Transport in weighted networks: optimal path and superhighways

Collaborators: Z. Wu, Y. Chen, E. Lopez, S. Carmi, L.A. Braunstein, S. Buldyrev, H. E. Stanley

Shlomo HavlinBar-Ilan UniversityIsrael

Wu, Braunstein, Havlin, Stanley, PRL (2006) Yiping, Lopez, Havlin, Stanley, PRL (2006) Braunstein, Buldyrev, Cohen, Havlin, Stanley, PRL (2003)

What is the research question?

• In complex network, different nodes or links have different importance in the transport process.

• How to identify the “superhighways”, the subset of the most important links or nodes for transport? Also important for immunization.

• Identifying the superhighways and increasing their capacity enables to improve transport significantly. Immunization them will reduce epidemics.

10

3

8

62

4

1 1550

30 Networks with weights, such as “cost”, “time”, “resistance” “bandwidth” etc. associated with links or nodes

Many real networks such as world-wide airport network (WAN), E Coli. metabolic network etc. are weighted networks.

Many dynamic processes are carried on weighted networks.

Weighted networks

Barrat, Vespiggnani et al PNAS (2004)

10

3

8

62

4

1 1550

30

The tree which connects all nodes with minimum total weight.

Union of all “strong disorder” optimal paths between any two nodes.

The MST is the part of the network that most of the traffic goes through

MST -- widely used in optimal traffic flow, design and operation of communication networks.

Minimum spanning tree (MST)

A

B

In strong disorder the weight of the path is determined by the largest weight along the path!

Optimal path – strong disorderRandom Graphs and Watts Strogatz Networks

CONSTANT SLOPE

0n - typical range of neighborhood

without long range links

0n

N- typical number of nodes with

long range links

31

~ Nlopt Analytically and Numerically

LARGE WORLD!!

Compared to the diameter or average shortest path or weak disorder

Nl log~min (small world)

N – total number of nodes

Braunstein, Buldyrev, Cohen, Havlin, Stanley,

Phys. Rev. Lett. 91, 247901 (2003);

18

0

0

0

15

0

712

0

Number of times a node (or link) is used by the set of all shortest paths between all pairs of nodes - betweenes centrality.

Measure the frequency of a node being used by traffic.

( ) MSTMST ~ 2MSTP C C δ δ− ≈

Newman., Phys. Rev. E (2001) D.-H. Kim, et al., Phys. Rev. E (2004) K.-I. Goh, et al., Phys. Rev. E (2005)

Centrality of MST: How to find the importance

of nodes in transport?

For ER, scale free and many real world networks

High centrality nodes

Minimum spanning tree (MST)

• IIC is defined as the largest component at percolation criticality.

• For a random scale-free or Erdös-Rényi graph, to get the IIC, we remove the links in descending order of the weight, until

is < 2. At , the system is at criticality. Then the largest connected component of the remaining structure is the

IIC. • The IIC can be shown to be a subset of the MST

kk /2≡κ2=κ

. R. Cohen, et al., Phys. Rev. Lett. 85, 4626 (2000)

Incipient percolation cluster (IIC)

MST

I I C

The IIC is a subset of the MST

Superhighways

Superhighways and Roads

MST and IIC

sters

Superhighways (SHW) and Roads

Mean Centrality in SHW and Roads

( ) ( )opt

3 / 1 3 4

1/ 3 4 and ER

λ λ λν

λ

⎧ − − < <= ⎨

>⎩

opt

MST~ ( )f g Nν< > l

The average fraction of pairs of nodes using the IIC

MST

IIC

ll

≡uSquare lattice

ERSF, λ= 4.5SF, λ= 3.5

ER,+ 2nd largest clusterER + 3nd largest cluster

How much of the IIC is used?

The IIC is only a ZERO fraction of the network of order N2/3 !!

Distribution of Centrality in MST and IIC

Theory for Centrality Distribution

l3

2

1/31/3 1/3

2 /3

for network at criticality

is number of nodes in MST within

~ for nodes in the IIC

Thus the number of nodes with centrality

larger than is

( ) ~

for all d

n

n

s

n

nSm C n n S n

s n−> ≈

l

l

l

l

ll l l

l l

: l

l

l

:

l4/3

ue to self-similarity. Thus,

( )IICp C C−:

For IIC inside the MST:

For the MST:1/3

1/3 2/

5/3 5/3

3( ) ~ ~ Thus,

( ) ~ ~MST

dmp C n C

dn

nNm C n n N n

n n

− −

−

=

> ≈l ll

ll

ll

l

Good agreement with simulations!

