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

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

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

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

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

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

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

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

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 2 2 2 3 3 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

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. Hołyst

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

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

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

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

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, 056110 (2004) J. Sienkiewicz, P. Fronczak, J. A. Hołyst

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

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, 066102 (2007) J. Sienkiewicz, P. Fronczak, J. A. Hołyst

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

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

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

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

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