**3. Collective Dynamics and Uniformity**

Let *ci*(*t*), *i* = 1, ... , *N*, *t* = 0, ... , *T* denote the multivariate time series of daily closing prices among our collection of *N* cryptocurrencies. Let *ri*(*t*) be the multivariate time series of log returns *i* = 1, . . . , *N*, *t* = 1, . . . , *T*, defined as

$$r\_i(t) = \log\left(\frac{c\_i(t)}{c\_i(t-1)}\right). \tag{1}$$

In this section, we analyze correlation matrices of log returns across rolling time windows of length *τ*; in this paper, we choose *τ* = 90 days. We standardize the log returns over such a window [*t* − *τ* + 1, *t*] by defining *Ri*(*s*)=[*ri*(*s*) − *ri* ]/*σ*(*ri*) where . denotes the average over the time window [*t* − *τ* + 1, *t*] and *σ* the associated standard deviation. Let **R** be the *N* × *τ* matrix defined by *Ris* = *Ri*(*s*) with *i* = 1, ... , *N* and *s* = *t* − *τ* + 1, ... , *t*, and then the correlation matrix **Ψ** is then defined as

$$\mathbf{\hat{Y}}(t) = \frac{1}{\tau} \mathbf{R} \mathbf{R}^T. \tag{2}$$

Individual entries describing the correlation between cryptocurrencies *i* and *j* can be written as

$$\Psi\_{i\bar{j}}(t) = \frac{1}{\pi} \frac{\sum\_{s=t-\tau+1}^{t} (r\_i(s) - \langle r\_i \rangle)(r\_{\bar{j}}(s) - \langle r\_{\bar{j}} \rangle)}{\left(\sum\_{s=t-\tau+1}^{t} (r\_i(s) - \langle r\_i \rangle)^2\right)^{1/2} \left(\sum\_{s=t-\tau+1}^{t} (r\_{\bar{j}}(s) - \langle r\_{\bar{j}} \rangle)^2\right)^{1/2}} \tag{3}$$

for 1 ≤ *i*, *j* ≤ *N*. We may analogously define the cross-correlation matrices for each individual decile by restricting *i* and *j* to be chosen from a set of indices corresponding to that decile.

All entries Ψ*ij* lie in the interval [−1, 1]. **Ψ** is a symmetric and positive semi-definite matrix with real and non-negative eigenvalues *λi*(*t*), so we order them as *λ*<sup>1</sup> ≥···≥ *λ<sup>N</sup>* ≥ 0. All the diagonal entries of **Ψ** are equal to 1, so the trace of **Ψ** is equal to *N*. Thus, we may normalize the eigenvalues by defining by *<sup>N</sup>*, to wit, *<sup>λ</sup>*˜ *<sup>i</sup>* = *<sup>λ</sup><sup>i</sup>* ∑*<sup>N</sup> <sup>j</sup>*=<sup>1</sup> *λ<sup>j</sup>* = *<sup>λ</sup><sup>i</sup> <sup>N</sup>* . We display the

rolling normalized first eigenvalue *λ*˜ <sup>1</sup>(*t*) for both the entire collection of cryptocurrencies and the 10 deciles in Figure 1.

In Figure 1a, we see particular periods of heightened collective correlation between cryptocurrencies. In particular, we see extended periods of high correlation in early 2020, coinciding with COVID-19 and the BitMEX crash, and toward the end of 2022, reflecting the tumultuous period around the collapse of FTX. These are perhaps the most significant moments of collective crisis in the cryptocurrency market in the last three years. These broad trends are reflected on a decile-by-decile basis in Figure 1b, where each individual decile exhibits a rise in collective correlations in these two periods.

To a nuanced extent, this is a signature of growing maturity in the cryptocurrency market. Specifically, crisis periods are observed; there is a fairly robust time differential between crises; collective correlations rise during crises and fall outside these periods; such effect is seen rather uniformly among different sectors of the market. However, we must remark that the extent of maturity does not coincide with more established markets such as the equity market. The time differential between peaks in collective correlations is still notably shorter than it is for equities; for example, the large time differential between the Dot-com bubble, the global financial crisis and the COVID-19 crash. Moreover, the strength of collective correlations between deciles varies significantly, despite their sharing temporal patterns. Some deciles, such as the third, exhibit significantly higher collective behaviors than others such as the second, fourth and ninth, whereas these behaviors are much more uniform for equity indices.

Next, we turn to an analysis of the leading *eigenvector* **v**<sup>1</sup> to complement our study of the leading eigenvalue. We analyze its *uniformity* via the following computation:

$$h(t) = \frac{|\langle \mathbf{v}\_1, \mathbf{1} \rangle|}{||\mathbf{v}\_1|| \, ||\mathbf{1}||},\tag{4}$$

where **<sup>1</sup>** = (1, 1, ... , 1) <sup>∈</sup> <sup>R</sup>*<sup>N</sup>* is a uniform vector of 1's. We may compute this for both the entire collection of cryptocurrencies and individual deciles, analogously to the leading eigenvalue. We observe that *h*(*t*) ≤ 1 with *h*(*t*) = 1 if and only if **v**<sup>1</sup> = **1** (up to scalar multiplication). In this case, all cryptocurrencies carry the same amount of variance in the "market effect" summarized by *λ*˜ <sup>1</sup>(*t*). This can be used to quantify the potential benefit of diversification: increased values of *h*(*t*) indicate increased interchangeability of

**Figure 1.** Normalized leading eigenvalue *λ*˜ <sup>1</sup>(*t*) of the cross-correlation matrix as a function of time, for (**a**) the entire collection of cryptocurrencies and (**b**) the ten deciles. Like the equity market, collective correlations spike during market crises, such as COVID-19, and the collapse of exchanges BitMEX and FTX.

We display the rolling uniformity of the first eigenvector *h*(*t*) for both the entire collection of cryptocurrencies and the 10 deciles in Figure 2. Unlike Figure 1, we observe a substantial difference compared to the equity market. In the case of the equity market, the uniformity for each sector and the entire market are consistent with the degree of collectivity. The degree of uniformity spikes during market crises such as the dot-com bubble GFC and COVID-19. This spike during market crises occurs for sectors (when studied independently) as well as across the entire market. The cryptocurrency market produces dramatically different findings to that of the equity market. Most notably, we see that there are substantial differences between the uniformity of independent sectors of the cryptocurrency market with that of the equity market. The cryptocurrency market clearly exhibits less uniformity during crises (which we see during the COVID-19 market crisis), and significantly higher variance between sectors of securities throughout our window of analysis. This is opposite to the finding of the equity market, where industry sectors exhibited more uniformity during crises. Another point to note is the stark contrast in how low the *h*(*t*) values reach when comparing the two asset classes. In the case of equities, there is a clear lower bound around the value of 0.75, while for cryptocurrencies we see two groups of cryptocurrencies reach values below 0.5 (with one reaching less than 0.3)

during our analysis window. Our analysis, therefore, suggests that we see less persistent and amplified uniformity among cryptocurrencies when compared to equities.

**Figure 2.** Uniformity *h*(*t*) of the leading eigenvector **v**<sup>1</sup> of the cross-correlation matrix as a function of time, for (**a**) the entire collection of cryptocurrencies and (**b**) the ten deciles. The results are dramatically different compared to the equity market, with numerous deciles exhibiting strikingly low uniformity scores over time.
