Fractals in Small-World Networks With Time Delay

TL;DR

This paper investigates how time-delay influences the properties of small-world networks, providing analytical and numerical insights into their fractal and multifractal characteristics.

Contribution

It introduces an analytical approach to analyze the effect of time-delay on small-world networks and characterizes their fractal and multifractal properties.

Findings

Time-delay affects the fractal dimension of small-world networks.

Small-world networks with time-delay exhibit multifractal properties.

Analytical expressions match numerical simulation results.

Abstract

The small-world networks recently introduced by Watts and Strogatz [Nature 393, 440 (1998)] has attracted much interests in studying the interesting properties of the networks without time-delay. However, a signal or influence travelling on the small-world networks often associated with time-delay features which are very common in biological and physical networks. We develop an analytical approach as well as numerical simulations to try to characterize the effect of time-delay on the properties of small-world networks. An analytical expression of the fractal dimension of the small-world networks is given and thus compared with the results from numerical simulations. Analysis shows that small-world networks with time-delay generally have the multifractals property.