# Pairwise and high-order dependencies in the cryptocurrency trading network

T. Scagliarini1111Corresponding author<EMAIL_ADDRESS>, G.
Pappalardo2, A.E. Biondo3,
A. Pluchino2,4, A. Rapisarda2,4,5, S. Stramaglia1,6

(1 Dipartimento Interateneo di Fisica, Università degli Studi Aldo Moro, Bari
and INFN, Italy
2 Dipartimento di Fisica e Astronomia, Università degli Studi di Catania,
Italy
3 Dipartimento di Economia e Impresa, Università degli Studi di Catania, Italy
4 INFN Sezione di Catania, Italy
5 Complexity Science Hub, Vienna
6 Center of Innovative Technologies for Signal Detection and Processing
(TIRES),
Università degli Studi Aldo Moro, Italy )

###### Abstract

In this paper we analyse the effects of information flows in cryptocurrency
markets. We first define a cryptocurrency trading network, i.e. the network
made using cryptocurrencies as nodes and the Granger causality among their
weekly log returns as links, later we analyse its evolution over time. In
particular, with reference to years 2020 and 2021, we study the logarithmic US
dollar price returns of the cryptocurrency trading network using both pairwise
and high-order statistical dependencies, quantified by Granger causality and
O-information, respectively. With reference to the former, we find that it
shows peaks in correspondence of important events, like e.g., Covid-19
pandemic turbulence or occasional sudden prices rise. The corresponding
network structure is rather stable, across weekly time windows in the period
considered and the coins are the most influential nodes in the network. In the
pairwise description of the network, stable coins seem to play a marginal role
whereas, turning high-order dependencies, they appear in the highest number of
synergistic information circuits, thus proving that they play a major role for
high order effects. With reference to redundancy and synergy with the time
evolution of the total transactions in US dollars, we find that their large
volume in the first semester of 2021 seems to have triggered a transition in
the cryptocurrency network toward a more complex dynamical landscape. Our
results show that pairwise and high-order descriptions of complex financial
systems provide complementary information for cryptocurrency analysis.

##  1 Introduction

The interest in cryptocurrencies increased a lot in the last few years, among
investors and researchers. Still recently, until approximately five-seven year
ago, the cryptocurrency markets were the object of interest of few
professionals, vastly academic, interested on applications, technology and
even speculation on DeFi (Decentralized Finance). As data shows, during the
year 2017, a huge increase in the USD price of Bitcoin (something like
$2300\%$, from about 800 in January to more than 19000 in December [1]) caused
un undoubtedly understandable interest. An even more impressive piece of
evidence was that, after just after one month, that same price fell down to
$6852$ USD - a fall of more than $64\%$ [2]. Also other cryptocurrencies
manifested what we could name “turbulence” (at best) but the very peculiar
aspect of their dynamics was the impressive volatility, the apparently
intrinsic instability of these financial assets.

Needless to say, their markets have become attractive as a very ambitious and
dangerous lottery, still able to provide wide returns, at a scale difficultly
comparable with “traditional” financial instruments. Since then, a proliferous
set of cryptocurrencies has developed and traded on exchanges (ETH since 2015,
REP, XRP and ETC since 2016, USDT, XLM and BCH since 2017 among many others,
see table 2 for a fast overview), thus giving rise to an entire “sector”, as
if they should behave similarly, all apparently representing the hottest
opportunity of finance.

In time, therefore, the predominant position of Bitcoin has diminished and the
financial potential of these cryptos is now so fragmented over a numerous set
of competing alternatives, that it is very unlikely that one of them can
assume any truly monetary role. One of the apparent elements of appeal of
cryptos is the absence of a centralized control over them. The basic idea of
the blockchain rests more on the side of tracking transactions: encryption
prevents any personal identification of market participants and this attracts
many forms of traders, investors, negotiators in a wide variety of clear and
unclear market transactions. Therefore, the market capitalization of the
crypto world has assumed notable dimensions going to $\$14$ billion in 2014 to
$\$831$ billion in 2018, still remaining very high after severe reductions in
2018, until 2020 at $\$192$ billion. On the aggregate side, it is easy to
collect data of cryptocurrencies, since most of the anonymized exchanges are
often publicly available.

Crises show periodically that financial markets are complex systems where
interactions among individual participants is much more relevant in terms of
dynamics that the individual characteristics of individual participants. One
of the most relevant reasons why micro-prudential supervision (e.g., Basel
Agreements) is almost vein in order to prevent the systemic risk is that it
fails to consider complexity features emerging from interconnectedness among
financial institutions and their resulting financial network. Then, given the
wild and unpredictable instability of the price of cryptocurrencies, a branch
of literature has flourished dealing with analysis of signals and precursors
that could be useful to check the existence of correlations and systemic
regularities.

In this work, we study the impact of cryptocurrency price returns using
statistical tools to detect the flows of information between time series. To
this purpose, we first show how, in general, two variables exchange
information in a pairwise way by Granger causality, which detects the
information flow from a source variable to a target one.

Granger causality is a statistical method aimed to quantify the gain in linear
predictability between time series. The idea of defining causality as a useful
tool for predictions 222This definition does not necessarily imply a true
causal link between time series, due to the possible presence of confounding
or unobserved variables [3]. For this reason, in this context causality must
be intended as a form of predictive causality. was introduced by Wiener [4]
and was later implemented by Granger [5, 6] in the context of linear
autoregressive models, that will be used in the present work as weight on a
directed cryptocurrency traded network where each nodes is a cryptocurrency
and each link states if a crypto may be useful on the price prediction of
another one. Since 1960, Grange Causality has become a standard tool in
econometrics and has recently received a lot of attention in many other
contexts, such as neuroscience [7] and climatology [8]. All these studies take
into account the information exchange between pairs of variables, possibly
conditioning over a set of confounding elements to remove spurious effects.
Recently, it has been shown that in many systems a simple pairwise approach
may not fully describe the entire multivariate structure of information
exchange, since higher-order effects play an important role in their
description [9].

