# Projecting XRP price burst by correlation tensor spectra of transaction
networks

Abhijit Chakraborty  Kyoto University, Graduate School of Advanced Integrated
Studies in Human Survivability, Kyoto, 606-8306, Japan  RIKEN
Interdisciplinary Theoretical and Mathematical Sciences Program, Saitama,
351-0198, Japan<EMAIL_ADDRESS>Tetsuo Hatsuda  RIKEN
Interdisciplinary Theoretical and Mathematical Sciences Program, Saitama,
351-0198, Japan  Yuichi Ikeda  Kyoto University, Graduate School of Advanced
Integrated Studies in Human Survivability, Kyoto, 606-8306, Japan

###### Abstract

Cryptoassets are becoming essential in the digital economy era. XRP is one of
the large market cap cryptoassets. Here, we develop a novel method of
correlation tensor spectra for the dynamical XRP networks, which can provide
an early indication for XRP price. A weighed directed weekly transaction
network among XRP wallets is constructed by aggregating all transactions for a
week. A vector for each node is then obtained by embedding the weekly network
in continuous vector space. From a set of weekly snapshots of node vectors, we
construct a correlation tensor. A double singular value decomposition of the
correlation tensors gives its singular values. The significance of the
singular values is shown by comparing with its randomize counterpart. The
evolution of singular values shows a distinctive behavior. The largest
singular value shows a significant negative correlation with XRP/USD price. We
observe the minimum of the largest singular values at the XRP/USD price peak
during the first week of January 2018. The minimum of the largest singular
value during January 2018 is explained by the decomposing the correlation
tensor in the signal and noise components and also by evolution of community
structure.

## Introduction

Cryptoassets represent value that one can transfer, store, or trade digitally.
It uses cryptography to protect data and distributed ledger technology to
record transactions. A blockchain technology, which is a form of secure
digital ledger, is used to store the records of crypto transactions. Recently,
cryptoassets have been very popular as an investment, but cryptoasset prices
are extremely volatile and unpredictable. The cryptomarket experienced severe
price fluctuations from 2017, December to 2018, January. The presence of
bubbles i.e., explosive price behavior in this asset has attracted attention
from the researchers. At present, there are many cryptoassets. Some well-known
of them are Bitcoin (BTC), Ethereum (ETH) and XRP. For the last one decade
complex network theory has been widely used to analyze cryptoasset transaction
data. Among the cryptoassets, networks of BTC and ETH transactions have been
studied extensively [1], which include different structural properties [2, 3],
temporal evolution [2, 4, 5, 6] and market effect [7, 8, 9] of these networks.
In contrast, XRP transactions have been less explored. Ripple Labs Inc., a US-
based technology company, created XRP as a native currency for the Ripple
network, a real-time gross settlement system, currency exchange and remittance
network with aims of fast transactions settle, low-cost transaction fees,
scalable, stable, sustainable and decentralized ledger system. P. Moreno-
Sanchez et al. uncovered community formation and clustering properties for the
Ripple network [10]. Y. Ikeda studied the structural properties of XRP
transaction network, which include the heavy tail nature of nodal strength
distributions, low value of clustering coefficient and significant triangular
motif [11]. More recently, the role of the most active nodes with respect to
outgoing and incoming flows for a duration including bubble/crash period has
been quantified for BTC and XRP transaction networks [12].

Any statistical dependence between a pair of variables can be studied by cross
correlation which is predictive and useful for empirical data. It can be
measured in different ways: One of the simple and well-known methods is the
Pearson correlation which represents the linear dependence between variables
and is defined for a pair of variables $x$ and $y$ with $n$ observations as
$\rho=\sum\limits_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y})/((n-1)\sigma_{x}\sigma_{y})$.
Here $\overline{x},\overline{y}$, and $\sigma_{x},\sigma_{y}$ indicate mean
and standard deviations of $x$ and $y$. The cross correlation methodology
armed with random matrix theory (RMT) is mostly applied to time series data.
The aim of such method is to analyze high dimensional data to find key factors
for the collective dynamics of many quantities. For example, it is used to
study the daily returns of different stocks [13, 14, 15] and foreign exchange
rates [16, 17], monthly macroeconomic data [18] , or different medical data
such as electroencephalogram, magnetoencephalography data recording [19].
Also, D. Kondor et al. applied principal component analysis on the matrices
obtained from the daily network snapshots to show the relationship between the
price of bitcoin with structural changes in the transaction network [20]. The
application of cross correlation on stock market data shows the relationship
between stock price changes and liquidity or trading volume [21]. Note that
these are time series of macro variables that emerge due to interactions of
different entities of the system. In most cases, we lack detail information of
the interactions at the micro level. For example, we do not have detail
information about the interaction among the individuals in stock markets.

