# Statistical properties of user activity fluctuations in virtual worlds

## TL;DR

This study analyzes user activity fluctuations in 95 virtual worlds, revealing power-law distributions, multifractal features, and correlations with activity levels, providing insights into online social dynamics.

## Contribution

It is the first comprehensive analysis of the statistical properties and multifractal nature of user activity fluctuations across multiple virtual worlds.

## Key findings

- Power-law tail distribution with an average exponent of 2.15.
- Multifractal features are present in all virtual worlds studied.
- Negative correlation between singularity width and maximum activity in some worlds.

## Abstract

User activity fluctuations reflect the performance of online society. We investigate the statistical properties of 1-min user activity time series of simultaneously online users inhabited in 95 independent virtual worlds. The number of online users exhibits clear intraday and weekly patterns due to human's circadian rhythms and week cycles. Statistical analysis shows that the distribution of absolute activity fluctuations has a power-law tail for 44 virtual worlds with an average tail exponent close to 2.15. The partition function approach unveils that the absolute activity fluctuations possess multifractal features for all the 95 virtual worlds. For the sample of 44 virtual worlds with power-law tailed distributions of the absolute activity fluctuations, the width of singularity $\Delta\alpha$ is negatively correlated with the maximum activity ($p$-value=0.070) and the time to the maximum activity ($p$-value=0.010). The negative correlations are not observed for neither the other 51 virtual worlds nor the whole sample of the 95 virtual worlds. In addition, numerical experiments indicate that both temporal structure and large fluctuations have influence on the multifractal spectrum. We also find that the temporal structure has stronger impact on the singularity width than large fluctuations.