High-order effects might be appreciated under two points of view: when
considering multiplets of variables, the joint effect can be greater than the
sum of single contributions, a condition known as _synergy_ ; alternatively,
the case in which the overall effect results lower than the sum of the parts,
is called _redundancy_. Stated differently, redundancy occurs when multiple
copies of the same information can be found in different parts of the system,
while synergy refers to the part of the information that is not stored in any
specific element, but rather in the joint state of a group of variables. A
trivial example is the case where $X_{1}$ and $X_{2}$ are independent binary
random variables and $X_{3}=\texttt{XOR}(X_{1},X_{2})$. In this situation,
neither $X_{1}$ nor $X_{2}$ contain any information about $X_{3}$; rather,
information is completely stored in their joint state.

In a real-world example, the study of how two market indices cooperate to
transfer information to a third one has shown non-trivial synergistic effects
[10] using the Partial Information Decomposition, a statistical framework that
decomposes information in the unique, redundant and synergistic part but has
the drawback of not being computationally feasible for more than three
variables [11]. A simple yet promising approach for quantifying high-order
dependencies in a system is the O-information [12], which quantifies the
balance between redundancy and synergy. Speaking qualitatively, in a complex
system, the synergy measures its capacity to make integration of information
whilst redundancy provides robustness to the system; decomposing interactions
between variables into synergistic and redundant components illuminates how
the system addresses the trade-off between robustness and integration. For
example, recent works focusing on the brain at the macroscale have identified
high synergy brain regions which support higher cognitive function [13] and
are affected by the aging process [14].

Many studies have investigated different aspects of the cryptocurrency time
series, from their correlation properties [15, 16] to the possibility of being
used for portfolio optimization [17, 18]. The spillover effect between the
major cryptocurrencies has been investigated by using VAR (Vector
Autoregressive Models, as in [19], for instance), finding that return and
volatility spillover is mainly driven by the Bitcoin and this effect was
continuously rising during the considered period, suggesting a growing
interdependence among cryptocurrencies. Other studies have investigated the
relationship among the most important coins using VAR and SVAR (Structural
VAR) Granger causality, finding that ”Ethereum is likely to be the independent
coin in this market, while Bitcoin tends to be the spillover effect recipient”
[20]. An example of direct use of Granger Causality test to study information
flow between two major cryptocurrencies has been performed [21], but this
study was limited only to Bitcoin and Litecoin. In [22], multivariate transfer
entropy, which is a model-free approach to study information flow [23], was
used to study a network of cryptocurrencies, using a greedy algorithm to
detect the most informative drivers. They found an increase in information
flow during the market turbulence of March 2020. It has been shown that
Granger Causality and transfer entropy are equivalent for Gaussian processes
[24]. Being pairwise methods, neither Granger Causality nor Transfer entropy
can assess high-order effects. To the best of our knowledge, there are no
investigations among cryptocurrencies using high-order dependencies. The
present work addresses to contribute in filling this gap, studying a high-
order generalization of Granger Causality. We will show that the pairwise
approach and the higher-order one give complementary information to describe
the cryptocurrency trading network.

The paper is organized as follows. In section 2 and section 3 we describe the
dataset and the methodology used for the analyses, respectively. In section 4
we present and discuss the results while, in section 5, we close the paper
drawing some conclusions.

##  2 Data and preprocessing

The dataset investigated in this study is freely available from Kraken crypto
Exchange [25] and contains traded prices in general between cryptocoins and
currencies.

In the following we will use different names for cryptocurrencies according to
their own properties. We refer more specific to a Coin to describe a
cryptocoin with its blockchain, to a Token to describe a cryptocoin backed by
a blockchain of another Coin. Also, we refer as Stablecoin to indicate to a
Cryptocoin which has its price “pegged” to the price of another asset (usually
the USD change value) and in general to a Fiat currency to any government-
issued currency such as Euro, Dollar or others as described in [26].

Each file of the dataset is organized in “pairs”, where one pair consist of a
cryptocurrency traded using a fiat currency, so it reports all trade on a
specific market, which collect bid and ask orders on a corresponding order
book, from cryptocurrency to fiat currency and vice-versa. Each line refers to
a single trade or transaction. Since it is possible to have more than one
transaction per second referring to the same timestamp, with different price
and volume, they are reported on separate lines.

For each pair, data provided for each trade have the following information:

  * •

_timestamp_ represents the time when the trade occurs, with resolution in
seconds. It is possible to have more than one trade with the same timestamp,
in this case volume and price may be different for each trade;

  * •

_price_ is the price at which the cryptocurrency was traded, in terms of the
corresponding fiat currency;

  * •

_volume_ is the amount of cryptocurrency exchanged on the corresponding trade;

Data were aggregated on the scale of minutes, considering the weighted mean
over volume for the minute price of the trade, and the sum of volumes for the
total volume exchanged at the current minute time. If there were no trade on a
minute, in order to avoid discontinued time series, we hold the previous last
price to fill the gap on the time series, setting the corresponding volume to
zero. Then, we computed the logarithmic returns of each time series of the
price, which will be divided into time series of weekly length, to see how the
results vary over time and to take into account effect due to the different
volume of transactions over the week (usually smaller volumes are traded
during weekends). We divided the considered period in weekly time windows,
each starting at 00:00:00 every Monday and terminating at 23:59:59 every
Sunday (with the exception of last window because last day available on the
dataset was Friday, 31 Dec 2021), thus obtaining $104$ windows, each
containing a total of 10080 points.

We remark that the analysis is performed just on the time series of
logarithmic returns; the time series of volumes are used here only to compare
the time evolution of information-theoretical quantities with the total volume
of exchanges in US dollars. A cryptocurrency can be traded on the Exchange
using more than one fiat, then there is the possibility to have more than one
pair for each crypto, one for each available fiat market (i.e. Bitcoin can be
traded in EUR, Dollar or other fiat currencies related to the country of the
user). Also, the Exchange allowed to use some crypto coins (i.e. Bitcoin,
Ethereum) as fiat, probably to speed up buying and selling operations.