In the case of XRP, we know the information for all transactions between
wallets. Using this high-quality micro level data, we develop a method of
cross correlation tensor which can be applied on dynamical XRP transaction
networks. Our method relates the structural properties of XRP networks to the
XRP/USD price. To calculate the correlation tensor, nodes of the networks are
converted to vectors using network embedding techniques [22, 23]. We use the
DeepWalk embedding technique [22], which uses a set of truncated random walks
to learn latent features. The latent features capture neighborhood information
and community memberships of the nodes. Using a network as the input, the
algorithm provides a latent representation in a continuous vector space as an
output. A generalization of the DeepWalk algorithm is node2vec [23], which
uses biased random walks to learn features that encode more complex relations
of nodes, such as functional relationships. Using the vector representation
for a subset of nodes present in every weekly XRP transaction networks, we
calculate correlation tensors for different time periods. We perform a double
singular value decomposition on the correlation tensors to obtain its spectra.
We compare the results with reference correlation tensors to understand the
significance. As a reference correlation tensor, one usually uses the RMT
method. However, as we are using a small time window to calculate the
correlation tensor, RMT method will not be suitable as reference. We use
randomized and reshuffled correlation tensors as reference correlation
tensors. The largest few singular values capture the impact of the bubble
period or crash in XRP price. The largest singular value is found to be
significant and has a strong negative correlation with XRP/USD price.
Furthermore, it provides an early indication for XRP/USD price, including
bubble periods or crashes.

Figure 1:  The variation of (a) total number of nodes (b) number of links per
node and (c) total transaction volume in millions XRP for each weekly network.
(d) The daily XRP/USD close price (source:
https://www.marketwatch.com/investing/cryptocurrency/xrpusd). The dotted grey
vertical lines represent the weekly windows.

## Results

Figure 2: The distributions of the elements of empirical, reshuffle and
randomize correlation tensors calculated for the week, November 06 to November
12, 2017. The data are averaged over 20 uncorrelated embedding of the
networks.

The weekly network obtained from XRP transactions between wallets evolves with
time. We focus our study on the duration 2017 October 02 to 2018 March for
which covers a bubble period in XRP price. It indicates $22$ weekly networks.
The number of nodes for each weekly network is shown in Fig. 1 (a). We observe
that the number of nodes of the weekly networks increases rapidly from
$45,169$ during the week of 2017, November 27 - December 04 to the peak value
$209,143$ during the week of 2018, January 01 - January 07 and later further
falls to $27,811$ during the week of 2018, February 26 to March 04. Fig. 1 (b)
shows the decline in the number of links per node indicating the average nodal
in- or out-degree from $2.15$ to $1.20$ during 2017, October 02 - 2018 March
04. We note that XRP transaction volume was very high for three weeks between
December 05 - December 24 and also for the week during 2018, January 22 -
January 29, as shown in Fig. 1 (c) . The three weeks of extremely high XRP
transaction volume may have contributed to the bubble formation in XRP/USD
during 2017, December 25 to 2018 January 07. The daily XRP/USD price from
October 2, 2017 to March 4, 2018 is shown in Fig. 1 (d). The XRP/USD price had
an extraordinary rise and fall between December 2017 and January 2018. This
indicates a bubble period for XRP. We consider this period for our study
because this is the most significant bubble period for the cryptoasset market.
The chart of XRP/USD prices for a more extended period is shown in Fig. S1.

Figure 3: Sorted singular values $\rho_{k}^{\gamma}$ of the empirical,
reshuffle and randomize correlation tensor for the week, November 06 to
November 12, 2017. The values represent average over $20$ uncorrelated
embeddings of the network.  Figure 4:  Evolution of the singular values for
empirical $\rho_{k}^{\gamma}$ and reshuffled $\tilde{\rho}_{k}^{\gamma}$
correlation tensors for different weeks. (a) Variation of the largest singular
value for empirical $\rho_{1}^{1}$ and reshuffled $\tilde{\rho}_{1}^{1}$
correlation tensors. (b) Variation of the second largest singular value for
empirical $\rho_{2}^{1}$, reshuffled $\tilde{\rho}_{2}^{1}$ correlation
tensors. (c) Variation of spectral gap for empirical
$(\rho_{1}^{1}-\rho_{2}^{1})$ and reshuffled
$(\tilde{\rho}_{1}^{1}-\tilde{\rho}_{2}^{1})$ correlation tensors. The error
bars indicate the standard deviation. The data are averaged over $20$
uncorrelated embeddings of the networks. The dotted grey vertical lines
represent the weekly windows.

