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Log-periodic oscillations due to discrete effects
in complex networks
Log-periodic oscillations due to discrete effects
in complex networks
Julian Sienkiewicz, Piotr Fronczak and Janusz A.
Holyst
Faculty of Physics and Centre for Excellence for
Complex Systems Research Warsaw University of
Technology, Poland
J. Sienkiewicz, P. Fronczak, J. A. Holyst
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Log-periodic oscillations due to discrete effects
in complex networks
How does ltkgt influence ltlgt?
How does ltlgt change vs. N for different ltkgt?
J. Sienkiewicz, P. Fronczak, J. A. Holyst
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Log-periodic oscillations due to discrete effects
in complex networks
Average path lengths play a crucial role in
networkology
Dept. of AstronomyUniv. of Florida
Asset tree SP 500
J. Sienkiewicz, P. Fronczak, J. A. Holyst
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Log-periodic oscillations due to discrete effects
in complex networks
In various works the logarithmic dependence of
ltlgt on N was observed.
More precisely (Fronczak et al. 2004)
J. Sienkiewicz, P. Fronczak, J. A. Holyst
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Log-periodic oscillations due to discrete effects
in complex networks
Lets suppose we need to bulid a computer network
J. Sienkiewicz, P. Fronczak, J. A. Holyst
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Log-periodic oscillations due to discrete effects
in complex networks
The same applies to public transport networks
J. Sienkiewicz, P. Fronczak, J. A. Holyst
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Log-periodic oscillations due to discrete effects
in complex networks
A simple way to get rid of this problem is to
introduce a following cost function (Schweitzer,
2003).

• Two contradicting demands
• fully connected network with shortest
  connections
• a tree with the smallest number of links

J. Sienkiewicz, P. Fronczak, J. A. Holyst
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Log-periodic oscillations due to discrete effects
in complex networks
However, for large ltkgt the relation is not so
straigthforward!!!
J. Sienkiewicz, P. Fronczak, J. A. Holyst
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Log-periodic oscillations due to discrete effects
in complex networks
Scaling of internode distances
In 2005 our group observed the following law in
public transport networks and then also in other
real-world networks as well as in such models as
Erdos-Renyi and Barabasi-Albert networks
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1 x 1 1
kikj ltlijgt
1 3.00
2 2.17
4 1.60
8 1.00
1 2 2 2 3 3
1 x 2 2
1 2
1 x 4 4
2 2 1
2 x 2 4
1 1 1
2 x 4 8
J. Sienkiewicz, P. Fronczak, J. A. Holyst
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Log-periodic oscillations due to discrete effects
in complex networks
Empirical data
Holyst et al. Phys. Rev. E 72, 026108 (2005),
Physica A 351, 167 (2005)
J. Sienkiewicz, P. Fronczak, J. A. Holyst
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Log-periodic oscillations due to discrete effects
in complex networks
Theoretical approach (1)
We need to find the mean shortest path between
node i of degree ki and node j of degree kj .
The probability, that after choosing a random
edge in the network and going in a random
direction we arrive at the node j is
So in average we need
trials to come to the node j.
J. Sienkiewicz, P. Fronczak, J. A. Holyst
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Log-periodic oscillations due to discrete effects
in complex networks
Theoretical approach (2)
If the distance between node i and the surface of
the tree is x, then on the surface there are (in
average)
Lets consider a tree with a branching factor ?.
nodes and the same number of edges (we neglect
loops). A condition for the tree to reach node j
is
This leads to the following equation
J. Sienkiewicz, P. Fronczak, J. A. Holyst
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Log-periodic oscillations due to discrete effects
in complex networks
Oscillations
In many networks with a high value of average
degree ltkgt we have observed oscillations along
the trend line
Spanish lang.ltkgt7
Actorsltkgt86
Food webltkgt26
Opole PTNltkgt50
ERltkgt40
BAltkgt40
J. Sienkiewicz, P. Fronczak, J. A. Holyst
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Log-periodic oscillations due to discrete effects
in complex networks
Hidden variables (1)
If 1) To each node we can assign a hidden
variable hi, drawn from a ?(h) distribution. 2)
The probability that an edge exists between a
random pair of nodes is proportional to the
product hihj Then Any degree probability
distribution P(k) can be written as
J. Sienkiewicz, P. Fronczak, J. A. Holyst
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Log-periodic oscillations due to discrete effects
in complex networks
Hidden variables (2)
One can prove that in a case of uncorrelated
networks pij(x) (the probability that there is
distance x between nodes i and j) is equal to
where
A. Fronczak et al. Phys. Rev. E 70, 056110 (2004)
J. Sienkiewicz, P. Fronczak, J. A. Holyst
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Log-periodic oscillations due to discrete effects
in complex networks
Hidden variables (3)
Summing the terms related to distances 1, 2, 3
we have
And finally we obtain the following expression
for ltlijgt
J. Sienkiewicz, P. Fronczak, J. A. Holyst
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Log-periodic oscillations due to discrete effects
in complex networks
Oscillations (1)
Depending on the value of ltkgt the part R may
have a different impact on ltlijgt plots for
?(h)(a-1)ma-1h-a a3 N10000 m2 (upper row)
m40 (lower row)
Sienkiewicz, P. Fronczak, Holyst Phys. Rev. E 75,
066102 (2007)
J. Sienkiewicz, P. Fronczak, J. A. Holyst
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Log-periodic oscillations due to discrete effects
in complex networks
Oscillations (2)
To obtain an approximative analytical solution we
use the following assumptions
SF a3 N10000 m40, inset oscillations for m2
i m40
J. Sienkiewicz, P. Fronczak, J. A. Holyst
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Log-periodic oscillations due to discrete effects
in complex networks
Oscillations (3)
To obtain an approximative analytical solution we
assume
SF N10000 m10
J. Sienkiewicz, P. Fronczak, J. A. Holyst
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Log-periodic oscillations due to discrete effects
in complex networks
Oscillations (4)
It is essential that after integrating the ltlijgt
over hidden variable distributions we obtain a
dependence of average path length ltlgt on network
size N, where the oscillations can also be
spotted
Red circles SF ltkgt8 a3
Red circles SF ltkgt60 a3
J. Sienkiewicz, P. Fronczak, J. A. Holyst
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Log-periodic oscillations due to discrete effects
in complex networks
Cost function
The effects of oscillations on average path
length may appear even in a simple problem of
cost optimization in the network.
The First term links maintaining cost The
Second term delays of information cost
Cost optimization in computer network Cost
optimization in public transport network
Applications
J. Sienkiewicz, P. Fronczak, J. A. Holyst
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Log-periodic oscillations due to discrete effects
in complex networks
Summary

• Both in network models and real-world networks
  we observe log-periodical oscillations over
  internode distances and average path length
• This effect takes place for networks with high
  density (high average degree)
• It is useful (and observable) even in very
  simple examples possible application in case of
  i.e. public transport networks or LAN

Acknowledgements
We are very thankful to Dr Agata Fronczak for
fruitful discussions.
J. Sienkiewicz, P. Fronczak, J. A. Holyst