Table 2 in Appendix A reports a brief description of all data available on
Kraken, grouped by years, number of cryptos and fiat, while table 3 shows
which tickers are included on the dataset and some of their features.

As the cryptocoin market grows, the number of cryptos and fiat listed on
Kraken increase to about hundred cryptocoins, traded up to 12 different
market, for a total of about 450 crypto-fiat pairs.

In our analysis, we focus on the period 2020-2021 since the number of cryptos
is large and the volume of the transaction is also large. It is possible to
have an overview of the data related to each cryptos for 2020-2021 on table 4.

##  3 Methods

###  3.1 Pairwise Granger Causality

In this section, we describe how to implement Granger Causality between pair
of variables in the context of linear autoregressive models.

Let us consider two stationary time series $X_{t}$ and $Y_{t}$. Under suitable
conditions [27], the target series $Y$ can be written as a weighted sum of its
past states and an error term $\epsilon^{\prime}_{t}$ (reduced model);
similarly, a second model can be built by adding the past states of the source
variable (full model):

$\displaystyle Y_{t}$
$\displaystyle=\sum_{m=1}^{p}a_{m}Y_{t-m}+\epsilon^{\prime}_{t}$ (1)
$\displaystyle Y_{t}$
$\displaystyle=\sum_{m=1}^{p}a_{m}Y_{t-m}+\sum_{m=1}^{q}b_{m}X_{t-m}+\epsilon_{t}~{}$
(2)

In other words, in the full model, the future states of the target are
predicted using its previous $p$ past states and the previous $q$ states of
the source, while in the reduced model only the target is involved.

The goodness of these models can be quantified by the variance of the error
term of the full model $\sigma^{2}_{f}=\langle\epsilon^{2}_{t}\rangle$ and of
the reduced model $\sigma_{r}^{2}=\langle\epsilon^{\prime}_{t}\rangle$, where
$\langle\cdot\rangle$ indicates the expected value. To test if the full model
improves the predictability of the target, we use the statistics

$\mathcal{F}_{X\rightarrow Y}=\ln\frac{\sigma_{r}^{2}}{\sigma^{2}_{f}}~{}.$
(3)

This choice has a number of useful properties [28], remarkably the invariance
under scaling transformation of the original time series and the asymptotic
chi-square distribution under the null hypothesis $\mathcal{F}_{X\rightarrow
Y}=0$. To build a bridge between Granger causality and the information-
theoretical methods for analyzing high-order dependencies of the next section,
we can consider the equivalence between Granger causality and transfer entropy
$\mathcal{T}$ under the Gaussian approximation [24].

Transfer entropy is a model-free version of Granger causality. Considering the
source states $\bm{X}^{-}_{t}=(X_{t-1},X_{t-2},\ldots,X_{t-p})$, at varying
$t$, as realizations of a stochastic variable $\bm{X}^{-}$, and analogously
$\bm{Y}_{t}^{-}=(Y_{t-1},Y_{t-2},\ldots,Y_{t-p})$ as realizations of the
target state variable $\bm{Y}^{-}$, $Y_{t}$ as realizations of the future of
the target y, considered as a stochastic variable, the transfer entropy is
defined as the mutual information between source and target variable
conditioned over the target past states:

$\mathcal{T}_{X\rightarrow Y}=I(y;\bm{X}^{-}|\bm{Y}^{-})$ (4)

The equivalence reads:

$\mathcal{F}_{X\rightarrow Y}=2\mathcal{T}_{X\rightarrow Y}$ (5)

The pairwise Granger causality has been computed for each pair using the
MATLAB toolbox MVGC [29]. The result of Granger causality produce an adjacency
matrix $A=\\{a_{ij}\\}$, where the element $a_{ij}$ represents the Granger
Causality from the cryptocoin $i$ toward the cryptocoin $j$, which can be used
as weighted links for our directed Cryptocurrencies Trading Network.

To find the most influencing and influenced nodes, one may define the
quantities out strength and in strength:

$k^{{\rm out}}_{i}=\sum_{j\neq i}a_{ij}$ (6)

$k^{{\rm in}}_{i}=\sum_{j\neq i}a_{ji}$ (7)

###  3.2 O-information

We now describe how to study high-order dependencies in a collection of random
variables $\bm{X}^{n}=(X_{1},X_{2},\ldots,X_{n})$. The first step is to extend
the concept of mutual information to the multivariate case: a popular way to
do it is through _total correlation_ (also known as multi-information)[30],
which quantifies the amount of high-order constraints and then is a measure of
redundant information. In terms of information-theoretical quantities, it can
be expressed as:

$\mathcal{T}(\bm{X}^{n})=\sum_{k=1}^{n}H(X_{k})-H(\bm{X}^{n}),$ (8)

where $H(X_{k})$ and $H(\bm{X}^{n})$ are the Shannon entropies, which quantify
the uncertainty of the k-th variable and of the whole system, respectively.
Another useful way of looking at this quantity is as a measure of the
”statistical constraints” acting on $X_{k}$: when $H$ is low, the system
explores more frequently small regions of the phase space and, in this sense,
the constraints are high. Conversely, when $H$ is maximum (a condition that
occurs when the distribution is uniform), then there is the minimum possible
amount of constraints acting on $X_{k}$.