We have embedded each of these weekly networks using the well-known DeepWalk
algorithm on $D=32$-dimensional space. For the details of the embedding
technique, see Method section. This gives a $D$-dimensional vector
$V_{i}^{\alpha}$ for each node of the networks. We use $i,j$ as node indices
and Greek letters $\alpha,\beta$ as components of the vectors on a $D$
dimensional space. In weekly networks of XRP transactions, we found $N=71$
nodes does at least one transaction every week. We call these $N=71$ nodes,
regular nodes. Each regular node of weekly networks in the embedding space is
represented by a D-dimensional vector time series $V_{i}^{\alpha}(t)$, where
$i=1,2,3,...N$, $t=1,2,3,...T$ and $\alpha=1,2,3,...D$. We have chosen $D=32$
for our study. Other values of $D$ give qualitatively similar results.

The correlation tensor between the different components of the regular nodes
is defined as

$M_{ij}^{\alpha\beta}(t)=\frac{1}{2\Delta T}\sum\limits_{t^{\prime}=t-\Delta
T}^{t+\Delta
T}\frac{[V_{i}^{\alpha}(t^{\prime})-\overline{V_{i}^{\alpha}}][V_{j}^{\beta}(t^{\prime})-\overline{V_{j}^{\beta}}]}{\sigma_{V_{i}^{\alpha}}\sigma_{V_{j}^{\beta}}},$
(1)

where $\sum$ is taken over $5$ weekly $\\{t-2,t-1,t,t+1,t+2\\}$ networks with
$\Delta T=2$ for our analysis. The $\overline{V_{i}^{\alpha}}$ and
$\sigma_{V_{i}^{\alpha}}$ represents mean and standard deviation of
$V_{i}^{\alpha}$ over a time window of $(2\Delta+1)=5$ weekly
$\\{t-2,t-1,t,t+1,t+2\\}$ networks. Note that lower the values of $\Delta T$,
more noise is present in the obtained correlation tensor. We also can not take
$\Delta T$ too large as we are studying the dynamical evolution of the
networks. We further discuss the dependence of the correlation tensor on the
time window in the SI Text 2.

To understand the significance of the empirical correlation tensor, we
consider the following two null hypotheses. The null hypothesis of the
components of embedding vectors for regular nodes, $V_{i}^{\alpha}$
(randomize), are independent, uniformly distributed, random variables within
$[-1,1]$. We obtained the randomized correlation tensor from $V_{i}^{\alpha}$
(randomize). Another method to randomize the empirical correlation tenor is to
remove the correlation present between the different components of the
embedding vectors, by reshuffling the positions of the components of embedding
vectors $V_{i}^{\alpha}$(reshuffle). We obtain the reshuffle correlation
tensor from $V_{i}^{\alpha}$ (reshuffle). The distributions of the elements of
empirical, randomize and reshuffle correlation tensor are shown in Fig. 2 for
the week of 2017, November 06 - November 12. We have taken the average of the
data for the distributions with $20$ different runs of the embedding
algorithm. The distributions from randomized and reshuffled correlation
tensors are qualitatively identical. This is because the auto correlation for
different components of the empirical embedding vectors vanishes when the lag
is more than one week. These two distributions are symmetrical around zero,
with an average value of the elements close to zero. The empirical correlation
differs from the randomized and reshuffle correlation tensor. It is an
asymmetric distribution having an average value of the elements $0.017$. The
correlation tensor has the dimension $N\times N\times D\times D$. Since we
have many elements in the correlation tensor, we uncover the crucial
information by diagonalizing it using a double singular value decomposition
(SVD) method.

A double SVD of the correlation tensor gives us $N\times D$ singular values
$\rho_{k}^{\gamma}$, where $i=1,2,3,...N$ and $\gamma=1,2,3,...D$. For the
details, refer to the Method section. We compare the singular values
$\rho_{k}^{\gamma}$ of the empirical correlation tensor with the singular
values of randomized correlation tensor $\rho_{k}^{\gamma}$ (randomize) and
reshuffled correlation tensor $\rho_{k}^{\gamma}$ (reshuffle). The comparison
is shown in Fig. 3 for the week of 2017, November 06 - November 12. This shows
that the largest singular value lies beyond the largest singular value of the
randomized correlation tensor. Also, the spectral gap
$(\rho_{1}^{1}-\rho_{2}^{1})$ in the empirical correlation tensor is
significantly large compared with its random counterpart.

