Hierarchical deposition and scale-free networks: a visibility algorithm approach

TL;DR

This paper investigates the hierarchical deposition of particles and its resulting scale-free networks using a visibility algorithm, revealing scale invariance, modularity, and the influence of randomness on network properties.

Contribution

It introduces a novel approach linking hierarchical deposition processes with scale-free network generation via a visibility algorithm, providing exact analytical results.

Findings

Deterministic model yields a scale-free network with specific degree exponents.

Random deposition results in a scale-free network with exponent approaching 3.

Network exhibits scale invariance and modular hierarchical structure.

Abstract

The growth of an interface formed by the hierarchical deposition of particles of unequal size is studied in the framework of a dynamical network generated by a horizontal visibility algorithm. For a deterministic model of the deposition process, the resulting network is scale-free with dominant degree exponent $\gamma_e = \ln{3}/\ln{2}$ and transient exponent $\gamma_o = 1$. An exact calculation of the network diameter and clustering coefficient reveals that the network is scale invariant and inherits the modular hierarchical nature of the deposition process. For the random process, the network remains scale free, where the degree exponent asymptotically converges to $\gamma =3$, independent of the system parameters. This result shows that the model is in the class of fractional Gaussian noise (fGn) through the relation between the degree exponent and the series' Hurst exponent $H$. Finally, we show through the degree-dependent clustering coefficient $C(k)$ that the modularity remains present in the system.