Another popular extension of mutual information is the _dual total
correlation_ [31]

$\mathcal{D}(\bm{X}^{n})=H(\bm{X}^{n})-\sum_{k=1}^{n}H(X_{k}|\bm{X}^{n}_{-k})$
(9)

where $\bm{X}^{n}_{-k}$ represents all the system minus the k-th variable.
$\mathcal{D}(\bm{X}^{n})$ may be interpreted as the amount of uncertainty that
can be explained only by observing more than one variable at once: for this
reason, dual total correlation is a measure of the synergy present in the
system at global level. Crucially, it can be easily shown that both
$\mathcal{T}(\bm{X}^{n})$ and $\mathcal{D}(\bm{X}^{n})$ are nonnegative [32]
and then can be used as proper measures of redundancy and synergy.

The following quantity, called O-information [12], represents the balance
between the redundant and the synergistic dependencies in $\bm{X}^{n}$:

$\displaystyle\Omega(\bm{X}^{n})$
$\displaystyle=\mathcal{T}(\bm{X}^{n})-\mathcal{D}(\bm{X}^{n})$ (10)
$\displaystyle=(n-2)H(\bm{X}^{n})+\sum_{k=1}^{n}\big{[}H(X_{k})-H(\bm{X}^{n}_{-k})\big{]}.$
(11)

Accordingly, if this quantity is greater than $0$, we say that the system is
redundancy-dominated, whilst if $\Omega<0$ the system is synergy-dominated.

Now, we introduce a target variable $y$ to study whether the high-order
dependencies between a multiplet of $n$ variables $\bm{X}^{n}$ with $y$ are
redundant or synergistic. We can express the O-information of the system
jointly formed by the multiplet and the target as follows

$\Omega(\bm{X}^{n}\cup y)=\Omega(\bm{X}^{n})+\Delta^{y}$ (12)

where $\Delta^{y}$ the contribution of a variable to the O-information and
reads:

$\Delta^{y}=(1-n)I(y;\bm{X}^{n})+\sum_{k=1}^{n}I(y;\bm{X}^{n}_{-k}),$ (13)

where $I(y;\bm{X}^{n})=H(y)+H(\bm{X}^{n})-H(y;\bm{X}^{n})$ denotes mutual
information.

Now we turn to consider $n$ time series $\\{X_{i,t}\\}$ and a target
$\\{Y_{t}\\}$, where $t=1,\ldots,T$ and $i=1,\ldots,n$; the informational
character of the information flow from the multiplet of X variables
(considered as sources) to the target, can be assessed introducing stochastic
variables $\bm{X}^{n-}=(X_{1}^{-},X_{2}^{-},\ldots,X_{n}^{-})$, $\bm{Y}^{-}$
and $y$ representing the sources, the target and the future of the target
respectively, whose realizations can be obtained from the samples multivariate
time series.

Hence, the dynamical version of the O-information, introduced in [33] for the
analysis of neural signals, reads

$d\Omega^{y}(\bm{X}^{n})=(1-n)I(y;\bm{X}^{n-}|\bm{Y}^{-})+\sum_{k=1}^{n}I(y;\bm{X}^{n-}_{-k}|\bm{Y}^{-}).$
(14)

This quantity can be also thought as a high-order version of the Granger
causality and its sign has the same interpretation of the O-information in
terms of redundancy and synergy.

In order to analyse the dataset, we proceed as follows. First, we fixed a
target variable $y$; then, we searched among the remaining $n-1$ variables for
the pair that maximize and minimize the quantity

$d\Omega^{y}_{\text{red}}(\bm{X}^{2})=\max_{i,j}d\Omega^{y}(\\{X_{i}X_{j}\\})$
$d\Omega^{y}_{\text{syn}}(\bm{X}^{2})=\min_{i,j}d\Omega^{y}(\\{X_{i}X_{j}\\})$

Then, $d\Omega^{y}_{\text{red}}(\bm{X}^{2})$ and
$d\Omega^{y}_{\text{syn}}(\bm{X}^{2})$ is the highest and lowest dynamic
O-information to the target $y$ from the best multiplets
${\bm{\tilde{}}{X}^{2}}=\\{X_{i^{*}}X_{j^{*}}\\}$ of size $n=2$. For sizes
$n>2$, we used a greedy approach: for $n=3$, we start from the
${\bm{\tilde{}}{X}^{2}}=\\{X_{i^{*}}X_{j^{*}}\\}$ found in the previous step,
and for the remaining variables $X_{k}$ we choose the one that maximizes and
minimize $d\Omega^{y}(\\{X_{i}X_{j}X_{k}\\})$. We repeated the same procedure
for $n=4$ and so on, and we stop at $n=5$.

##  4 Results and Discussion

In this section we present and discuss the results obtained with both the
pairwise Granger causality approach and the O-information methodology,
evaluated among the logarithmic returns time series of cryptocurrencies
exchanged in US dollars (USD) on the same time window.

###  4.1 Pairwise Granger Causality

For each one of the 104 weekly windows (from 30 Dec, 2019 to 31 Dec, 2021),
Granger causality (GC) analysis was conducted using Equation (3) for all the
$n(n-1)/2$ pair of variables active in that window, which become nodes of the
cryptocurrency traded network. For each pair, the optimum model order $p$ was
chosen using the Bayesian information criterion [34]. A link between a pair of
nodes is created only if the GC strength overcomes the significance threshold
of $1\%$.

A first visual representation of the structure of the USD network averaged
over all the time windows is shown in Figure 1, where the biggest nodes
represent the most influential elements and the color scale of the labels
indicates how much a node is influenced by all the others (for a better
visualization, only the first $5\%$ of links sorted according to their GC
strength is reported in the figure). This kind of graph is very effective in
giving an overall idea about the relative importance of cryptocurrencies in
the whole considered period (the network evolution across weekly time windows
will be addressed later).

Figure 1: Average USD network. Each link $\overline{a}_{ij}$ represents the
Granger Causality averaged over the 104 weekly windows from 28 Dec, 2019 to 30
Dec, 2021. Node size represents the total out-strength $k^{{\rm
out}}_{i}=\sum_{j}\overline{a}_{ij}$, while the label color (from black to
red) is the total in-strength $k^{{\rm in}}_{i}=\sum_{j}\overline{a}_{ij}$,
that is how much that node is influenced by the rest of the network.