The temporal variation of the largest singular values of empirical
$\rho_{k}^{\gamma}(t)$ and reshuffle $\tilde{\rho}_{k}^{\gamma}(t)$
correlation tensor for different weeks is shown in Fig. 4. It is evident that
the largest singular value $\rho_{1}^{1}$ is well above the largest singular
values $\tilde{\rho}_{1}^{1}$ of the randomized correlation tensor for all
weeks. However, the second largest singular value $\rho_{2}^{1}$ lies below
the randomized counterpart. Similar to the largest singular value
$\rho_{1}^{1}$, the spectral gap $(\rho_{1}^{1}-\rho_{2}^{1})$ appears
significantly higher than that of the randomized case. Moreover, we observe
$\tilde{\rho}_{k}^{\gamma}(t)$ remains approximately constant with time. In a
stark contrast, the largest two empirical singular values and as well as a few
more singular values (not shown) vary distinctively for different weeks.

Figure 5:  The comparison of daily XRP/USD price (black curves) with (a) the
largest singular value $\rho_{1}^{1}$, (b) the second largest singular value
$\rho_{2}^{1}$ and (c) the spectral gap $(\rho_{1}^{1}-\rho_{2}^{1})$ of
correlation tensors for different weeks (blue curves). The dotted grey
vertical lines represent the weekly windows.

To investigate the relationship between singular values $\rho_{k}^{\gamma}$
and XRP/USD price, we compare the variation of XRP/USD daily price with the
largest singular value $\rho_{1}^{1}$, the second largest singular value
$\rho_{2}^{1}$ and the spectral gap $(\rho_{1}^{1}-\rho_{2}^{1})$ in Fig. 5
(a)-(c). To quantify the dependence, we measure the correlation between the
weekly XRP/USD price and the two largest singular values $\rho_{1}^{1}$,
$\rho_{2}^{1}$ respectively. The weekly XRP/USD price indicates the average
daily closing price of XRP/USD for the week. Let us denote the weekly XRP/USD
price as $\overline{\rm XRP/USD}$. The Pearson correlation between
$\rho_{1}^{1}(t)$ and $\overline{\rm XRP/USD}(t+1)$ is found $r=-0.908$ and
p-value $=1.912\times 10^{-7}$. The Pearson correlation between $\rho_{2}^{1}$
and $\overline{\rm XRP/USD}(t+1)$ is found $r=0.847$ and p-value $=9.22\times
10^{-6}$. Furthermore, we perform a multi-linear regression of $\overline{\rm
XRP/USD}(t+1)\sim C_{0}+C_{1}\rho_{1}^{1}(t)+C_{2}\rho_{2}^{1}(t)$, gives
$C_{0}=27.91,C_{1}=-0.033,C_{2}=0.008$ and only the variable $\rho_{1}^{1}(t)$
is significant. We found $R^{2}=0.8091$ and p-value $=1.581\times 10^{-6}$,
indicating that the $80\%$ variation of $\overline{\rm XRP/USD}(t+1)$ can be
explained by the largest singular value $\rho_{1}^{1}(t)$. We also observe
there are significant correlation between singular values and weekly XRP/USD
price with two weeks lead $\overline{\rm XRP/USD}(t+2)$ and three weeks lead
$\overline{\rm XRP/USD}(t+3)$, which is described in detail in SI Text 3.
Correlation between $\overline{\rm XRP/USD}(t+3)$ and $\rho_{1}^{1}(t)$ is
found to be $r=-0.68$ and p-value $=0.001$. This indicates that the largest
singular value $\rho_{1}^{1}$ can give an early signal for XRP/USD price.

We also observe that the minimum of $\rho_{1}^{1}(t)$ appears during the week
of 2017 December 25 - December 31. The decomposition of the correlation tensor
$M_{ij}^{\alpha\beta}$ into signal $M_{ij}^{\alpha\beta}(\rm signal)$and noise
component $M_{ij}^{\alpha\beta}(\rm noise)$can explain the reason for this
minima. The decomposition of correlation tensor $M_{ij}^{\alpha\beta}$ for the
week of 2017, November 06 - November 12 is shown in Fig. 6 (a), which
indicates that the distribution of the elements of the signal component is
much wider than the distribution of the elements of the noise component. Fig.
6 (b) shows the distributions of the elements of the signal component for
three different weeks 2017, November 06 - November 12, 2018, January 01-
January 07 and 2018, February 12 - February 18. This reflects the fact that
the distribution for 2018, January 01- January 07 has a largest peak at zero
and it is narrower than the other distributions, indicating that the
dependence among the components of node vectors decreases during this time.
The peakedness of each distribution can be quantified by its fourth moment,
which is known as kurtosis. The different moments for the elements of the
signal component of correlation tensor are tabulated in Table 1. Clearly, the
peakedness and spread of the distribution during 2018, January 01- January 07
are relatively higher and thinner, respectively. As the average dependence
among the components of the node vectors decreases during 2018, January 01-
January 07, $\rho_{1}^{1}(t)$ has the minima during this period.

