*5.1. The Response of Network Structures to Noise and Trend Effects*

Given the fact that there are noise and trends in the cryptocurrency market, we examine whether these factors affect the cryptocurrency network structure. Since we have four datasets corresponding to four timescales (e.g., 30 min, 6 h, 12 h and 24 h), we use both metric-related methods and visualization for all available datasets to discover the discrepancy between original and cleaned (after removing noise and trends) datasets.

To show the difference between two network structures, we choose two such metrics to measure the connection strength in a network of cryptocurrencies:

• Residuality Coefficient [93]: This compares the relative strength of the connections above and below a threshold distance value. In this experiment, we use the highest distance value ensuring connectivity of the MST as the threshold, denoted *L*:

$$R = \frac{\Sigma\_{\left[d\_{ij} > L\right]} d\_{ij}^{-1}}{\Sigma\_{\left[d\_{ij} \le L\right]} d\_{ij}^{-1}} \tag{4}$$

• MST-based mean distance [111]: this calculates the average distance of the MST:

$$M = \frac{1}{N - 1} \Sigma\_{d\_{ij} \in MST} d\_{ij} \tag{5}$$

An increase in these means that cryptocurrencies are further from each other. By contrast, cryptocurrencies are closer to each other if these metrics decrease. Note that although both metrics are used to examine the connection strength of cryptocurrencies, the Residuality coefficient is known to be more vulnerable to the links between cryptocurrencies in different groups, i.e., if the connection strength between cryptocurrencies in different groups increases, the Residuality coefficient will decrease dramatically, and vice versa; the connections between cryptocurrencies within one group do not affect the Residuality coefficient much [112]. On the other hand, Mean distance is more vulnerable to the

links between cryptocurrencies belonging to one group, as it mainly uses the connections within a group to find the average value and ignores the connections between different groups [111].

Table 4 shows the results of the two metrics using different levels of granularity. It is clear that both Residuality coefficients and Mean distance values increase significantly when the effects of noise and trend are dismissed. This phenomenon remains unchanged in different timescales, implying that this is a genuine characteristic of the cryptocurrency market. Furthermore, a visualization of network structures before and after cleaning is shown in Figure 2 to reinforce our finding. As can be seen, the topological structure changes after the noise and trends are removed. Moreover, what happens in each time window is that the number of communities decreases after removing these effects. From these figures and illustrations, we can conclude that the connections between cryptocurrencies are caused mainly by the noise and trend effects. That is, these factors result in different cryptocurrencies becoming closer to each other and forming a group. This phenomenon can be explained by low values for Residuality coefficients and Mean distance values in the original data compared to the cleaned data. A value less than unity of the prior metric means that there are few connections greater than the threshold *L*. Moreover, a small value of the latter metric means that cryptocurrencies within a group are closer to each other. In summary, each group of the network is compact with strong links inside, which helps the community detection algorithm to easily cluster them. In other words, the difference between different groups is clear because the links between different groups are weak, i.e., the ones greater than *L*. However, after cleaning the correlation matrix, cryptocurrencies that are closely related to each other through noise and trend become further away, i.e., the strong links between some cryptocurrencies are broken. This causes our metrics to increase dramatically, which means that the network structure starts to expand, forming a sparse network. For example, the Residuality coefficient of the second time window in the 30 min original data is 0.28, while it is 20 times higher after cleaning the effects of noise and trends. This fact is also true for the rest of our dataset. The result is in line with [20]; these authors did not consider the noise effect but, with the removal of trends, they found that the correlation between the 80 most liquid cryptocurrencies from 1 January 2020 to 1 October 2021 decreased.

**Table 4.** Cryptocurrency network connection strength through three time windows measured by Residuality Coefficient and Mean Distance. Four different granularity levels are considered, each with datasets, including original and cleaned dataset after removing noise and trend effects.


(**e**) Time window 3, original (**f**) Time window 3, cleaned

**Figure 2.** Cryptocurrency network structures using daily data. For each time window, Louvain method is applied to both original and cleaned data to detect existing communities. The illustrations on the left and right hand side are for the original and cleaned data, respectively, for 3 time windows referring to normal, downturn and recovery times, respectively.
