Scaling behavior of the contact process in networks with long-range connections

TL;DR

This study investigates the contact process on networks with long-range connections, revealing phase transition behavior, critical exponents influenced by network structure, and the impact of initial conditions on scaling laws.

Contribution

It provides numerical analysis of phase transitions and critical exponents in long-range connected networks, highlighting the effects of network inhomogeneity and initial seed placement.

Findings

Absorbing phase transition occurs at finite infection rate.

Critical exponents depend on network structure and initial seed location.

Log-periodic oscillations indicate discrete scale invariance.

Abstract

We present simulation results for the contact process on regular, cubic networks that are composed of a one-dimensional lattice and a set of long edges with unbounded length. Networks with different sets of long edges are considered, that are characterized by different shortest-path dimensions and random-walk dimensions. We provide numerical evidence that an absorbing phase transition occurs at some finite value of the infection rate and the corresponding dynamical critical exponents depend on the underlying network. Furthermore, the time-dependent quantities exhibit log-periodic oscillations in agreement with the discrete scale invariance of the networks. In case of spreading from an initial active seed, the critical exponents are found to depend on the location of the initial seed and break the hyper-scaling law of the directed percolation universality class due to the inhomogeneity of the networks. However, if the cluster spreading quantities are averaged over initial sites the hyper-scaling law is restored.