Figure 6: The decomposition of the correlation tensor in the signal component and noise component. (a) Distributions for the elements of the correlation tensor, signal component and noise component for 2017, November 06 - November 12. (b) Distributions for signal components of the correlation tensors before the bubble period ( 2017, November 06 - November 12), during the bubble period (2018, January 01- January 07) and after the bubble period (2018 February 12 - February 18). The legends indicate different components of correlation tensor in (a), and different periods in (b).  Figure 7:  Evolution of regular nodes within the community structure. (a) Evolution of the number communities of the weekly XRP network that contains at least one regular node (blue curve). (b) Evolution of the maximum number of regular nodes in a community of the weekly XRP network (blue curve). The black curves in both the panels represent the daily XRP/USD price. The dotted grey vertical lines represent the weekly windows.  Table 1:  The mean (for the absolute values of the elements), standard deviation and kurtosis of the signal components of correlation tensors for three different weeks. The values within the brackets indicate standard deviations of the quantities.  | 2017, November 06 - November 12 | 2018, January 01- January 07 | 2018, February 12 - February 18
---|---|---|---
$\langle{|M_{ij}^{\alpha\beta}(\rm signal)|}\rangle$ | $0.343(0.006)$ | $0.324(0.005)$ | $0.334(0.005)$
standard deviation | $0.419(0.006)$ | $0.401(0.005)$ | $0.409(0.005)$
kurtosis | $2.447(0.035)$ | $2.538(0.032)$ | $2.500(0.029)$

To understand why the average dependence among the components of node vectors
decreases during the bubble period, 2018, January 01- January 07, we
investigate the change in community structure of the regular nodes. We use
well known Infomap algorithm [24] to detect the communities in the entire
weekly directed weighted networks. We observe the evolution of the number of
communities of the XRP weekly network that contains at least one regular node
in Fig. 7 (a). It is observed that the number of such communities decreases
from around $40$ to around $20$ during the week 2017 December 04 - December 10
and the week 2017 December 18 - December 24. At the same duration, we observe
that the maximum number of regular nodes in a community increases from around
$10$ to around $50$, as shown in Fig. 7 (b). We further delve deeper into the
community structure and observe the evolution of each regular node within the
communities of the weekly networks. The evolution is shown in SI Fig. S3\- S6.
It shows that a big community of regular nodes forms during 2017, December 18
to December 24, as shown in SI Fig. S4\- S5. This big community got fragmented
in subsequent weeks. Moreover, the large community of the regular nodes
remains almost intact during the non-bubble period. This distinctive change in
the community structure might be related to the decrease in the average
dependence among the components of node vectors or the appearance of minima
for the largest singular value $\rho_{1}^{1}(t)$ during the bubble period
compared with non-bubble period.

Finally, using RMT [25, 26, 27, 28], we show the relationship between the
largest singular value and the standard deviation for the correlation tensor
with normally distributed elements in Fig. 8. For the detail calculation, see
the Method section. It also shows the deviation of the largest singular values
for randomized correlation tensors calculated with different time windows. It
is observed that as we increase the time window to measure the randomized
correlation tensor, the largest singular value $\rho_{1}^{1}$ approaches the
theoretical value $\rho_{1}^{1}=2\sqrt{N}D\sigma$.

Figure 8: Plot for the largest singular value $\rho_{1}^{1}$ with standard
deviation of the correlation tensor elements $\sigma$. Black circles represent
singular values for the randomized correlation tensors with time window
$(2\Delta T+1)=5,15,30$ and $50$ (from right to left). Red triangles represent
the largest singular values for correlation tensors, where the elements are
drawn from a normal distribution with mean zero and standard deviation
$\sigma$. The blue line represents $\rho_{1}^{1}=2\sqrt{N}D\sigma$ which is
the analytical expression for the singular values as given in Eq. 11.

