Limited Path Percolation in Complex Networks Eduardo Lpez Los - - PowerPoint PPT Presentation

SLIDE 1

Outline

“Limited path length percolation in complex networks”, López, Parshani, Cohen, Carmi and Havlin, Phys. Rev. Lett. (in press). cond-mat/0702691.

References

Limited Path Percolation in Complex Networks

Eduardo López

Los Alamos Nat. Lab. LA-UR 07-0432

Collaborators

Shlomo Havlin Reuven Cohen Shai Carmi Roni Parshani

• Motivation. Percolation and its effects.
• Presentation of new limited path length percolation model
• Scaling theory of new model and results
• Targeted percolation, theory and results.
• Conclusions

SLIDE 2

Motivation: How to go from Salem to Boston?

But in Boston, harsh winter! Weatherman wrong: Bigger storm!

Question: How many roads need to be closed before most people cannot get to work?

On a sunny day many paths

Answer: from Percolation theory

SLIDE 3

What is percolation theory?

p i, j distance S(p): # connected nodes l’ij>lij due to removal

i j lij l’ij

=pc l’’ij> l’ij Ndf/d =1 lij N <1 l’ij P∞N <pc most disconnec. log N

p: occupied fraction of links P∞: probability of random node to be in largest cluster pc: connectivity threshold df: fractal dim., d dim.

Transition: disconnected connected Theory to determine connectivity in systems

l’’ij ↓

Log N Log S p<pc p=pc p>pc Slope df /d Slope 1 L

• g

a r i t h m i c

1 1

c

p Logarithmic Fractal Linear

∞

P

SLIDE 4

Motivation: What’s the problem with percolation?

• Salem-Boston connected

with any path! Commute time: 120min

• Long or short paths OK

Commute time: 240min

• There is practical limit

to connectivity ⇔ longer paths not useful.

Answer: sometimes percolation accepts useless paths.

• Percolation finds critical

percentage pc of roads needed to keep cities connected.

• Percolation increases path

lengths (and time), i.e., smaller p⇒longer path. All day driving! Commute time: 60-70min Commute time: 50min

Storm hits

SLIDE 5

Social contact network

SLIDE 6

New percolation model applied to complex networks

• Definition of connection: i and j are connected if l’ij ≤ alij

Results: New limited path percolation transition

• Is there a critical occupation above which Sa~N?

c

p p ~ =

• Notation:

Sa(p): Largest cluster size at occupation p, length condition a

• Find new critical occupation

c c

p p > ~

• Scaling theory
• Critical point is now a critical range:

( )

c c a

p p p p a N S ~ ) , ( , ~ < < =δ δ

δ

• Below and above range, behavior is

similar to regular percolation:

( )

c a

p p N S < log ~

( )

c a

p p N S ~ ~ >

1 1

c

p

c

p ~ Logarithmic Fractal Linear LP perc. R e g . p e r c .

∞

P

SLIDE 7

• Developed in the 1960’s by Erdős and Rényi. (Publications of the

Mathematical Institute of the Hungarian Academy of Sciences, 1960).

Theory of model networks: Erdős-Rényi

• Define k as the degree (number of links of a node), and ‹k›

is average number of links per node over the network. a) Complete network

! ) ( k k e k P

k k −

=

Construction

• Distribution of degree is Poisson-like (exponential)

c) Realization of network b) Annihilate links with probability

φ − 1

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = 1 N k φ

node i degree of i, ki=2

• N nodes and each pair connected with probability φ.

degree of j, kj=3 node j

SLIDE 8

Outline of scaling theory for Limited Path Percolation Example: Erdős-Rényi

• Before percolation, typical path length l ~ log N/log <k>
• Tree approx. ⇒ Sa ~ (κ -1)l = (p<k>) a log N/log <k> = Nδ
• Scaling exponent 0 ≤ δ ≡ a(1+log p/log <k>) ≤ 1
• δ ≤1 because Sa cannot exceed N
• Solving δ = 1 ⇒

a a c

k p

/ ) 1 (

~

−

=

k N a log / log

1 + k p

1

~

− ∞ →

= ⎯ ⎯→ ⎯ k p p

c a c

• Usual percolation recovered with a→∞:
• After percolation, local structure is tree-like, with

branching factor

1 + = k p κ

SLIDE 9

Comparison of phase diagram of regular & Limited Path Percolation (Erdős-Rényi)

Regular percolation Limited path percolation Communicating Non-communicating

Limited path percolation predicts a larger communication threshold.

