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Transport in weighted networks:
superhighways and roads
Shlomo Havlin
Bar-Ilan University
Israel
Collaborators: Z. Wu, Y. Chen,
E. Lopez, S. Carmi, L.A.
Braunstein, S. Buldyrev, H. E.
Stanley
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?
• Identifying the superhighways and increasing
their capacity enables to improve transport
significantly.
Weighted networks
10
 Networks with weights, such as
30
6
“cost”, “time”, “bandwidth” etc.
associated with links or nodes
2
3
4
1
 Many real networks such as
50
15
8
Barrat et al PNAS (2004)
world-wide airport network
(WAN), E Coli. metabolic
network etc. are weighted
networks.
 Many dynamic processes are
carried on weighted networks.
Minimum spanning tree (MST)
10
3
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.
30
6
2
4
B
1

50
15
8
A
In strong disorder the weight
of the path is determined by the
largest weight along the path!
Optimal path – strong disorder
Random Graphs and Watts Strogatz Networks
CONSTANT SLOPE
N – total number of nodes
l opt ~ N
1
3
Analytically and Numerically
LARGE WORLD!!
Compared to the diameter or
average shortest path or weak
disorder
lmin ~ log N
(small world)
n 0 - typical range of neighborhood
without long range links
Braunstein, Buldyrev, Cohen, Havlin, Stanley,
Phys. Rev. Lett. 91, 247901 (2003);
N
- typical number of nodes with
n0
long range links
Betweeness Centrality of MST
0
0

Number of times a node (or link)
is used by the set of all shortest
paths between all pairs of node.

Measure the frequency of a node
being used by traffic.
15
18
12
0
7
0
PMST  C  ~ C MST  MST  2
0
For ER, scale free and real world networks
Newman., Phys. Rev. E (2001)
D.-H. Kim, et al., Phys. Rev. E (2004)
K.-I. Goh, et al., Phys. Rev. E (2005)
Minimum spanning tree (MST)
High centrality nodes
Incipient percolation cluster (IIC)
• 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
 k /k
2
is < 2. At   2, 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
. R. Cohen, et al., Phys. Rev. Lett. 85, 4626 (2000)
MST and IIC
MST
Superhighways
and Roads
The IIC is a subset of the MST
IIC
Superhighways
Superhighways (SHW) and Roads
ster
Mean Centrality in SHW and Roads
The average fraction of pairs of nodes using the IIC
 f ~ g (
MST
N
 opt
)
   3 /    1
3  4
 opt  
1/ 3
  4 and ER

How much of the IIC is used?
Square lattice
 IIC
u
 MST
ER + 3nd largest cluster
ER,+ 2nd largest cluster
ER
SF, λ= 4.5
SF, λ= 3.5
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
For IIC inside the MST:
n
3
for network at criticality
n is number of nodes in MST within
S ~
2
for nodes in the IIC
Thus the number of nodes with centrality
larger than n is
m(C  n )
1/ 3
n
S n1/ 3
 2 / 3 S ~ n 1/ 3
S
n
for all due to self-similarity. Thus,
pIIC (C ) C 4 / 3
For the MST:
m(C  n ) ~ n
1/ 3
N n1/ 3

N ~ n 2 / 3 Thus,
n
n
pMST (C ) 
dm
~ n 5 / 3 ~ C 5 / 3
dn
Good agreement with simulations!
Application: improve flow in the network
Comparison between two strategies:
sI: improving capacity of all IIC links--highways
sII: improving the highest centrality links in MST (same number as sI).
Assume: multiple sources and sinks: randomly choose n pairs
of nodes as sources and other n nodes as sinks
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
Application: compare two strategies
current flow and maximum flow
n=50
n=250
n=500
sI: improve the IIC links.
sII: improve the high C
links in MST.
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
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!!
Applications: compare 2 strategies
current flow and maximum flow
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)
resistance/capacity = eax, with a = 40 (strong disorder)
Multiple sources and sinks: randomly choose n pairs of nodes
as sources and other n nodes as sinks
Two strategies to improve flow, F, of the network:
sI: improving the IIC links.
sII: improving the high C links in MST.
[4]. Using the push-relabel algorithm by Goldberg. http://www.avglab.com/andrew/soft.html
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
Strong Disorder
 4
Theoretically
+
•
Collaborators: Eduardo Lopez and Shlomo Havlin
Numerically
 N (  3) /( 1)
 1
l opt ~  N 3 log N
 13
 N
3  4
4
4
LARGE WORLD!!
Numerically
lopt ~ log  1 N
SMALL WORLD!!
Weak Disorder
lopt ~ log N for all 
Diameter – shortest path
Braunstein, Buldyrev, Cohen, Havlin, Stanley,
Phys. Rev. Lett. 91, 247901 (2003);
Cond-mat/0305051
lmin
 3
log N

~ log N / log log N
 3
 log log N
2 3

2 3
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).
7
4
3
source
11
20
5
destination
2
Optimal Path: the path with lowest total weight.
(If all weights the same, the shortest path is the optimal path)
Previous
1
Most extensively studied
w  [1, e a )
results: P(w) 
weight distribution
aw
(Generated by an exponential function)
a small:
Weak disorder : all the weights along

the optimal path contribute to the total
weight along the optimal path ( wopt ).