## Conclusion

In this work, we have used all the direct transactions between XRP wallets in
a week to construct a weekly weighted directed network. Using the deep walk
method, we have embedded weekly snapshots of the network in vector space. Deep
walk encodes community information into nodal vectors using truncated random
walks on the network. Once can explore other methods of embedding, such as
node2vec, to study other regularities of the network. From the weekly
snapshots of the regular nodal vector components, we calculate the correlation
tensor at different time periods. We have used a double SVD to remove
redundant information from the correlation tensor. The significance of our
result is shown by comparing it with the results of a randomized counterpart.
The evolution of the largest singular value $\rho_{1}^{1}$ shows distinctive
behavior and it correlates significantly with the XRP/USD price. The
decomposition of the correlation tensor into signal and noise components shows
that correlation in the signal component decreases significantly during the
bubble period. We explain this decrease in the correlation during the bubble
period by the evolution of community structure that shows disruptive behavior
during the bubble period. The disruptive behavior of community structure has
also been observed in the foreign exchange market during economic downturns
[17].

In summary, we have developed a method of correlation tensor spectra from XRP
transactions. This method provides crucial insights into the bubble period of
the XRP price. This method is very general and can be used to analyze the
transactions of other assets.

## Data and methods

### Data description

We collected all the direct transactions between different XRP wallets from
October 2, 2017 to March 4, 2018, which were recorded as ledger data using the
Ripple Transaction Protocol. We grouped these data into $T=22$ weekly XRP
networks where wallets are the nodes and a direct transaction from a source
wallet to a destination wallet forms a directed link between them. The total
transaction volume between the wallets represents link-weight. See [11] for
the structural properties of the XRP transaction network.

### Network embedding

We embedded weighted directed weekly networks using well-known node2vec method
[23] with parameters $p=q=1$, which is a special case of the node2vec method
and represents the DeepWalk method. Based on a natural language model, these
embedding methods capture structural regularities in the network.
Particularly, the DeepWalk method encodes the community structure in the
vector representation of the nodes. It uses a truncated random walks to
extract the neighborhood information of nodes by generating a sequence of
nodes $S=\\{V_{1},V_{2},V_{3},...V_{S}\\}$ that is equivalent to a sentence in
natural language. Furthermore, it applies SkipGram algorithm [29] to map each
node $V_{j}$ to its vector representation $\Phi(V_{j})\in\mathcal{R}^{D}$ by
maximizing the co-occurrence probability of its neighbours in the random walk.

### Double singular value decomposition

To find the spectrum of the correlation tensor, we perform a double SVD in the
following way:

We conduct the diagonalization of $M_{ij}^{\alpha\beta}$ in terms of
$(ij)$-index and $(\alpha\beta)$-index successively by the bi-unitary
transformation or equivalently SVD.

The first step is

$M_{ij}^{\alpha\beta}=\sum\limits_{k=1}^{N}L_{ik}\sigma_{k}^{\alpha\beta}R_{kj},$
(2)

and the second step is

$\sigma_{k}^{\alpha\beta}=\sum\limits_{\gamma=1}^{D}\mathcal{L}^{\alpha\gamma}\rho_{k}^{\gamma}\mathcal{R}^{\gamma\beta}.$
(3)

Then, altogether we have

$M_{ij}^{\alpha\beta}=\sum\limits_{k=1}^{N}\sum\limits_{\gamma=1}^{D}\rho_{k}^{\gamma}(L_{ik}R_{kj})(\mathcal{L}^{\alpha\gamma}\mathcal{R}^{\gamma\beta}).$
(4)

Here $\rho_{k}^{\gamma}$ is the $N\times D$ generalized singular values. Also,
note that all singular values are real and positive because the correlation
tensor $M$ is real.

### Reference correlation tensors

We examine the properties of the empirical correlation tensor by comparing the
same quantity between the original correlation tensor and reference
correlation tensors, which is analogous to random matrix theory. To get the
reference correlation tensors, we have used two different techniques to obtain
a reshuffled correlation tensor and a randomized correlation tensor in the
following way:

#### Reshuffle correlation tensor

We reshuffle the components of embedding vectors $V_{i}^{\alpha}(t)$ within
the time window $(2\Delta T+1)$. Then, calculate the correlation tensor using
Eq. 1 with the reshuffled embedding vector components.

#### Randomize correlation tensor

We assign uniform random numbers between $[-1,1]$ to the components of
embedding vectors and calculate the correlation tensor using Eq. 1.