To see more in detail which cryptocurrencies transfer the most information to
the rest of the system and which receive the most information from the others,
it is useful to look at the ranking of the average in strength and out
strength, calculated with Equation (6). These information are shown,
respectively, in the top and bottom panels of Figure 2. The different classes
of cryptocurrencies - token, coin and stable coins - are represented using
different colors. As can be seen in the top panel, the most influential
cryptocurrencies belong to the class “coin” (the ones that have their own
blockchain, colored in red). Unsurprisingly, the most influential coins are
the eldest and most capitalized, such as Bitcoin (XBT), Ethereum (ETH), and
Litecoin (LTC). Among the topmost influenced elements (bottom) we find younger
and less capitalized “coins”, such as Tron (TRX), EOSIO (EOS) and Dash (DASH).
Stable coins seem to play a marginal role in the pairwise description of the
network, both for transferred (EUR, DAI, GBP) and received information (EUR,
USD).

Figure 2: The most influential and influenced nodes in the USD network of
pairwise Granger causality. On the y-axis is indicated the total in-strenght
and out-strength . Yellow bars indicates stable-coins, red bars coins and blue
bars tokens. Only the first $70$ elements are shown.

To get a visual hint of how the information flow changes over time, in Figure
3 we depict the structure of the USD network in several weekly windows (for a
period going from 28 Dec, 2020 to 22 Mar, 2021), selected for being stable
windows or characterized by the presence of crashes or turbulence. For
example, an anomalous activity involving several currencies with high out
strength (biggest nodes) can be appreciated in the week starting on 22 Feb,
2021, while during the majority of the other weeks the information flow is
visibly much lower. It is worth to notice that the number of nodes in the
network slightly increases for subsequent windows, since new cryptocurrencies
enter in the market/dataset.

|  |
---|---|---
|  |
|  |
|  |

(a) 28-Dec-2020  (b) 04-Jan-2021

(c) 11-Jan-2021 (d) 18-Jan-2021

(e) 25-Jan-2021 (f) 01-Feb-2021

(g) 08-Feb-2021 (h) 14-Feb-2021

(i) 22-Feb-2021 (j) 01-Mar-2021

(k) 08-Mar-2021 (l) 15-Mar-2021

Figure 3: The USD network of pairwise Granger causality for weekly time
windows. Node size represents the total out strength of that node or, in other
words, how much that node influences the network, while the label color (from
black to red) is the total in strength, that is how much that node is
influenced by the rest of the network.

In this regard, it is interesting to investigate if there is some correlation
between the age of the cryptocurrencies (i.e. the number of time windows in
the USD dataset where it is present) and their in and out strength. As can be
seen in the first row of Table 4.1, actually it seems that such a correlation
does exist and is also quite consistent (expecially for the out strength).

[tabular=c—c—c, table head=, table foot=, no head, late after line=
, late after first line=
,]./figures/birth_corr.csv

Table 1: Table shows the Spearman correlation $\rho$ between the in-strength
and the out-strength of a fiat with its “age” (that is, the number of windows
in which it was active in the dataset). Two asterisks indicate that $\rho$ is
significant at $p<0.01$, one asterisk at $p<0.05$.

Repeating the same procedure also for the other datasets (where the cryptos
are exchanged in different fiats, i.e. EUR, Bitcoin or Ethereum), it results
that the correlation stays quite high only when EUR is adopted as fiat (second
row in Table 4.1) while it rapidly disappear for the other fiat currencies
(third and fourth rows). This can be explained by the fact that when a given
fiat is firstly adopted for the exchange, it is still quite unpopular, and
then its connections with the rest of the system from the point of view of
Granger causality are still weak. Conversely, as time passed, its importance
grows and this is reflected in the values of information flow.

Finally, it is also interesting to look at the correlation between pair of USD
networks of Granger causality as they appear in any couple of windows $h$ and
$k$. To this aim, in Figure 4 we report the Pearson correlations
$\rho(A^{h},A^{k})$ between the adjacency matrices $A^{h}$ and $A^{k}$, for
each $h,k=1,2,\ldots,104$. Correlation $\rho$ can be seen as a measure of
similarity between networks: high values of $\rho$ in two different windows
indicate that, on average, the information flows between the cryptocurrencies
are similar in the corresponding two periods. If we look at values close to
the main diagonal, we see that for consecutively windows the correlation is
very high, often above $0.9$; also, inspecting windows more separate in time
(that is, looking at values far from the main diagonal) correlation still
remains quite high, indicating that the structure of the Granger causality
network is quite stable or, in other words, that the information flow between
the various nodes does not change much over time.

Analogous correlation matrices calculated for the other datasets (EUR, XBT and
ETH) are reported for completeness in Figure 9 of Appendix B (of course, as
expected for the reasons explained above, correlations between networks traded
in XBT and ETH are much less significant with respect to EUR and USD).

Figure 4: The Pearson correlation between adjacency matrices at different time
windows. The element $c_{ij}=\rho(A^{i},A^{j})$ of the matrix is the Pearson
correlation between the vectorized adjacency matrices at windows $i$ and $j$.
High values of $c_{ij}$ indicate that the structure of the network at windows
$i$ and $j$ is very similar. Diagonal elements, indicating self-correlation,
are colored in gray.

Summarizing the results obtained with the Granger pairwise analysis, we can
say that, on one hand, as one could expect, for a specific time window nodes
influence/get influenced each other according to their out/in strength. On the
other hand, the structure of the network stays quite stable over consecutive
time windows, since the in/out strength values of nodes remain comparable. In
this respect, even if in correspondence of important events or shock in the
market the structure of the network seems to change, increasing the number of
links and their weight, when the effect of those events vanishes the network
come back to its original state.

###  4.2 O-information

Let us now describe the results of the analysis of the USD dataset from the
point of view of high-order dependencies, using the dynamical O-information
introduced in Section 3.2.