Comparison between two strategies:sI: improving capacity of all IIC links--highwayssII: improving the highest centrality links in MST (same number as sI). BOTH, SAME COST

Application: improve flow in the network

We study two transport problems:

•Current flow in random resistor networks, where each link of the network represents a resistor. (Total flow, F: total current or conductance)

•Maximum flow problem from computer science, where each link of the network has an upper bound capacity. (Total flow, F: maximum possible flow into network)

Result: sI is better

Assume: multiple sources and sinks: randomly choose n pairs of nodes as sources and other n nodes as sinks

sII: improve the high C links in MST.

sI: improve the IIC links.

Two types of transport• Current flow: improve the

conductance• Maximum flow: improve

the capacity

F0: flow of original network.

FsI : flow after using sI.FsII: flow after using sII.

N=2048, <k>=4

n=50

n=250

n=500

Application: compare two strategiescurrent flow and maximum flow

Summary

• MST can be partitioned into superhighways which carry most of the traffic and roads with less traffic.

• We identify the superhighways as the largest percolation cluster at criticality -- IIC.

• Increasing the capacity of the superhighways enables to improve transport significantly. The superhighways of order N2/3 -- a zero fraction of the the network!! Wu, Braunstein, Havlin, Stanley, PRL (2006)

Two strategies to improve flow, F, of the network: sI: improving the IIC links.sII: improving the high C links in MST.

Two transport problems:• Current flow in random resistor networks, where each link of

the network represents a resistor. (Total flow, F: total current or conductance)

• Maximum flow problem in computer science[4], where each link of the network has a capacity upper bound. (Total flow, F: maximum possible flow into network)

Multiple sources and sinks: randomly choose n pairs of nodes as sources and other n nodes as sinks

resistance/capacity = eax, with a = 40 (strong disorder)

[4]. Using the push-relabel algorithm by Goldberg. http://www.avglab.com/andrew/soft.html

Applications: compare 2 strategiescurrent flow and maximum flow

Universal behavior of optimal paths in weighted networks with general disorder

Yiping Chen

Advisor: H.E. Stanley

Y. Chen, E. Lopez, S. Havlin and H.E. Stanley “Universal behavior of optimal paths in weighted networks with general disorder” PRL(submitted)

Scale Free – Optimal Path

⎪⎪⎩

⎪⎪⎨

⎧

>

=

<<−−

4

4log

43

~

31

31

)1/()3(

λ

λ

λλλ

N

NN

N

lopt

Theoretically

+

Numerically

Numerically 32log~ 1 <<− λλ Nlopt

Strong Disorder

Weak Disorder

λallforNlopt log~

Diameter – shortest path

⎪⎩

⎪⎨

⎧

<<=

>

32loglog

3loglog/log

3log

~min

λ

λ

λ

N

NN

N

l

LARGE WORLD!!

SMALL WORLD!!

Braunstein, Buldyrev, Cohen, Havlin, Stanley, Phys. Rev. Lett. 91, 247901 (2003); Cond-mat/0305051

4=λ

• Collaborators: Eduardo Lopez and Shlomo Havlin

Motivation:

Different disorders are introduced to mimic the individual properties of links or nodes (distance, airline capacity…).

Weighted random networks and optimal path:

Weights w are assigned to the links (or nodes) to mimic the individual properties of links (or nodes).

Optimal Path: the path with lowest total weight.

(If all weights the same, the shortest path is the optimal path)

4

20

7

113

5

2

source

destination

L

l

optdL~l )2(22.1 Ddopt =

L~l

Previous results:Previous results: ),1[1

)( aewaw

wP ∈=

Y. M. Strelniker et al., Phys. Rev. E 69, 065105(R) (2004)

Strong disorder : is dominated by the highest weight along the path.

optw

⇓

Weak disorder : all the weights along the optimal path contribute to the total weight along the optimal path .

)( optw

⇓

Most extensively studied weight distribution

small:

large:

a

a

(Generated by an exponential function)

Needed to reflect the properties of real world.

Ex: • exponential function----quantum tunnelling

effect• power-law----diffusion in random media • lognormal----conductance of quantum dots• Gaussian----polymers

)(wP

M

Unsolved problem: General weight distribution

Questions:Questions:1. Do optimal paths for different weight distributions show similar behavior?

2. Is it possible to derive a way to predict whether the weighted network is in strong or weak disorder in case of general weight distribution?

3. Will strong disorder behavior show up for any distributions when distribution is broad?

Theory: On lattice

7w

5w8w

3w 9w

Suppose the weight follows distribution

1w

Sw

w−≡1

1

2

lLL wwwwopt +++= 21

1+> ii wwwhere

1: , dominates the total cost (Strong limit)

012 →ww

0: , cannot dominate the total cost (Weak limit)

112 →ww

w

2w4w

6w S

S

Using percolation theory:

)2(3/4 D=νStructural & distributional parameter

Percolation exponent

1w

1w

ν/1−≅ ALS

Assume S can determine the strong or weak behavior.