### Signal and noise components of the correlation tensor

Eq. 4 can be used for dimensionality reduction to capture important
relationships in the data and it can be written as

$M_{ij}^{\alpha\beta}=M_{ij}^{\alpha\beta}({\rm
signal})+M_{ij}^{\alpha\beta}({\rm noise}),$ (5)

where

$M_{ij}^{\alpha\beta}({\rm
signal})=\rho_{1}^{1}(L_{i1}R_{1j})(\mathcal{L}^{\alpha,1}\mathcal{R}^{1,\beta}),$
(6)

and

$M_{ij}^{\alpha\beta}({\rm
noise})=\sum\limits_{k=1}^{N}\sum\limits_{\gamma=1}^{d}\rho_{k}^{\gamma}(L_{ik}R_{kj})(\mathcal{L}^{\alpha\gamma}\mathcal{R}^{\gamma\beta})-\rho_{1}^{1}(L_{i1}R_{1j})(\mathcal{L}^{\alpha,1}\mathcal{R}^{1,\beta}).$
(7)

The signal component is calculated using the singular values that lie above
the largest singular values of the randomized correlation tensor. In our
study, we find that only $\rho_{1}^{1}$ lies above the largest singular values
of the randomized correlation tensor.

### Singular values of the correlation tensor with normally distributed
elements

Let us first consider the case of a $p\times q$ random matrix $R$, where the
elements of R are normally distributed with zero mean and standard deviation
$\sigma_{R}$. The singular values $s$ of such a matrix $R$ follows the
following distribution [25, 26]

$P(s)=\frac{1}{\pi
s\sigma_{R}^{2}}\sqrt{(s_{max}^{2}-s^{2})(s^{2}-s_{min}^{2})}$ (8)

for $s_{min}<s<s_{max}$ and zero otherwise, where

$s_{max}=\sqrt{2}\sigma_{R}\sqrt{\frac{(p+q)}{2}+\sqrt{pq}}$ (9)

and

$s_{min}=\sqrt{2}\sigma_{R}\sqrt{\frac{(p+q)}{2}-\sqrt{pq}}$ (10)

When $p=q=N$ and $\sigma_{R}=\sigma$, $s_{max}=2\sigma\sqrt{N}$. Now consider
the case of the $(N\times N\times D\times D)$ random correlation tensor
$M_{ran}$, where its elements are normally distributed with zero mean and
standard deviation $\sigma$ . The random correlation tensor $M_{ran}$ contains
$D^{2}$ number of $(N\times N)$ random matrix $R$. Analogous to Eq.2, SVD of
the random correlation tensor will give $\sigma_{k}^{\alpha\beta}(ran)$, where
$\sigma_{1}^{\alpha\beta}(ran)$ is a $(D\times D)$ matrix, where its elements
are distributed with mean $2\sigma\sqrt{N}$. A further SVD of
$\sigma_{1}^{\alpha\beta}(ran)$ corresponding to Eq.3, will give [27]

$\rho_{1}^{1}=2\sigma\sqrt{N}D+O(\sqrt{D}).$ (11)

We show how the largest singular values of randomized correlation tensors
deviate from the largest singular values of correlation tensors where its
elements are drawn from a normal distribution with mean zero and standard
deviation $\sigma$ in Fig. 8.

### Data availability

We collected the data from the ripple data API at https://xrpl.org/data-
api.html#payment-objects.

## References

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## Acknowledgements

We thank the members of the Kyoto Univ.- RIKEN blockchain study group for
discussions. This work is partially supported by Ripple Impact Fund
2022-247584 (5855).

## Author contributions statement

All authors conceived and designed the study. Y.I. developed the original
conceptualization. T.H. formulated a double singular value decomposition. A.C.
analyzed the data. All authors discussed the results and contributed to the
manuscript. A.C. wrote the paper. All authors reviewed the manuscript.

## Supplementary Information

### SI Text 1: The daily XRP/USD price

Here we show the daily XRP price between May 18 , 2017 and September 28, 2022
in Fig. S1. It can be observed that the most significant bubble formation and
crash of XRP/USD rate occurred at the end of 2017 and beginning of 2018.

SI Fig. S1:  The daily XRP/USD close price (source:
https://www.marketwatch.com/investing/cryptocurrency/xrpusd. The region
between the red dotted lines indicates the period that we consider for our
analysis.

### SI Text 2: Dependence of the correlation tensor on window size

The distributions for the elements of the randomized correlation tensors
become narrower as the time window $(2\Delta T+1)$ increases, which is shown
in Fig.S2 (a). We also show that the largest singular value decreases as the
window size increases in Fig.S2 (b). Moreover, the gap between smaller
singular values and larger singular values exists for small window sizes, but
vanishes as window size becomes large.