A first interesting observation can be made about Figure 5, where it is shown
the ranking of the presence of each class of cryptocurrency in the best
multiplets, both for redundancy and synergy. Here, on the contrary of the
pairwise case, one can observe that stable coins are the variables which
belong to the highest number of synergistic information circuits, hence they
play a major role for high order effects.

Figure 5: Most influential nodes for synergy and redundancy in the USD
dataset. The bar plot counts the number of times a cryptocurrency is found in
the best multiplets built as explained in the text. Yellow bars indicates
stable-coins, red bars coins and blue bars tokens. Only the first $60$
elements are shown.

We get other insights by observing the composition of the best multiplets as
function of their size. In Figure 6 we report the typical fraction of elements
belonging to the 3 different classes of cryptocurrencies for both redundancy
(left panel) and synergy (right panel). Concerning the redundant multiplets,
the composition do not depend on the size but only on the class. The highest
fraction of elements in these multiplets are tokens, which are also the most
common in the original dataset (60/99): high redundancy can be explained from
the fact that these cryptocurrency don’t have a own blockchain, but are built
on top of another crypto-coin. Notice that the lowest fraction are stable
coins, which are the least common in the dataset (6/99).

Figure 6: We depict the typical fraction of coin, token and stable-coin in the
optimal multiplets $\bm{\tilde{X}}^{n}$ as a function of their size $n$, for
both redundancy (left) and synergy (right). In the redundant case, these
weights are similar to the original dataset (composed of 60 tokens, 33 coins
and 6 stable-coins), while for synergistic multiplets coins are the most
common and the stable-coins are much more present.

The situation is very different for the synergistic multiplets (right panel),
where the size plays a greater role: actually, the fraction of stable coins
increases with the size, while that of the other classes tends to decrease.
Interestingly, from size $3$ onwards the best multiplets are populated much
more often by coins and, in particular, by the stable coins, even if they are
very few in the dataset.

These findings are quite surprising since stable coins have a rather marginal
activity in the pairwise Granger causality, as can be seen in Figure 2. This
result confirms that taking into account high-order dependencies through
O-information provides insights that cannot be retrieved with a pairwise
analysis.

###  4.3 Complementarity of pairwise and high-order descriptions

In order to better highlight the complementarity between the pairwise and the
high-order descriptions, in Figure 7 we make a comparison between the average
values of the pairwise and high-order indicators analyzed so far for the USD
dataset. The second and third panel (from the top) address the O-information,
showing the values of $d\Omega_{\bm{\tilde{X}^{5}}\rightarrow y}$ averaged
over each target as function of the $104$ weekly windows, for redundancy and
synergy respectively. In the bottom panel, we depict the analogous time
behavior of Granger causality $\mathcal{F}_{X\rightarrow Y}$ averaged over all
the possible pairs. Finally, in the top panel, we show the behavior of total
exchanges in US dollars, which can be meant as the total volume of
transactions which have been performed during each weekly window.

Figure 7: From top to bottom: the total volume (in US dollar) exchanged in
each time window; the average value of redundancy and synergy and the average
value of Granger causality $\overline{\mathcal{F}}$ calculated for each time
window. The 10-weeks moving average of the signal is represented in red.

It is quite clear the presence of three successive stages characterized by a
different behaviour of the O-information indicators: (i) the first phase, from
January to December 2020, shows high redundancy of the market and low synergy;
(ii) the second phase, from January to June 2021, is characterized by a fall
of the redundancy and a rise of the synergy; (iii) finally, a third phase can
be located between July and December 2021 and is characterized by a stationary
dynamics of both synergy and redundancy, which stay relatively high. Since
synergy may be interpreted as an indicator of the system’s complexity, this
result suggests that, during the first semester of 2021, the cryptocurrencies
network has undergone a transition toward a more complex dynamical landscape.
The transition seems to be triggered by the large volume of transactions
occurred in the first semester of 2021.

On the other hand, the behaviour of the Granger causality seems insensitive to
such a transition, remaining quite homogeneous during the whole considered
period, with the exception of sudden peaks in correspondence of important
events - like Covid turbulence in 2020, days of sudden rise in prices etc.
Notice, in particular, the peak of 22 Feb 2021, which also appears in the
synergy panel and coincides with the first minimum of redundancy. It could be
related with the week of anomalous activity already observed in Figure 3,
maybe the effect of some market movements and announcements of Elon Musk about
cryptos and their adoption/dismiss as payment method - as reported by
financial press 333See for example:
https://enterprise.press/stories/2021/02/22/what-the-markets-are-doing-
on-22-february-2021-32943/.

A global perspective about the time behaviour of both the O-information and
pairwise indicators for each cryptocurrency of the USD dataset can be
appreciated in Figure 8, where we report the same quantities of the previous
figure but without averaging over all the crypto. In particular, in the left
and central panel it is shown the redundancy and synergistic value of
dynamical O-information from the best multiplet, respectively. In the right
panel, it is shown the Granger causality out strength of each crypto for each
weekly time window. Notice that in such a visualization it becomes clear that
only about thirty cryptos (the first from the top) are always present in the
dataset for the whole considered time period: the others, ordered according to
their age, show an increasing number of weekly windows (colored in gray) in
which they are not yet listed.

Focusing on the highest and lowest values of redundancy and synergy, a
behaviour analogous to that one observed in Figure 7 can be easily detected
also at this disaggregated level, and the same holds for the Granger
causality, thus further confirming the complementarity of the pairwise and the
high-order approach.

Figure 8: The redundant (left) and synergistic (center) dynamical
O-information toward various targets, where each point represents the value of
$d\Omega_{{\bm{\tilde{}}{X}^{5}}\rightarrow y}$ of the best multiplet of
sources ${\bm{\tilde{}}{X}^{5}}$ that conveys the highest and lowest
O-information toward the target $y$. We also depict the out-strength Granger
causality (right) for each weekly window. Regions in grey represents windows
where that cryptocurrency was not yet listed.

