Transport in weighted networks:

1
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)
2
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.

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Weighted networks

• Networks with weights, such as cost, time,
  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.

Barrat et al PNAS (2004)
4
Minimum spanning tree (MST)

• 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.

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

5
Optimal path strong disorder Random Graphs and
Watts Strogatz Networks
N total number of nodes
Braunstein, Buldyrev, Cohen, Havlin, Stanley,
Phys. Rev. Lett. 91, 247901 (2003)
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Betweeness Centrality of MST

• 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.

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)
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Minimum spanning tree (MST)
High centrality nodes
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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
• is lt 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

. R. Cohen, et al., Phys. Rev. Lett. 85, 4626
(2000)
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MST and IIC
MST
Superhighways and Roads
The IIC is a subset of the MST
I I C
Superhighways
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Superhighways (SHW) and Roads
ster
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Mean Centrality in SHW and Roads
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The average fraction of pairs of nodes using the
IIC
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How much of the IIC is used?
Square lattice
ER 3nd largest cluster
ER, 2nd largest cluster
ER
SF, ? 4.5
SF, ? 3.5
The IIC is only a ZERO fraction of the network of
order N2/3 !!
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Distribution of Centrality in MST and IIC
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Theory for Centrality Distribution
For IIC inside the MST
For the MST
Good agreement with simulations!
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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
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Application compare two strategiescurrent flow
and maximum flow
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

n50
n250
F0 flow of original network. FsI flow after
using sI. FsII flow after using sII.
n500
N2048, ltkgt4
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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!!

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Applications compare 2 strategiescurrent 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 science4,
  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
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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)
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Scale Free Optimal Path
Strong Disorder

• Collaborators Eduardo Lopez and Shlomo Havlin

LARGE WORLD!!
SMALL WORLD!!
Weak Disorder
Diameter shortest path
Braunstein, Buldyrev, Cohen, Havlin, Stanley,
Phys. Rev. Lett. 91, 247901 (2003)
Cond-mat/0305051
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Motivation
Different disorders are introduced to mimic the
individual properties of links or nodes
(distance, airline capacity).
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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)

7
4
3
source
11
20
5
destination
2
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Previous results
Most extensively studied weight distribution
(Generated by an exponential function)
small
Weak disorder all the weights along the optimal
path contribute to the total weight along the
optimal path .
L
large
Strong disorder is dominated by the
highest weight along the path.
Y. M. Strelniker et al., Phys. Rev. E 69,
065105(R) (2004)
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Unsolved problem General weight distribution

• 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

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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?
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Theory On lattice
Suppose the weight
follows distribution
(Total cost)
where
We define
0 , cannot
dominate the total cost (Weak limit)
1 , dominates the
total cost (Strong limit)
L
Assume S can determine the strong or weak
behavior.
Using percolation theory
Percolation exponent
Structural distributional parameter
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General distributions studied in simulation

• Power-law
• Power-law with additional
• parameter
• Lognormal
• Gaussian

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My simulation result on 2D-lattice
Answer to questions 1 and 2
-0.22
L the linear size of lattice
Strong
Weak
the length of optimal path
Y. Chen, E. Lopez, S. Havlin and H.E. Stanley
Universal behavior of optimal paths in weighted
networks with general disorder PRL(submitted)
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Erdos-Rényi (ER) Networks
For each pair of nodes, they have probability p
to be connected

• Definition

A set of N nodes
p
My simulations on ER network show the same
agreement with theory.
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Distributions that are not expected to have
strong disorder behavior
Answer to question 3

• Gaussian
• Exponential

( 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,
can not be large, and no strong
disorder will show up.
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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

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Theory On lattice
Suppose
follows distribution
where
S goes small and are
comparable (Weak)
S goes large
(Strong)
Percolation applies
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Percolation Theory
Percolation properties
Percolation threshold (0.5 for 2D square
lattice)
In finite lattice with linear size L
The first and second highest weighted bonds in
optimal path will be close to and follow
its deviation rule.
Strong disorder and percolation behave in the
similar way
Thus
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• From percolation theory

The result comes from percolation theory
Transfer back to original disorder distribution
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Test on known result
Apply our theory on disorder distribution
, we get
percolation threshold percolation exponent
(Constants for certain structure)
In 2D square lattice
To have same behavior by keeping fixed, we
get
constant
Compatible with the reported results.
(The crossover from strong to weak disorder
occurs at )
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Scaling on ER network

• Percolation at criticality on Erdos-Rényi(ER)
  networks is equivalent to percolation on a
  lattice at the upper critical dimension
  .

Virtual linear size
(N number of nodes)
Percolation exponent in ER network
( is now depending on number of nodes in ER
network)
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Simulation result on ER networks
In ER network, the percolation exponent
(Nnumber of nodes)
log-log
log-linear
Strong
Weak

Strong
Weak

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