results of Simulations and Theory
Random Breakdown (Immune)
- no critical threshold a spanning cluster
always exists
- a critical threshold exists
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• Conclusions and Applications
• Generalized random graphs
  , ?gt4 Erdos-Renyi, ?lt4 novel topology.
• Distance in scale free networks dloglogN -
  ultra small world      
• Scale free networks (2lt?lt3) are resilient to
  random breakdown. 
•  Scale free networks are sensitive to intentional
  attack on the most 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 resilience to
  random breakdown and/or intentional attack. 

36

• Conclusions and Applications
• Generalized random graphs
  , ?gt4 Erdos-Renyi, ?lt4 novel topology novel
  physics.
• Distance in scale free networks ?lt3 dloglogN -
  ultra small world, while for ?gt3 dlogN (small
  world).        
• Scale free networks (2lt?lt3) are robust to random
  breakdown. 
•  Scale free networks are vulnerable to
  intentional attacks on the most 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 in ?lt4
  than in ?gt4 different universality classes! THE
  UNIVERSALITY CLASS DEPENDS ON THE WAY CRITICALITY
  REACHED!
•   Optimize networks to have maximum robustness
  to both random breakdown and intentional attacks. 

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Scale Free Optimal Path
Strong Disorder
Diameter shortest path
LARGE WORLD!!
SMALL WORLD!!
Weak Disorder
For
Braunstein, Buldyrev, Cohen, Havlin, Stanley,
Phys. Rev. Lett. 91, 247901 (2003)
Cond-mat/0305051
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KA/B
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Critical Threshold
Binomial Distribution Two Gaussians
SIR (Susceptible-Infected-Removed) model Scale
Free
r the infection rate ? - infection time ?
2.5 ? 3.5 Top random immunization Bottom
Acquaintance Immunization
d distance between Gaussians one at k3
second at k3d Top- random immunization Bottom-
Acquaintance immunization variance
2 variance 8
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Percolation in Directed Networks
General example .
Newman et al PRE (2001) Dorogortev Mendes PRE
(2001) General condition for giant component
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Percolation in Scale Free Directed Networks
Critical concentration
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Directed Scale Free Critical Exponents
Directed SF networks without correlation Directed SF networks with correlation


Non-Directed SF networks


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Scale Free Networks in Euclidean Space
Principle Minimum total length of links
Rozenfeld, Cohen, ben-Avraham, Havlin Phys. Rev.
Lett. 89, 218701 (2002)
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Shortest path
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We find that not only critical threshold but also
critical exponents are different ! THE
UNIVERSALITY CLASS DEPENDS ON THE WAY CRITICALITY
REACHED
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Fractal Dimensions
From the behavior of the critical exponents the
fractal dimension of scale-free graphs can be
deduced.
Far from the critical point - the dimension is
infinite - the mass grows exponentially with the
distance. At criticality - the dimension is
finite for ?gt3 .
(upper critical dimension)
The dimensionality of the graphs depends on the
distribution!
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Fractal Dimensions
From the behavior of the critical exponents the
fractal dimension of scale-free graphs can be
deduced.
Far from the critical point - the dimension is
infinite - the mass grows exponentially with the
distance. At criticality - the dimension is
finite for ?gt3 .
l
-
ì
2
l
lt
4
ï
ï
l
-
3
Chemical dimension

d
í
l
ï
l
³
24
ï
î
Fractal dimension
(upper critical dimension)
The dimensionality of the graphs depends on the
distribution!
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Cohen et al, PRL 85, 4626 (2000) PRL 86, 3862
(2001)
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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. Melloy 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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Percolation theory
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For Barabasi-Albert scale free networks
61

• Conclusions and Applications
• Distance in scale free networks dloglogN -
  ultra small world      
• Scale free networks (2lt?lt3) are resilient to
  random breakdown. 
•  Scale free networks are sensitive to intentional
  attack on the most 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 resilience to
  random breakdown and/or intentional attack. 

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Efficient Immunization Strategies Acquaintance
Immunization
Critical Threshold Scale Free

• Acquaintance Immunization
• Random immunization is inefficient in scale free
  graphs, while targeted immunization requires
  knowledge of the degrees.
• In Acquaintance Immunization one immunizes
  random neighbors of random individuals.
• One can also do the same based on n neighbors.
• The threshold is finite and no global knowledge
  is necessary.

