Relations between anomalous diffusion and fluctuation scaling: The case of ultraslow diffusion and time-scale-independent fluctuation scaling in language

TL;DR

This study links fluctuation scaling and anomalous diffusion by analyzing Japanese blog data, revealing that time-scale-independent FS corresponds to ultraslow, logarithmic diffusion, thus providing a new way to detect such diffusion in real systems.

Contribution

The paper demonstrates a direct relationship between fluctuation scaling and ultraslow diffusion, validated with large-scale real data, bridging theoretical models and empirical observations.

Findings

Time-scale-independent FS corresponds to logarithmic diffusion.

The relationship between FS and ultraslow diffusion is confirmed in real data.

Detection of FS can serve as an indicator of ultraslow diffusion.

Abstract

Fluctuation scaling (FS) and anomalous diffusion have been discussed in different contexts, even though both are often observed in complex systems. To clarify the relationship between these concepts, we investigated approximately three billion Japanese blog articles over a period of six years and analyzed the corresponding Poisson process driven by a random walk model with power-law forgetting, which reproduces both the anomalous diffusion and the FS. From the analysis of the model, we have identified the relationship between the time-scale dependence of FS and characteristics of anomalous diffusion and showed that the time-scale-independent FS corresponds to essentially a logarithmic diffusion (i.e., a kind of ultraslow diffusion). In addition, we confirmed that this relationship is also valid for the actual data. This finding may contribute to the discovery of actual examples of ultraslow diffusion, which have been nearly unobserved in spite of many mathematical theories, because we can detect the time-scale-independent FS more easily and more distinctly than through direct detection of the logarithmic diffusion based on the mean squared displacement.