Log-periodic oscillations due to discrete effects in complex networks

TL;DR

This paper investigates how discretization in complex networks causes log-periodic oscillations in internode distances and average path length, supported by analytical derivations, simulations, and real-world examples.

Contribution

It reveals the impact of discrete effects on network metrics and introduces analytical models to explain observed oscillations and their influence on network optimization.

Findings

Discretization causes log-periodic oscillations in network distances.

Analytical expressions match numerical simulations and real-world data.

Discrete effects can lead to nontrivial solutions in network optimization.

Abstract

We show that discretization of internode distribution in complex networks affects internode distances l_ij calculated as a function of degrees (k_i k_j) and an average path length <l> as function of network size N. For dense networks there are log-periodic oscillations of above quantities. We present real-world examples of such a behavior as well as we derive analytical expressions and compare them to numerical simulations. We consider a simple case of network optimization problem, arguing that discrete effects can lead to a nontrivial solution.