Synchronization in Scale Free networks: The role of finite size effects

TL;DR

This paper investigates how finite size effects influence synchronization in Scale Free networks, revealing a crossover in fluctuation behavior at a system size dependent on the degree distribution exponent.

Contribution

It introduces a novel crossover behavior in fluctuations for Scale Free networks and provides a function modeling this transition, enhancing understanding of finite size effects.

Findings

Identification of a crossover between logarithmic and constant fluctuation regimes.

Fluctuations decrease with increasing λ for large system sizes.

Synchronization improves as λ increases above a critical size.

Abstract

Synchronization problems in complex networks are very often studied by researchers due to its many applications to various fields such as neurobiology, e-commerce and completion of tasks. In particular, Scale Free networks with degree distribution $P(k)\sim k^{-\lambda}$, are widely used in research since they are ubiquitous in nature and other real systems. In this paper we focus on the surface relaxation growth model in Scale Free networks with $2.5< \lambda <3$, and study the scaling behavior of the fluctuations, in the steady state, with the system size $N$. We find a novel behavior of the fluctuations characterized by a crossover between two regimes at a value of $N=N^*$ that depends on $\lambda$: a logarithmic regime, found in previous research, and a constant regime. We propose a function that describes this crossover, which is in very good agreement with the simulations. We also find that, for a system size above $N^{*}$, the fluctuations decrease with $\lambda$, which means that the synchronization of the system improves as $\lambda$ increases. We explain this crossover analyzing the role of the network's heterogeneity produced by the system size $N$ and the exponent of the degree distribution.