Diffusive Persistence on Disordered Lattices and Random Networks

TL;DR

This paper investigates how the topology of disordered and random networks influences diffusive persistence, revealing power-law behaviors at the percolation threshold in 2D lattices and complex dynamics in Erdős-Rényi networks.

Contribution

It provides a detailed analysis of diffusive persistence in disordered and random networks, highlighting the impact of structural transitions on persistence scaling laws.

Findings

Power-law decay of persistence in 2D disordered networks above percolation threshold.

Altered scaling exponent at the percolation threshold due to structural transition.

No simple power-law scaling observed in Erdős-Rényi networks above percolation threshold.

Abstract

To better understand the temporal characteristics and the lifetime of fluctuations in stochastic processes in networks, we investigated diffusive persistence in various graphs. Global diffusive persistence is defined as the fraction of nodes for which the diffusive field at a site (or node) has not changed sign up to time $t$ (or in general, that the node remained active/inactive in discrete models). Here we investigate disordered and random networks and show that the behavior of the persistence depends on the topology of the network. In two-dimensional (2D) disordered networks, we find that above the percolation threshold diffusive persistence scales similarly as in the original 2D regular lattice, according to a power law $P(t,L)\sim t^{-\theta}$ with an exponent $\theta \simeq 0.186$, in the limit of large linear system size $L$. At the percolation threshold, however, the scaling exponent changes to $\theta \simeq 0.141$, as the result of the interplay of diffusive persistence and the underlying structural transition in the disordered lattice at the percolation threshold. Moreover, studying finite-size effects for 2D lattices at and above the percolation threshold, we find that at the percolation threshold, the long-time asymptotic value obeys a power-law $P(t,L)\sim L^{-z\theta}$ with $z\simeq 2.86$ instead of the value of $z=2$ normally associated with finite-size effects on 2D regular lattices. In contrast, we observe that in random networks without a local regular structure, such as Erd\H{o}s-R\'enyi networks, no simple power-law scaling behavior exists above the percolation threshold.