~L
a large:
L
Strong disorder : wopt is dominated
by the highest weight along the path.

d opt
~L
d opt  1.22 (2 D)
Y. M. Strelniker et al., Phys. Rev. E 69, 065105(R) (2004)
Unsolved problem:
General weight distribution P(w)
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

Questions:
Do optimal paths for different weight
distributions show similar behavior?
1.
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
Suppose the weight
w follows distribution P(w)
wopt  w1  w2    w (Total cost)
where wi  wi 1
w7
w2
w4
w8 w6
w1
w3 w9
L
We define
w5
w2
 1 S
w1
0: w2 w1  1,
w1cannot dominate the total cost (Weak limit)
S
S
1: w2 w1  0,
w1 dominates the total cost (Strong limit)
Assume S can determine the strong or weak behavior.
Using percolation theory:
S  AL1/
Structural & distributional parameter
Percolation exponent
  4 / 3 ( 2 D)
General distributions studied in
simulation
• Power-law
P ( w) 
• Gaussian
P( w) ~
x [0,1)
f ( x)  x a
x [1  ,1)
w
a
• Power-law with additional
w1/ a 1
P( w) 
parameter
a
• Lognormal
f ( x)  x a
1 / a 1
e
P(w) ~ e
 (ln w ) 2 / 2 2
w
 w2 /(2 2 )
Answer to questions 1 and 2:
My simulation result on 2D-lattice
-0.22
S   L / A
L the linear size of lattice
the length of optimal
path
Strong:
 ~ L1.22
Weak:
~L
S  AL1/  S   L / A
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:
For each pair of nodes, they
have probability p to be
connected
A set of N nodes
p
My simulations on ER network show the same agreement with theory.
Answer to question 3:
Distributions that are not expected to
have strong disorder behavior
• Gaussian
P( w) ~ e
 w2 /( 2 2 )
• Exponential
P( w) ~e
w
A

pc
1
2 erf ( pc )e
[ erf 1 ( pc )]2
pc
A
 (1  pc ) ln( 1  pc )
( pc the percolation threshold, constant for certain network structure)
A is independent of

which describes the broadness of distribution.
No matter how broad the distribution is,
large, and no strong disorder will show up.
S  AL1/can not be
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?
Theory: On lattice
Suppose w follows distribution P( w)
w  f ( x)
x  0,1
w
x  f ( w)   P( w)dw
1
f ( x7 )
0
wopt  f ( x1 )  f ( x2 )    f ( x )
f ( x2 )
f ( x4 )
f ( x6 )
f ( x8 )
f ( x3 )
f ( x5 )
f ( x1 )
f ( x9 )
where f ( xi )  f ( xi 1 )
 f ( x2 ) 
 f ( x1 ) 1 
  f ( x1 )( 2  S )
f ( x1 ) 

d (ln f )
S
( x1  x2 )
dx x  x1
S goes small: f ( x1 ) and f ( x2 ) are comparable (Weak)
S goes large: f ( x1 )  f ( x2 ) (Strong)

Percolation applies
Percolation Theory
Percolation properties:
Percolation threshold pc(0.5 for 2D square lattice)
In finite lattice with linear size L:
 p ~ pc L1/   4 / 3 (2D)
The firstdisorder
and secondand
highest
weighted
Strong
percolation
bonds in optimal path will be close to pc
the similar
andbehave
follow itsin
deviation
rule. way
c
w
P(w)
Thus
xi
x   P( w)dw
0
x1  pc
x2  pc
 x ~ pc L1/
 x ~ pc L1/
1
w
wi
x1

x
2
pc
2
~ L1/
From percolation theory
The result comes from
percolation theory
x1  x2
~ L1/
pc
Transfer back to original
disorder distribution
w  f ( x)
x  0,1
w
x  f (w)   P(w)dw
1
0


d (ln f )
S
( x1  x2 )
dx x  x1

d (ln f )
S  pc
dx
L1/
x  pc

pc L1/
S
 AL1/
wc P( wc )
where

wc
0
P( w)dw  pc
Test on known result 1
Apply our theory on disorder distribution P( w) 
w  [1, e a ),
aw
we get
pc
1/
S  apc L

percolation threshold
percolation exponent
(Constants for certain structure)
In 2D square lattice
pc  0.5
  4/3
To have same behavior by keeping S fixed, we get
L / a  constant
Compatible with the reported results.
(The crossover from strong to weak disorder occurs at L / a  1 )
Scaling on ER network
Percolation at criticality on Erdős-Rényi(ER)
networks is equivalent to percolation on a lattice at
the upper critical dimension
.
dc  6
Virtual linear size
L~N
1/ 6
(N = number of nodes)
Percolation exponent in ER network
1/
S  AL

 AN
1/ 6
  1/ 2
 AN
1/ 3
( S is now depending on number of nodes in ER network)
In ER network, the percolation exponent   1
Simulation result on ER networks
S  AN 1/ 3  S 1  N 1/ 3 / A
(N=number of nodes)
log-linear
From early report:
log-log
Strong:  ~ N 1/ 3
Weak:  ~ log N

Strong:  ~ S 1
Weak:  ~ log S 1
L.A. Braunstein et al. Phys. Rev. Lett. 91, 168701 (2003)