## 3 Methods
### The Allan Factor  
The Allan Factor (AF) is a tool commonly employed to analyze the temporal dynamics of various natural phenomena [36]. Compared to the coefficient of variation (\(C_{\triangledown}\)), it provides a more refined assessment of the temporal structure within a sequence. While \(C_{\triangledown}\) can distinguish between Poissonian, clustered, or regular event patterns, the AF additionally identifies the specific temporal scales at which these behaviors occur. To compute the AF, the time axis is partitioned into contiguous, equally sized intervals of duration \(\tau\), referred to as the timescale. Within each interval, the number of events is counted, yielding a sequence \(\{N_{k}(\tau)\}\) where \(N_{k}\) denotes the number of events falling in the \(k\)-th window [37]. The Allan Factor is then defined by the following formula:  
\[AF(\tau)=\frac{<(N_{k+1}(\tau)-N_{k}(\tau))^{2}>}{2<N_{k}(\tau)>} \tag{3}\]  
where \(<..>\) indicates the average value. A Poissonian sequence, whose events are uncorrelated and independent, is characterized by the AF being rather flat for all theinvestigated timescales assuming a value around 1, except for very large timescales due to finite-size effects [38]. A decrease in the AF at a specific timescale indicates that the sequence exhibits periodic behavior at that scale [39]. Conversely, a time-clustered sequence is characterized by a scale-dependent AF; in particular, for a fractal (self-similar) process, the AF increases with the timescale \(\tau\) following a power-law relationship:  
\[AF(\tau)\approx\tau^{\kappa}. \tag{4}\]  
A power-law behavior of the AF indicates that the sequence is characterized by correlation among the events [37]. The exponent \(\alpha\) is called the fractal exponent and quantifies the intensity of the time clusterization or time correlation. Thus, if the null hypothesis \(\alpha=0\) is not rejected against the alternative hypothesis \(\alpha>0\), the sequence of interevent intervals \((T_{i})\) is assumed to follow a Poissonian behavior. Conversely, if there is sufficient evidence to reject the null hypothesis \(\alpha=0\), the sequence \((T_{i})\) is assumed to exhibit a time-clustered pattern.  
In order to investigate the significance of the AF, random sequences can be generated with two surrogate methods: (1) the Poissonian surrogate method, consisting of generating Poisson-distributed sequences with the same rate as the original one; and (2) the random permutating method [40], consisting of generating random shuffles of the interevent times of the original sequence; with the second surrogate method, the distribution of the original interevent times does not change. After calculating the AF for each surrogate, the distribution of AF values for each timescale \(\tau\) is computed. For instance, by setting a confidence level of 97.5%, the 97.5% confidence curve corresponds to the envelope of all 97.5 th percentiles. If the AF is significantly above the Poissonian confidence level, it indicates a notable deviation from Poissonian behavior. Similarly, if it is well above the confidence curve derived from random shuffles of interevent times, this suggests that the clustering behavior is not driven by the probability density of interevent times.