1. Log-periodic oscillations due to discrete effects in complex networks
Julian Sienkiewicz, Piotr Fronczak and Janusz A. Hołyst
Faculty of Physics and
Centre for Excellence for Complex Systems Research
Warsaw University of Technology, Poland
J. Sienkiewicz, P. Fronczak, J. A. Hołyst

2. How does <k> influence <l>?
Log-periodic oscillations due to discrete effects in complex networks
How does <k> influence <l>?
How does <l> change vs. N for different <k>?
J. Sienkiewicz, P. Fronczak, J. A. Hołyst

3. Dept. of Astronomy Univ. of Florida
Log-periodic oscillations due to discrete effects in complex networks
Average path lengths play a crucial role in „networkology”…
Dept. of Astronomy Univ. of Florida
Asset tree – S&P 500
J. Sienkiewicz, P. Fronczak, J. A. Hołyst

4. More precisely (Fronczak et al. 2004):
Log-periodic oscillations due to discrete effects in complex networks
In various works the logarithmic dependence of <l> on N was observed.
More precisely (Fronczak et al. 2004):
J. Sienkiewicz, P. Fronczak, J. A. Hołyst

5. Let’s suppose we need to bulid a computer network…
Log-periodic oscillations due to discrete effects in complex networks
Let’s suppose we need to bulid a computer network…
But we don’t want to allow for either of presented situations…
J. Sienkiewicz, P. Fronczak, J. A. Hołyst

6. The same applies to public transport networks…
Log-periodic oscillations due to discrete effects in complex networks
The same applies to public transport networks…
J. Sienkiewicz, P. Fronczak, J. A. Hołyst

7. Two contradicting demands:
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. Hołyst

8. However, for large <k> the relation is not so straigthforward!!!
Log-periodic oscillations due to discrete effects in complex networks
However, for large <k> the relation is not so straigthforward!!!
J. Sienkiewicz, P. Fronczak, J. A. Hołyst

9. Scaling of internode distances
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: 1 4 2
lij
kikj
1 x 1 = 1 3
kikj
<lij> 1
3.00 2
2.17 4
1.60 8
1.00
1 x 2 = 2
1 x 4 = 4
1 2
2 x 2 = 4
2 2 1
2 x 4 = 8
1 1 1
J. Sienkiewicz, P. Fronczak, J. A. Hołyst

10. Log-periodic oscillations due to discrete effects in complex networks
Empirical data
Holyst et al. Phys. Rev. E 72, (2005), Physica A 351, 167 (2005)
J. Sienkiewicz, P. Fronczak, J. A. Hołyst

11. Theoretical approach (1)
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. Hołyst

12. Theoretical approach (2)
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):
Let’s consider a tree with a branching factor κ.
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. Hołyst

13. Spanish lang. <k>=7
Log-periodic oscillations due to discrete effects in complex networks
Oscillations
In many networks with a high value of average degree <k> we have observed oscillations along the trend line:
Spanish lang. <k>=7
Actors <k>=86
Food web <k>=26
Opole PTN <k>=50
ER <k>=40
BA <k>=40
J. Sienkiewicz, P. Fronczak, J. A. Hołyst

14. 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 ρ(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. Hołyst

15. Log-periodic oscillations due to discrete effects in complex networks
Hidden variables (2)
One can prove that in a case of uncorrelated networks p*ij(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, (2004)
J. Sienkiewicz, P. Fronczak, J. A. Hołyst

16. 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 <lij>:
J. Sienkiewicz, P. Fronczak, J. A. Hołyst

17. Log-periodic oscillations due to discrete effects in complex networks
Depending on the value of <k> the part R may have a different impact on <lij> plots for ρ(h)=(a-1)ma-1h-a a=3 N=10000 m=2 (upper row) m=40 (lower row):
Sienkiewicz, P. Fronczak, Hołyst Phys. Rev. E 75, (2007)
J. Sienkiewicz, P. Fronczak, J. A. Hołyst

18. Log-periodic oscillations due to discrete effects in complex networks
To obtain an approximative analytical solution we use the following assumptions:
SF a=3 N=10000 m=40, inset: oscillations for m=2 i m=40
J. Sienkiewicz, P. Fronczak, J. A. Hołyst

19. Log-periodic oscillations due to discrete effects in complex networks
To obtain an approximative analytical solution we assume:
SF N=10000 m=10
J. Sienkiewicz, P. Fronczak, J. A. Hołyst

20. Log-periodic oscillations due to discrete effects in complex networks
It is essential that after integrating the <lij> over hidden variable distributions we obtain a dependence of average path length <l> on network size N, where the oscillations can also be spotted:
Red circles: SF <k>=8 a=3
Red circles: SF <k>=60 a=3
J. Sienkiewicz, P. Fronczak, J. A. Hołyst

21. Cost function Applications:
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. Hołyst

22. We are very thankful to Dr Agata Fronczak for fruitful discussions.
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. Hołyst