Evolution equation for a model of surface relaxation in complex networks

TL;DR

This paper derives an analytical evolution equation for surface relaxation in complex networks, revealing nonlinear effects due to heterogeneity that explain the divergence of surface roughness with system size.

Contribution

It introduces a new analytical model for surface relaxation in complex networks, accounting for heterogeneity-induced nonlinearities absent in Euclidean lattice models.

Findings

Nonlinear terms arise in the evolution equation due to network heterogeneity.

Logarithmic divergence of saturation roughness with system size for degree exponent λ<3.

Analytical results align with previous numerical findings.

Abstract

In this paper we derive analytically the evolution equation of the interface for a model of surface growth with relaxation to the minimum (SRM) in complex networks. We were inspired by the disagreement between the scaling results of the steady state of the fluctuations between the discrete SRM model and the Edward-Wilkinson process found in scale-free networks with degree distribution $ P(k) \sim k^{-\lambda}$ for $\lambda <3$ [Pastore y Piontti {\it et al.}, Phys. Rev. E {\bf 76}, 046117 (2007)]. Even though for Euclidean lattices the evolution equation is linear, we find that in complex heterogeneous networks non-linear terms appear due to the heterogeneity and the lack of symmetry of the network; they produce a logarithmic divergency of the saturation roughness with the system size as found by Pastore y Piontti {\it et al.} for $\lambda <3$.