##  5 Conclusions

In this work, we have analysed the logarithmic USD price returns, building a
network of traded cryptocurrency by defining the Granger causality between
each pairs of cryptocoin in the years 2020 and 2021 as links. We used both
pairwise and high order statistical dependencies, as measured by Granger
causality and O-information, with a dataset consisting of two classes of
currencies (coins and tokens) and six stable coins, for a total of 99 time
series.

We found that, in terms of Granger causality, the network structure is rather
stable across time, referring to weekly time windows in the considered period
since, according to our analysis, the set of most influential cryptocoins
presents small changes over time. The structure of the network may change due
to relevant events or shocks within a specific period, but in a short time it
tends to restore its original condition.

Turning to consider high-order dependencies, we have analysed how multiplets
of cryptocurrencies carry redundant and synergistic information toward the
future states of other variables. In synergistic multiplets, as the size of
the multiplets increases, the same does the fraction of stable coin on it,
despite their marginal role pointed out by the pairwise analysis. On the other
hand, in the redundant case, stablecoins fraction is less than 10% and the
multiplets composition does not vary while increasing their size.

We obtained other insights by looking at the temporal trends of the pairwise
and high-order indicators. In particular, we compared Granger causality,
redundancy and synergy with the time evolution of the total amount of
exchanges in US dollars, and we found three successive stages characterized by
different behaviour of these indicators. The first period, between January to
December 2020, is characterized by high redundancy and low synergy among
cryptocurrencies. Both change directions on the second phase, starting from
January to June 2021, where redundancy fall and synergy rise up. At the last
phase, located between July and December 2021, both indicators maintains a
stationary dynamics.

These results suggest that pairwise and high-order descriptions of complex
financial systems provide complementary information, thus, considered
together, they represent a very promising tool for the analysis of
cryptocurrency trading networks or similar ones.

Conclusively, considering the amount of resources and the relevance of trading
activities related to cryptocurrencies, the analysis of their dynamics is a
relevant field of investigation aimed at stabilizing their markets. At the end
of the day, cryptocurrencies are not financial instruments with an intrinsic
value representing productive assets. They have been conceived as mediums of
exchange, in markets where their quotation is totally self-referenced and
driven by their adoption in transactions (often of unknown type and content).

Nonetheless, the attracting appeal of positive price jumps and bubbles induces
investors to underestimate potential losses. Our analysis of the network of
cryptocurrencies might be useful to unveil that while drivers of instability
cannot be removed in general, specially in the absence of intrinsic –
fundamental – values, quotations and prices become extremely risky and
unpredictable. This, on a normative side, could inspire policies aimed at
fostering stability of markets, anchored more to produced values than to
financial speculation.

Matter of factly, some attempts have been done to create a formal market
structure for some crypto (i.e. lending platforms such as CoinLoan or
Hodlnaut, Binance and Crypto.com as an example of crypto-debit payments card
and even Sandbox virual world, where each player can buy objects using
cryptocoins and even earning coins by playing the game). The need for a more
“readable” orientation, aimed at providing priors on the market configuration
and on its physiology, shows that despite a financial artefact may reveal
appealing in block-chain transactions, it might be perceived as dangerous and
too risky.

Our analysis goes to the direction of investigating and understanding what
moves the dynamics of such highly volatile crypto currencies, which can be
useful from a policy standpoint in order to foster stability and prevent the
destruction of relevant financial values.

##  Appendix A Tables

In Tables 2, 3 and 4 we provide more details on the global Kraken dataset used
in the analysis.

Year | number of crypto | number of fiat
---|---|---
2013 | 2 | 2
| (XBT, LTC) | (EUR, USD)
2014 | 2 | 4
|  | (+ GBP,JPY only for XBT)
2015 | 3 | 5
| (+ ETH) | (+ CAD for XBT and ETH)
2016 | 9 | 7
| (+ZEC,XLM,REP,XRP,XDG,ETC) | (+XBT,ETH as fiat)
2017 | 16 | 7
| (+ USDT,DASH,XMR,GNO,MLN,EOS,BCH) |
2018 | 19 | 7
| (+ ADA,QTUM,XTZ) |
2019 | 30 | 10
| (…) | (+ USDT,CHF,DAI)
2020 | 60 | 11
| (…) | (+AUD)
2021 | 99 | 12
| (…) | (+DOT)
Table 2: Number of cryptocoins and fiat available on Kraken during years. At
the beginning of the launch of the Kraken platform, there were available only
two cryptos (Bitcoin as XBT and Litecoin as LTC) traded using 2 fiat (Euro and
Dollar, using EUR and USD as corresponding tickers). As far as the platform
grow up, there were listed more cryptos and fiat, up to 2021 when there are
more than 90 cryptos available, and traded using 12 fiat. Since 2016, Bitcoin
and Ethereum were also used as “fiat”, to simplify the process of buying and
selling.

