Scaling in directed dynamical small-world networks with random responses

TL;DR

This paper introduces a dynamical small-world network model with directed links and random responses, revealing how spreading length, time, and collective behavior scale with network parameters, applicable to social and natural systems.

Contribution

The study presents a novel directed small-world network model incorporating random responses, and uncovers universal scaling laws for spreading dynamics and collective behavior.

Findings

Scaling laws for spreading length and time as p^- ln N

Duple scaling form of the collective variable S

Universal functions describing the dynamics

Abstract

A dynamical model of small-world network, with directed links which describe various correlations in social and natural phenomena, is presented. Random responses of every site to the imput message are introduced to simulate real systems. The interplay of these ingredients results in collective dynamical evolution of a spin-like variable S(t) of the whole network. In the present model, global average spreading length \langel L >_s and average spreading time <T >_s are found to scale as p^-\alpha ln N with different exponents. Meanwhile, S behaves in a duple scaling form for N>>N^*: S ~ f(p^-\beta q^\gamma t'_sc), where p and q are rewiring and external parameters, \alpha, \beta, \gamma and f(t'_sc) are scaling exponents and universal functions, respectively. Possible applications of the model are discussed.