Cohen et al cond-mat/0207387
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Percolation and Immunization of Complex Networks

Shlomo Havlin



Eugene Stanley Boston University Armin Bunde
Giessen University
Daniel ben -Avraham
Clarkson University

Laszlo Barabasi
Notre Dame University

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DNA
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New York Times
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Percolation in Scale Free Directed Networks
Critical concentration
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Directed Scale Free Critical Exponents
Directed SF networks without correlation Directed SF networks with correlation


Non-Directed SF networks


Schwartz, Cohen, ben-Avraham Barabasi, Havlin PRE
RC (2002)
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Distance in Scale Free Networks
Small World
Cohen, Havlin, Phys. Rev. Lett. (2003)
Cond-mat/0205476 (2002)
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Percolation theory
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Results of Simulations and Theory

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Results of Simulations and Theory
Random Breakdown (Immune)
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Networks in Physics
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Efficient Immunization Strategy Theory
Acquaintance Immunization

• Acquaintance Immunization
• Random immunization is inefficient in scale free
  graphs, while targeted immunization requires
  knowledge of the degrees.
• In Acquaintance Immunization one immunizes
  random neighbors of random individuals.
• One can also do the same based on n neighbors.
• The threshold is finite and no global knowledge
  is necessary.

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• Conclusions and Applications
• Distance in scale free networks dloglogN -
  ultra small world      
• Scale free networks (2lt?lt3) are resilient to
  random breakdown. 
•  Scale free networks are sensitive to intentional
  attack on the most 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! 
•  Method for embedding Scale Free Networks in
  Euclidean space - dmin lt 1
• Large networks can have their connectivity
  distribution optimized for maximum resilience to
  random breakdown and/or intentional attack. 

81

• Conclusions and Applications
• Generalized random graphs
  , ?gt4 Erdos-Renyi, ?lt4 novel topology.
• Distance in scale free networks ?lt3 dloglogN -
  ultra small world, ?gt3 dlogN.       
• Scale free networks (2lt?lt3) are resilient to
  random breakdown. 
•  Scale free networks are sensitive to intentional
  attack on the most highly connected nodes. 
•  Efficient immunization is possible without
  knowledge of topology, using Acquaintance
  Immunization. 
•  The critical exponents for scale-free (?lt4)
  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 resilience to
  random breakdown and/or intentional attack. 

82

• Conclusions and Applications
• Scale Free network Tomography -
• Distance in scale free networks dloglogN -
  ultra small world       
• Optimal distance strong disorder Random
  Graphs and WS
• Scale free
• Scale Free networks (2lt?lt3) are resilient to
  random breakdown. 
•  Scale Free networks are sensitive to intentional
  attack on the most 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! 
•  Method for embedding Scale Free Networks
  embedded in Euclidean space - dmin lt 1
• Large networks can have their connectivity
  distribution optimized for maximum resilience to
  random breakdown and/or intentional attack. 

Large World

Large World Small World
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• Conclusions and Applications
•  
• The Internet is resilient to random breakdown. 
•  The Internet is sensitive to intentional attack
  on the most highly connected nodes. 
•  Distance in scale free networks dloglogN -
  ultra small world       
•  Efficient immunization is possible without
  knowledge of topology, using a strategy of
  Acquaintance Immunization. 
•  The critical exponents for scale-free directed
  and non-directed networks are different than
  those in exponential networks different
  universality class! 
•  Method for embedding Scale Free Networks in
  Euclidean space - dmin lt 1
• Large networks can have their connectivity
  distribution optimized for maximum resilience to
  random breakdown and/or intentional attack. 

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Armin Bunde
Giessen University
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Percolation and Immunization of Complex Networks

Shlomo Havlin




Daniel ben - Avraham
Clarkson University
Laszlo Barabasi
Notre Dame University

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Giessen University
Armin Bunde
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Giessen University
Armin Bunde
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Analytically and Numerically
LARGE WORLD!!
Compared to the diameter or average shortest path
or weak disorder
(small world)