We define

L

)(wP(Total cost)

General distributions studied in simulation

• Power-law

• Power-law with additional

parameter

• Lognormal

• Gaussian

a

wwP

a 1/1

)(−

=

Δ=

−

a

wwP

a 1/1

)(

w

ewP

w 22 2/)(ln

~)(σ−

)2/( 22

~)( σwewP −

)1,0[)( ∈= xxxf a

)1,1[)( Δ−∈= xxxf a

My simulation result on 2D-lattice

νν ALS /≅−

22.1~ Ll

ννν ALSALS //1 ≅→≅ −−

Strong: L~lWeak:

-0.22

L the linear size of latticelthe length of optimal path

Answer to questions 1 and 2:

Y. Chen, E. Lopez, S. Havlin and H.E. Stanley “Universal behavior of optimal paths in weighted networks with general disorder” PRL(submitted)

Erdős-Rényi (ER) Networks

Definition:

A set of N nodesp

For each pair of nodes, they have probability p to be connected

My simulations on ER network show the same agreement with theory.

Distributions that are not expected to have strong disorder behavior

)1ln()1( cc

c

pp

pA

−−−=wewP σ−~)(

21 )]([1 )(2 cperfc

c

eperf

pA −−−

=π

• Gaussian

• Exponential

)2/( 22

~)( σwewP −

A is independent of which describes the broadness of distribution.

No matter how broad the distribution is, can not be large, and no strong disorder will show up.

σν/1−≅ ALS

cp( the percolation threshold, constant for certain network structure)

Answer to question 3:

Summary of answers to 3 questions

1. Do optimal paths in different weight distributions show similar behavior?Yes

2. Is it possible to derive a way to predict whether the weighted network is in strong or weak disorder in case of general weight distribution? Yes

3. Will strong disorder behavior show up for any distributions when distribution is broad?No

Theory: On lattice[ )

∫ ′′=≡

∈≡− w

wdwPwfx

xxfw

wPw

0

1 )()(

1,0)(

)(

)( 4xf

)( 7xf

)( 5xf

)( 8xf

)( 3xf )( 9xf

Suppose

follows distribution

)( 6xf

)( 2xf

)( 1xf

)()(ln

)2)(()(

)(1)(

21

11

21

1

xxdx

fdS

Sxfxf

xfxf

xx

−≡

−=⎥⎦

⎤⎢⎣

⎡+≅

=

)()()( 21 lLL xfxfxfwopt +++=

)()( 1+> ii xfxfwhere

SS goes large: (Strong))()( 21 xfxf >>SS goes small: and are comparable (Weak)

)( 1xf )( 2xf

⇒Percolation applies

Percolation Theory

νν σσ /1/1

21

~~21

−−

≅≅

LpLp

pxpx

cxcx

cc

Strong disorder and percolation behave in

the similar way

∫ ′′≡w

wdwPx0

)(

w

)(wP

cp

ix

iw

In finite lattice with linear size L: νσ /1~ −Lpcpc

ν/121 ~ −−L

p

xx

c

Thus

Percolation threshold (0.5 for 2D square lattice)

Percolation properties:

)2(3/4 D=νThe first and second highest weighted bonds in optimal path will be close to and follow its deviation rule.

cp

⇓

From percolation theory

∫ =

==

⇓

−−

cw

c

cc

c

pdwwPwhere

ALwPw

LpS

0

/1/1

)(

)(ν

ν

)()(ln

21

1

xxdx

fdS

xx

−==

ν/1)(ln −

=

=

⇓

Ldx

fdpS

cpxc

The result comes from percolation theory

ν/121 ~ −−L

p

xx

c

}[ )

∫ ′′=≡

∈≡− w

wdwPwfx

xxfw

0

1 )()(

1,0)( }Transfer back to original disorder distribution

Test on known result

5.0=cp

),1[1

)( aewaw

wP ∈=

3/4=ν

Apply our theory on disorder distribution , we get percolation

threshold percolation exponent

ν/1−= LapS c

To have same behavior by keeping fixed, we get

=νaL /

S

constantCompatible with the reported results.

(The crossover from strong to weak disorder occurs at )1/ ≈νaL

νcp

In 2D square lattice

(Constants for certain structure)

Scaling on ER networkPercolation at criticality on Erdős-Rényi(ER) networks is equivalent to percolation on a lattice at the upper critical dimension .

6=cd

6/1~ NL

2/1=ν

3/16/1/1 −−− === ANANALS νν⇓

Virtual linear sizePercolation exponent in ER network

(N = number of nodes)

( is now depending on number of nodes in ER network)S

Simulation result on ER networks1=νANSANS /3/113/1 =→= −−

N

N

log~

~ 3/1

l

lStrong:

Weak:

log-loglog-linear

In ER network, the percolation exponent

From early report:

L.A. Braunstein et al. Phys. Rev. Lett. 91, 168701 (2003)

(N=number of nodes)

1

1

log~

~−

−

S

S

l

l⇒ Strong:

Weak: