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Transcript of Complex Networks - Shlomo Havlinhavlin.biu.ac.il/PCS/lec10.pdf · Complex Networks • Network is a...
![Page 1: Complex Networks - Shlomo Havlinhavlin.biu.ac.il/PCS/lec10.pdf · Complex Networks • Network is a structure of N nodes and 2M links (or M edges) • Called also graph – in Mathematics](https://reader033.fdocuments.in/reader033/viewer/2022051722/5a9cb9df7f8b9a7f278b6412/html5/thumbnails/1.jpg)
Complex Networks
• Network is a structure of N nodes and 2M links (or M edges)
• Called also graph – in Mathematics
• Many examples of networks
Internet: nodes represent computers
links the connecting cables
Social network: nodes represent people
links their relations
Cellular network: nodes represent molecules
links their interactions
• Weighted networks each link has a weight determining the strength or cost of the link
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Social Networks- Stanley Milgram (1967)
Nodes: individuals
Links: social relationship (family/work/friendship/etc.) Six Degrees of Separation
John Guare)1992(
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Map showing the world-wide internet traffic
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Skitter data depicting a macroscopic snapshot of Internet connectivity, with selected backbone ISPs (Internet Service Provider) colored separately
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Hierarchical topology of the international web cache
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Image of Social links in Canberra, Australia
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Network of protein-protein interactions. The color of a node signifies the phenotypic effect of removing the corresponding protein(red, lethal; green, non-lethal; orange, slow growth; yellow, unknown).
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Complex systemsMade of
many non-identical elementsconnected by diverse interactions.
NETWORK
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Degree distribution P(k) -- k- degree of a node
Diameter or distance – Average distance between nodes--d
Clustering Coefficient c(k)=
How many of my friends are also friends?
Centrality or Betweeness -- b
Number of times a bond or a node is relatively used for the shortest path
Critical Threshold: The concentration of nodes that are removed and the network collapses
Network Properties
1)/2-k(kneighborsk between links of no.
B
D
C
A
111)(
31
62)(
3)(
=−−
=
==
=
NNBCb
Dc
ABd
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Random Graph Theory
• Developed in the 1960’s by Erdos and Renyi. (Publications of the Mathematical Institute of the Hungarian Academy of Sciences, 1960).
• Discusses the ensemble of graphs with N vertices and M edges (2M links). • Distribution of connectivity per vertex is Poissonian (exponential),
where k is the number of links :
P k eck
ck
( )!
= −, c k
MN
= =2
• Distance d=log N -- SMALL WORLD
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More Results
• Phase transition at average connectivity, : k < 1 No spanning cluster (giant component) of order logN k > 1 A spanning cluster exists (unique) of order N
k = 1 The largest cluster is of order N 2 3/
k = 1
• Size of the spanning cluster is determined by the self-consistent equation:
P e k P∞
−= − ∞1
• Behavior of the spanning cluster size near the transition is linear:
β)( ppP c −∝∞ , 1=β , where p is the probability of deleting a site,
1 1/cp k= −
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Percolation on a Cayley Tree
• Contains no loops
• Connectivity of each node is fixed (z connections)
• Behavior of the spanning cluster size near the transition is linear:
β)( ppP c −∝∞ , 1=β
• Critical threshold:
11cp
z=
−
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In Real World - Many Networks are non-Poissonian
P k e
kk
kk
( )!
= −
( )
0ck m k KP k
otherwise
λ−⎧ ≤ ≤⎪= ⎨⎪⎩
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New Type of NetworksPoisson distribution
Exponential Network
Power-law distribution
Scale-free Network
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Networks in Physics
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Internet NetworkFaloutsos et. al., SIGCOMM ’99
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Metabolic network
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Jeong et all Nature 2000
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Jeong et al, Nature (2000)
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More Examples• Trust networks: Guardiola et al (2002)
• Email networks: Ebel etal PRE (2002)
Trust
Trust
-2.9
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Erdös Theory is Not Valid Distribution
Generalization of Erdös Theory: Cohen, Erez, ben-Avraham, Havlin, PRL 85, 4626 (2000)
Epidemiology Theory: Vespignani, Pastor-Satoral, PRL (2001), PRE (2001)
Modelling: Albert, Jeong, Barabasi (Nature 2000) Cohen, Havlin,Phys. Rev. Lett. 90, 58701(2003)
Infectious disease Malaria 99%Measles 90-95%Whooping cough 90-95%Fifths disease 90-95%Chicken pox 85-90%Mumps 85-90%Rubella 82-87%Poliomyelitis 82-87%Diphtheria 82-87%Scarlet fever 82-87%Smallpox 70-80%INTERNET 99%
Critical concentration
Stability and Immunization
Critical concentration 30-50%
11cpk
= −
Distance
Almost constant (Metabolic Networks,
Jeong et. al. (Nature, 2000))
Nd log~
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Experimental Data: Virus survival
(Pastor-Satorras and Vespignani, Phys. Rev Lett. 86, 3200 (2001))
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Erdös-Rényi model(1960)
- Democratic
- Random
PPááll ErdErdööss(1913-1996)
Connect with probability p
p=1/6N=10
⟨k⟩ ~ 1.5 Poisson distribution
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Scale-free model(1) GROWTH :At every time step we add a new node with m edges (connected to the nodes already present in the system).