Length factor a Concentration p 1 Logarithmic Phase (δ=0) Fractal Phase (δ<1) Linear Phase (δ=1) ER Transition line pc ~ pc= <k>-1 1

∞

Concentration p Linear phase Logarithmic phase Fractal point pc=<k> -1 1

1 1

c

p

Logarithmic Fractal Linear

∞

P

Regular percolation

SLIDE 10

10

3

10

4

10

5

10

6

N

10

2

10

3

10

4

10

5

Sa

a=1.0 1.1 1.2 1.4 1.7

10

4

10

5

N

10

2

10

3

10

4

Sa (a) ER <k>=3.0 (random)

Results for Sa~Nδ (Erdős-Rényi)

Log N Log S p<pc p=pc p>pc Slope df /d Slope 1 Logarithmic

Regular Percolation Limited path percolation

⎪ ⎩ ⎪ ⎨ ⎧ < = > ) ( log ) ( ) ( ~

3 / 2 c c c

p p N p p N p p N S

⎪ ⎩ ⎪ ⎨ ⎧ < ≤ ≤ > ) ( log ) ~ ( ) ~ ( ~

c c c c

p p N p p p N p p N S

δ

k pk a log log ≡ δ

a a c

k p

/ ) 1 (

~

−

=

1 −

= k p c

SLIDE 11

Complex Networks

Poisson distribution Erdős-Rényi Network

• λ

m K Scale-free distribution Scale-free Network

SLIDE 12

Some basic network properties

Erdős-Rényi networks Scale-free networks

• Narrow range of typical degree

k k k k k + ≤ ≤ −

• Wide range of typical degree

( )

1 / 1 min min −

≤ ≤

λ

N k k k

• Small diameter
• Small or ultra-small diameter

N D ln ~

] 3 [ ln ~ ] 3 2 [ ) ln(ln ~ > < < λ λ N D N D

(kmin is minimum degree)

SLIDE 13

Scaling theory for limited path percolation on scale-free networks

• For λ>3:

( ) [ ]

( )(

) a

a

• c

p a a

p N S

• −

− +

− =

1 1 log / log 1

1 ~ ~ κ

κ

• For 2<λ<3:

Tree approximation invalid. Networks are ultra-small:

) 2 log( log log ~ ' ) 2 log( log log ~ − −

∞

λ λ N P l N l

Therefore:

1 log log log log ~ '

∞ → ∞

→ =

N

N N P l l a

SLIDE 14

Phase Diagram of Limited Path Percolation on scale-free networks

Communicating Non-communicating

Length factor a Concentration p 1

~ =

c

p

1 L i n e a r p h a s e (δ = ) SF 2<λ<3

∞

Length factor a Concentration p 1 Logarithmic Phase (δ=0) Fractal Phase (δ<1) Linear Phase (δ=1) SF λ>3 Transition line pc ~ pc=(κo-1)-1 1

∞

SLIDE 15

Results for Sa~Nδ Scale-free

10

4

10

5

N

10

3

10

4

Sa

λ=2.2 2.3 2.4

(c) SF (2<λ<3) (random) 1

10

3

10

4

N

10

2

10

3

10

4

Sa

a=1.01 1.1 1.2 1.5

10

4

10

5

N

10

2

10

3

10

4

10

5

Sa (b) SF (λ=3.5) (random)

SLIDE 16

Targeted attacks on scale-free networks

• Scale-free networks have sensitive nodes (hubs) with large k.
• Examples: Airline hubs, central communication nodes,

disease super-spreaders.

Model for targeted percolation

• p: fraction of lowest degree nodes present.
• In targeted percolation (no length

restriction) pc is large: pc=1 (λ→2) pc close to 1 (λ>2) Network falls apart with few node removals.