SI Fig. S2: (a) Distributions for the elements of the randomized correlation
tensor calculated for different window size. The legends indicate window sizes
$(2\Delta T+1)$. (b) Sorted singular values for the randomized correlation
tensors.

### SI Text 3: Multilinear regression model for XRP/USD

We perform the following multilinear regression models for the weekly XRP/USD
price and report the results in SI Table S1. To calculate the weekly
$\overline{\rm XRP/USD}$ rate, we take $7$-day average of the daily close
price of XRP/USD using the following expression

$\overline{\rm XRP/USD}(t)=\sum\limits_{t^{\prime}=t-3}^{t+3}{\rm
XRP/USD}(t^{\prime})$

Model 1 : $\overline{\rm XRP/USD}(t+1)$ $\sim$ intercept + the largest
singular value $\rho_{1}^{1}(t)$ \+ the second largest singular value
$\rho_{2}^{1}(t)$.

The model 1 explains $81\%$ of the variance (adjusted $R^{2}=0.809$) and the
significant covariates is only the largest singular value $\rho_{1}^{1}$.

Model 2 : $\overline{\rm XRP/USD}(t+2)$ $\sim$ intercept + the largest
singular value $\rho_{1}^{1}(t)$ \+ the second largest singular value
$\rho_{2}^{1}(t)$. The model 2 explains $76\%$ of the variance (adjusted
$R^{2}=0.758$) and the significant covariates is only the largest singular
value $\rho_{1}^{1}$.

Model 3 : $\overline{\rm XRP/USD}(t+3)$ $\sim$ intercept + the largest
singular value $\rho_{1}^{1}(t)$ \+ the second largest singular value
$\rho_{2}^{1}(t)$. The model 3 explains only 43% of the variance (adjusted
$R^{2}=0.433$) and the largest singular value $\rho_{1}^{1}$ is weakly
significant.

Model 4 : $\overline{\rm XRP/USD}(t+3)$ $\sim$ intercept + the largest
singular value $\rho_{1}^{1}(t)$. The model 4 explains only 47% of the
variance (adjusted $R^{2}=0.466$) and the largest singular value
$\rho_{1}^{1}$ is significant.

Table S1:  Multilinear regression table for the weekly XRP/USD price.

Variables | Model 1 | Model 2 | Model 3 | Model 4
---|---|---|---|---
Intercept | $27.914$ |  $36.502$ * | $36.796$ |  $28.489$**
The largest singular value, $\rho_{1}^{1}$ |  $-0.033$ ** |  $-0.038$ ** | $-0.035$ |  $-0.029$ **
The second largest singular value, $\rho_{2}^{1}$ | $0.008$ | $0.0002$ | $-0.006$ |
Observations | $18$ | $18$ | $17$ | $17$
Adjusted $R^{2}$ | $0.809$ | $0.758$ | $0.433$ | $0.466$
p value of F test | $1.58\times 10^{-6}$ | $9.34\times 10^{-6}$ | $0.007$ | $0.001$

Significance codes for p value: $0\leq***\leq 0.001\leq**\leq 0.01\leq*\leq
0.05$

### SI Text 4: Evolution of regular nodes in the communities of the weekly XRP
transaction networks

We use the infomap algorithm to uncover the communities for the weekly
networks of XRP transactions. The evolution for the regular nodes of XRP
transaction networks within the communities is shown in Fig.S3-S6, .

SI Fig. S3: Alluvial diagram for regular nodes showing the evolution of these
nodes within the communities of $5$ consecutive weekly networks from 2017
October 23 to 2017 November 26. Communities are arranged according to their
size and the number indicates the community index.  SI Fig. S4: Alluvial
diagram for regular nodes showing the evolution of these nodes within the
communities of $5$ consecutive weekly networks from 2017 December 11 to 2018
January 14. Communities are arranged according to their size and the number
indicates the community index.  SI Fig. S5: Alluvial diagram for regular nodes
showing the evolution of these nodes within the communities of $5$ consecutive
weekly networks from 2017 December 18 to 2018 January 21. Communities are
arranged according to their size and the number indicates the community index.
SI Fig. S6: Alluvial diagram for regular nodes showing the evolution of these
nodes within the communities of $5$ consecutive weekly networks from 2018
January 29 to 2018 March 04. Communities are arranged according to their size
and the number indicates the community index.