(2) PREFERENTIAL ATTACHMENT :The probability Π 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)
jj
ii k
kkΣ
=Π )(
P(k) ~k-3
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Shortest Paths in Scale Free Networks
loglog
. 2
log 3logloglog
2
3
3
d const
NdN
d N
d N
λ
λ
λ
λ
= =
= =
=
= <
>
<
(Bollobas, Riordan, 2002)
(Bollobas, 1985)(Newman, 2001)
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)
Ultra Small World
Small World
( ) ~P k k λ−
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Model of Stability Random Breakdown (Immune)
Nodes are randomly removed (or immune)with probability p
The Internet is believed to be a randomly connected scale-free network, where
( ) , 2.5P k k λ λ−∝ ≈
Where does the phase transition occur ?
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Model for Stability Intentional Attack (Immune)
The fraction, p , of nodes with the highest connectivity are removed (or immune).
Is this fundamentally different from random breakdown?
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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THEORY FOR ANY DEGREE DISTRIBUTION Condition for the Existence of a Spanning Cluster
If we start moving on the cluster from a single site, in order thatthe cluster does not die out, we need that each site reached willhave, on average, at least 2 links (one “in” and one “out”).
But, by Bayes rule:
P k i jP i j k P k
P i jii i( )
( ) ( )( )
↔ =↔
↔
Exponential graph:
kk
k kk
2 2
2=+
=
⇒ =k 1
Cayley Tree:
pzc =
−1
1
We know that P i j kk
Nii( )↔ =− 1 and P i j
kN
( )↔ =− 1
Combining all this together:
κ ≡ =kk
2
2 (for every distribution) at the critical point.
This means:
k i j k P k i ji i iki
↔ = ↔ ≥∑ ( ) 2 , where i j↔ means that site i is
connected to site j .
i
Cohen et al, PRL 85, 4626 (2000): PRL 86, 3862 (2001)
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Percolation for Random Breakdown
If percolation is considered the connectivity distribution changes according
to the law: '
( ) ( ) (1 )k k k
k k
kP k P k p p
k′−
′>
⎛ ⎞′= −⎜ ⎟⎝ ⎠
∑
Calculating the condition κ =2 gives the percolation threshold:
0
11 ,1cp
κ− =
− where 20
00
.k
kκ = compared to 01 1/cp k= − < > for Erdos Renyi
and
For scale-free distribution with lower cutoff m , and upper cutoff K , gives 13 3
10 2 2
2 , .3
K m K NK m
λ λλ
λ λλκλ
− −−
− −
− −⎛ ⎞= ⎜ ⎟− −⎝ ⎠∼
For scale-free graphs with 3λ ≤ the second moment diverges. No critical threshold!
Network is stable (or not immuned) even for 1.p →
11 for Cayley tree1cp
z= −
−
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Percolation for Intentional Attack (Immune)
Attack has two kinds of influence on the connectivity distribution: • Change in the upper cutoff
Can be calculated by ( )K
k KP k p
=
=∑ ,
or approximately: 1/(1 )K mp λ−= .
• Change in the connectivity of all other sites due to possibility of a broken link (which is different than in random breakdown). The probability of a link to be removed can be calculated by:
0
1 ( )K
k Kp kP k
k =
= ∑ ,
or approximately: (2 ) /(1 )p p λ λ− −= .
Substituting this into: 11
1cpk
− =− ,
where 3 3
2 2
23
K mK m
λ λ
λ λ
ακα
− −
− −
− −⎛ ⎞= ⎜ ⎟− −⎝ ⎠, gives the critical threshold.
There exists a finite percolation threshold even for networks resilient to random error!
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Efficient Immunization Strategies:
Acquaintance Immunization
Critical Threshold Scale Free
Cohen et al. Phys. Rev. Lett. 91 , 168701 (2003)
0
2
0
22
0
1p 11
:
1p 1
c
c
K
kK
kF or P oisson
k k kK
k k
k
= −−
≡
+= =
= −
cp
Random
Acquaintance
Intentional
General result:
robust
vulnerable
Poor immunization
Efficient immunization
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Critical ExponentsUsing 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.
P p pc∞ −~ ( ) β
1 2 33
1 3 43
1 4
λλ
β λλ
λ
⎧ < <⎪ −⎪⎪= < <⎨ −⎪⎪ >⎪⎩
(known mean field)
Size of the infinite cluster:
n ss ~ − τ
2 3 42
2 .5 4
λ λλτ
λ
−⎧ <⎪⎪ −= ⎨⎪ ≥⎪⎩
(known mean field)
Distribution of finite clusters at criticality:
Even for networks with , where and are finite, the critical exponents change from the known mean-field result . The order of the phase transition and the exponents are determined by .
1=β3 4λ< < k 2k
3k
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Random Graphs – Erdos Renyi(1960)
Largest cluster at criticality
21
23
4
4
f
f c
dd dS R N N
S N
λλ λ
λ
−− ≤
≥
∼ ∼ ∼
∼
23S N∼
Scale Free networks
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 λ>3 .2 43
2 4ld
λ λλ
λ
−⎧ <⎪⎪ −= ⎨⎪ ≥⎪⎩
Chemical dimension:
dS ∼22 43
4 4fd
λ λλ
λ
−⎧ <⎪⎪ −= ⎨⎪ ≥⎪⎩
Fractal dimension:
fdS R∼ 12 43
6 4cd
λ λλ
λ
−⎧ <⎪⎪ −= ⎨⎪ ≥⎪⎩
Embedding dimension:
(upper critical dimension)
The dimensionality of the graphs depends on the distribution!