transport in weighted networks: optimal path and superhighways
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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:
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