Question: What happens for limited path percolation?

hub

SLIDE 17

Scaling theory for limited path targeted percolation on scale-free networks

• For λ>3:
• For 2<λ<3:

Tree approximation valid again after percolation:

( )

( ) ( )

2 log 1 log 2

log ~

− − λ κ a a

N S

( ) ( )

) , , ( ~ ~ ~

1 log 1 log

• c

c a a

a p p N S

• κ

κ

κ κ

=

− −

Any finite a fails to produce transition to linear phase:

1 ~ =

c

p

SLIDE 18

Communicating Non-communicating

Length factor a 1

1

Concentration p

~ =

c

p

L

• g

a r i t h m i c p h a s e SF 2<λ<3

∞

Length factor a Concentration p 1 Logarithmic Phase (δ=0) Fractal Phase (δ<1) Linear Phase (δ=1) SF λ>3 T r a n s i t i

• n

l i n e pc ~ pc=(κo-1)-1 1

∞

Random

Phase Diagram of Limited Path Percolation Scale-free targeted removal

SLIDE 19

Results for Sa~Nδ Scale-free targeted removal

10

4

10

5

N

10

2

10

3

10

4

10

5

Sa

1.2 1.5 2.0

10

4

10

5

N

10

3

10

4

Sa (d) SF (λ=3.5) (targeted)

0.50 0.55 0.60 0.65 0.70 0.75

Log (Log N)

2.75 3.25 3.75 4.25

Log Sa Slope=6.0+/−0.1 (d) SF (λ=2.3) (targeted)

SLIDE 20

Erdős-Rényi Scale-free (λ>3) Scale-free (2≤λ≤3)

c

p ~

δ Sa

a a

k

/ ) 1 ( −

( )

a a

• /

) 1 (

1

−

− κ

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + k p a log log 1

Quantity

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + ) 1 log( log 1

• p

a κ

1

Differences in Limited Path Percolation due to network structure and removal method at pc ≤p≤ pc ~

Nδ Nδ Nδ

Random removal Targeted removal

) , , ( ~

• c a

p κ κ

c

p ~

δ Sa 1 ) 1 log( ) 1 log( − −

• a

κ κ

) 2 log( ) 1 log( 2 − − λ κ a

Nδ (log N)δ

• Transition

Transition No Transition

SLIDE 21

Scaling function for Sa

• For Erdös-Rényi, and scale-free λ>3 with random and targeted

removal, there are two phases above and below

c

p ~

• Therefore:

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛

∞ δ δ

N p c N P f N p c Sa ) ( ) ( ~

δ

N p c Sa ) ( ~

) ~ (

c c

p p p < <

N P Sa

∞

~

) 1 ~ ( ≤ < p pc

⎩ ⎨ ⎧ >> << 1 ., 1 , ~ ) ( x cnst x x x f c(p) ≡ co [p (κο -1)+1]/[p(κο -1) -1]

• Two limits:

i) ii)

SLIDE 22

10

−6

10

−4

10

−2

10 10

2

10

4

P N/(c N

δ)

10

−6

10

−4

10

−2

10

Sa/(cN

δ)

(a) ER <k>=3.0 (random) 1

10

−6

10

−4

10

−2

10 10

2

P N/(cN

δ)

10

−6

10

−4

10

−2

10

Sa/(cN

δ)

(b) SF (λ=3.5) (random) 1

Results for scaling of Sa

10

−4

10

−2

10 10

2

10

4

P N/(cN

δ)

10

−4

10

−2

10 10

2

Sa/(cN

δ)

(c) SF (λ=3.5) (targeted) 1

SLIDE 23

Conclusions

• We find a new percolation transition at

which implies when lengths are constrained, more connections are necessary to percolate. Transition preserves path length scaling.

( )

c a a c

p p > − =

− / ) 1 (

1 ~ κ

• We define a new percolation model which takes into

account the length restriction of useful paths.

• We encounter two typical phases: i) power-law with Sa ~ Nδ,

and ii) a linear phase Sa ~ N.

• This model is important in real-world applications such as

epidemics, data transfer, and transportation.

SLIDE 24

Conclusions

• We find a new percolation transition at

which implies when lengths are constrained, more connections are necessary to percolate. Transition preserves path length scaling.

( )

c a a c

p p > − =

− / ) 1 (

1 ~ κ

• We define a new percolation model which takes into

account the length restriction of useful paths.

• We encounter two typical phases: i) power-law with Sa ~ Nδ,

and ii) a linear phase Sa ~ N.

• Few models of percolation exist. Our model is an innovative

new approach to percolation with great opportunities for research.

• This model is important in real-world applications such as

epidemics, data transfer, and transportation.

SLIDE 25

Phase Diagram of Limited Path Percolation Scale-free targeted removal

Concentration p

1

Length factor a

1

Concentration p Linear phase (δ=1) Fractal phase (δ<1) Transition line p

~ c(a)(targeted)

Transition line p

~ c(a)(random)

SF λ>3 (κo−1)

−1

Logarithmic phase

Length factor a 1 SF 2<λ<3

Logarithmic phase

∞ 1 ~ =

c

p

Communicating Non-communicating

SLIDE 26

Timeline of percolation theory

Gelation or how the egg hardens: Flory(1941) and Stockmayer(1943). Flow through a random medium: Broadbent and Hammersley(1957).

Tree percolation Directed percolation Invasion percolation Limited path percolation

Displacement of fluid by another: Wilkinson and Willemsen (1983).

Bootstrap percolation

Ferromagnets: Pollak and Reiss (1975).

Percolation

Communications and epidemics: López et al. (2007).

1940 1960 1980 2000 Year 1 2 # models per decade

Tree percolation Percolation Bootstrap & directed percolation Invasion percolation Limited path percolation

Steady state chemical reactions: Schlögl (1972).

SLIDE 27

Degree: 2 3 5 2 3 3

Molloy-Reed Algorithm for scale-free Networks

λ −

k k P ~ ) (

Create network with pre-specified degree distribution P(k)

1) Generate set of nodes with pre-specified degree distribution form 3) Randomly pair copies excluding self-loops and double connections: 4) Connect network: Example: 2) Make ki copies of node i:

SLIDE 28

Theory: Properties of scale-free networks

• Network size with branching factor κο:

~ (κο-1)l (λ>3); variable (2<λ<3)

• Typical distance l between nodes:

( ) ( ) ( ) ( )

3 2 2

• log

log log ; 3 1 log log ~ < < > − λ λ λ κ N N l

• Branching factor at occupation p:

( ) ( )

3 for 1 1 > − = − λ κ κ

• p
• Branching factor:

κο=<k2>/<k>=cons. (λ>3); incres. (2<λ<3)

• Percolation thresholds:

pc= (κο-1) -1 (λ>3); 0 (2<λ<3)

• Nodes connected at p=pc:

S ~ N (λ-3)/(λ-1) (λ>3); N (2<λ<3)

SLIDE 29

Erdös-Rényi

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = =

−

k p a N S k p

a a a c

log log 1 , ~ , ~

/ ) 1 (

δ

δ

Summary of theoretical results

Scale-free (λ>3)

( ) ( )⎟

⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + = − =

−

1 log log 1 , ~ , 1 ~

/ ) 1 (

• a

a a

• c

p a N S p κ δ κ

δ

Scale-free (2<λ<3)

N S p

a c

~ , ~ =

SLIDE 30

Summary of theoretical results

Targeted removal on scale-free networks λ>3 2<λ<3

) 1 log( ) 1 log( , ~ ), , , ( ~ ~ − − = =

• a
• c

c

a N S a p p κ κ δ κ κ

δ

) 2 log( ) 1 log( 2 , ) (log ~ , 1 ~ − − = = λ κ δ

δ

a N S p

a c

SLIDE 31

Motivation: Where else does percolation fail?

• Communications such as data packet routing:

Rerouting makes sense if new path is short Internet

Message route Communication problems require data rerouting

Long paths compound error + reduce performance + security

• Infectious diseases:

Flu decays over time/season. Increase of immunity in population.

• Communications such as data packet routing:

Long paths ineffective.

• Transportation:

Long commute times prohibitive.

• Other important extensions like path cost considerations.

SLIDE 32

Motivation: Where else does percolation fail?

• Communications such as data packet routing:

Rerouting makes sense if new path is short Internet

Message route Communication problems require data rerouting

Long paths compound error + reduce performance + security

• Infectious diseases:

Flu decays over time/season. Increase of immunity in population.

• Communications such as data packet routing:

Long paths ineffective.

• Transportation:

Long commute times prohibitive.

• Other important extensions like path cost considerations.

SLIDE 33

Complex Networks

Poisson distribution Erdős-Rényi Network Scale-free Network

• λ

m K Scale-free distribution

SLIDE 34

SLIDE 35

Complex Networks

Poisson distribution Erdős-Rényi Network

• λ

m K Scale-free distribution Scale-free Network

SLIDE 36

Complex Networks

Poisson distribution Erdős-Rényi Network

• λ

m K Scale-free distribution Scale-free Network

SLIDE 37

Phase Diagram of Limited Path Percolation Scale-free

1

Length factor a

1

Concentration p Linear phase (δ=1) Fractal phase (δ<1) Logarithmic phase Transition line p

~ c(a)(random)

SF λ>3 (κo−1)

−1

Concentration p 1

Linear phase ∞

1 SF 2<λ<3 Length factor a

~ =

c

p

Communicating Non-communicating

SLIDE 38

Comparison of phase diagram of regular &Limited Path Percolation (Erdős-Rényi)

Concentration p

Linear phase Logarithmic phase Fractal line

<k> -1 1

1

Length factor a

1

Concentration p Linear phase (δ=1) Fractal phase (δ<1) Transition line p

~ c(a)

<k>

−1

ER Logarithmic Phase

Regular percolation Limited path percolation Epidemic No epidemic Limited path percolation predicts a larger epidemic threshold.

SLIDE 39

Theory: Erdős-Rényi network properties

• Nodes connected with branching factor κο:

1 + κο + κο(κο-1) + κο(κο-1)2 ~ (κο-1)l

• Typical distance l between nodes:

( )

k N N l

• log

/ log 1 log / log ~ = − κ

• Branching factor:

κο=<k2>/<k>=<k>+1

• Percolation threshold:

pc=<k>-1

• Nodes connected at p=pc:

S ~ N2/3

l shells

• Degree distribution:

κο: typ. # links/node

• Typical length at p=pc:

l ~ N1/3

SLIDE 40

New percolation model applied to complex networks

• Definition of connection: i and j are connected if l’ij ≤ alij

Results: New limited path percolation transition

• Is there a critical occupation above which Sa~N?

c

p p ~ =

• Notation:

Sa(p): Largest cluster size at occupation p, length condition a

• Find new critical occupation

c c

p p > ~

• Analytical scaling theory
• Critical point is now a critical range:

( )

c c a

p p p p a N S ~ ) , ( , ~ < < =δ δ

δ

• Below and above range, behavior is

similar to regular percolation:

( )

c a

p p N S < log ~

( )

c a

p p N S ~ ~ >

1 p 1 P Logarithmic Fractal Linear

• Reg. Perc.

LP perc. pc ~ pc

SLIDE 41

What is percolation theory?

p i, j distance S(p): # connected nodes l’ij>lij due to removal

i j lij l’ij

=pc l’’ij> l’ij Ndf/d =1 lij N <1 l’ij P∞N <pc most disconnec. log N

p: occupied fraction of links P∞: probability of random node to be in largest cluster pc: connectivity threshold df: fractal dim., d dim.

Transition: disconnected connected Theory to determine connectivity in systems

l’’ij ↓

Log N Log S p<pc p=pc p>pc Slope df /d Slope 1 L

• g

a r i t h m i c

1 p P pc 1

Logarithmic Linear Fractal

SLIDE 42

Comparison of phase diagram of regular & Limited Path Percolation (Erdős-Rényi)

Regular percolation Limited path percolation Communicating Non-communicating

Limited path percolation predicts a larger communication threshold.

Length factor a Concentration p 1 Logarithmic Phase (δ=0) Fractal Phase (δ<1) Linear Phase (δ=1) ER Transition line pc ~ pc= <k>-1 1

∞

Concentration p Linear phase Logarithmic phase Fractal point pc=<k> -1 1