Ticker | Name | website | type | Consensus mechanism
---|---|---|---|---
1INCH | 1inch | https://1inch.io/ | TOKEN |
AAVE | Aave | https://aave.com/ | TOKEN |
ADA | Cardano | https://cardano.org | COIN | Proof of Stake
ALGO | Algorand | http://algorand.foundation/ | COIN | Proof of Stake
ANKR | Ankr | https://www.ankr.com/ | TOKEN |
ANT | Aragon | https://aragon.org/ | TOKEN |
ATOM | Cosmos | https://cosmos.network/ | COIN | Proof of Stake
AVAX | Avalanche | https://avax.network/ | COIN | Proof of Stake
AXS | Axie Infinity Shards | https://axieinfinity.com/ | TOKEN |
BADGER | Badger DAO | https://app.badger.finance/ | TOKEN |
BAL | Balancer | https://balancer.finance/ | TOKEN |
BAND | Band Protocol | https://bandprotocol.com/ | TOKEN |
BAT | Basic Attention Token | https://basicattentiontoken.org/ | TOKEN |
BCH | Bitcoin Cash | http://bch.info/ | COIN | Proof of Work
BNC | Bifrost | https://bifrost.finance/ | TOKEN |
BNT | Bancor | https://bancor.network/ | TOKEN |
CHZ | Chiliz | https://www.chiliz.com/ | TOKEN |
COMP | Compound | https://compound.finance/governance/comp | TOKEN |
CQT | Covalent | https://www.covalenthq.com/ | TOKEN |
CRV | Curve | https://www.curve.fi/ | TOKEN |
CTSI | Cartesi | https://cartesi.io/ | TOKEN |
DAI | Dai | http://www.makerdao.com/ | TOKEN - USD StableCoin |
DASH | Dash | https://www.dash.org/ | COIN | Hybrid – PoW & PoS
DOT | Polkadot | https://polkadot.network/ | COIN | Nominated Proof of Stake
DYDX | dYdX | https://dydx.community/ | TOKEN |
ENJ | Enjin Coin | https://enjin.io/ | TOKEN |
EOS | EOSIO | https://eos.io | COIN | Delegated Proof of Stake
ETC | Ethereum Classic | https://ethereumclassic.org/ | COIN | Proof of Work
ETH | Ethereum | https://www.ethereum.org/ | COIN | Proof of Work
ETH2.S | Ethereum 2.0 Staking |  |  | Proof of Stake
EUR |  |  | FIAT |
EWT | Energy Web | https://www.energyweb.org/ | COIN | Proof of Authority
FIL | Filecoin | https://filecoin.io/ | COIN | Proof-of-Replication and Proof-of-Spacetime
GBP |  |  | FIAT |
GHST | Aavegotchi | https://aavegotchi.com/ | TOKEN |
GNO | Gnosis | https://gnosis.io/ | TOKEN |
GRT | The Graph | https://thegraph.com/ | TOKEN |
ICX | Icon | https://icon.community/ | COIN | Proof of Stake
INJ | Injective | https://injective.com/ | TOKEN |
KAR | Karura | http://karura.network/ | TOKEN |
KAVA | Kava | https://www.kava.io/ | TOKEN |
KEEP | Keep Network | https://keep.network/ | TOKEN |
KILT | Kilt Protocol | https://kilt.io/ | TOKEN |
KNC | Kyber Network | https://kyber.network/ | TOKEN |
KSM | Kusama | https://kusama.network/ | COIN | Nominated Proof of Stake
LINK | Chainlin | https://chain.link/ | TOKEN |
LPT | Livepeer | https://livepeer.org/ | TOKEN |
LRC | Loopring | https://loopring.org/ | TOKEN |
LSK | Lisk | https://lisk.com/ | COIN | Delegated Proof of Stake
LTC | Litecoin | https://litecoin.org/ | COIN | Proof of Work
LUNA | Terra | https://terra.money/ | COIN | Proof of Stake
MANA | Decentraland | https://decentraland.org/ | TOKEN |
MATIC | Polygon | https://polygon.technology/ | COIN | Proof of Stake
MINA | Mina | https://minaprotocol.com/ | COIN | Zk-SNARK
MIR | Mirror Protocol | https://mirror.finance/ | TOKEN |
MKR | Maker | https://makerdao.com/ | TOKEN |
MLN | Enzyme Finance | https://enzyme.finance/ | TOKEN |
MOVR | Moonriver | https://moonbeam.network/networks/moonriver/ | TOKEN |
NANO | Nano | https://nano.org/en | COIN | Delegated Proof of Stake
OCEAN | Ocean | https://oceanprotocol.com/ | TOKEN |
OGN | Origin | https://www.originprotocol.com/ | TOKEN |
OMG | OMG Network | https://omg.network/ | TOKEN |
OXT | Orchid | https://www.orchid.com/ | TOKEN |
OXY | Oxygen | https://www.oxygen.org/ | TOKEN |
PAXG | PAX Gold | https://www.paxos.com/paxgold/ | TOKEN |
PERP | Perpetual Protocol | https://perp.com/ | TOKEN |
PHA | Phala | https://phala.network/ | TOKEN |
QTUM | Qtum | https://qtum.org/ | COIN | Mutualized Proof of Stake
RARI | Rarible | https://app.rarible.com/rari | TOKEN |
RAY | Raydium | https://raydium.io/ | TOKEN |
REN | Ren | https://renproject.io/ | TOKEN |
REP | Augur | http://www.augur.net/ | TOKEN |
REPV2 | Augur v2 | http://www.augur.net/ | TOKEN |
SAND | Sandbox | https://www.sandbox.game/en/ | TOKEN |
SC | Siacoin | https://sia.tech/ | COIN | Proof of Work
SDN | Shiden | https://shiden.astar.network/ | TOKEN |
SHIB | Shiba Inu | https://shibatoken.com/ | TOKEN |
SNX | Synthetix | https://www.synthetix.io/ | TOKEN |
SOL | Solana | https://solana.com/ | COIN | Proof of Stake
SRM | Serum | https://projectserum.com/ | TOKEN |
STORJ | Storj | https://storj.io/ | TOKEN |
SUSHI | Sushi | https://sushi.com/ | TOKEN |
TBTC | tBTC | https://tbtc.network/ | TOKEN |
TRX | Tron | https://tron.network/ | COIN | Delegated Proof of Stake
UNI | Uniswap | https://uniswap.org/blog/uni/ | TOKEN |
USD |  |  | FIAT |
USDC | USD Coin | https://www.centre.io/usdc | TOKEN - USD StableCoin |
USDT | Tether | https://tether.to/ | TOKEN - USD StableCoin |
WAVES | Waves | https://waves.tech/ | COIN | Leased Proof of Stake
WBTC | Wrapped Bitcoin | https://wbtc.network/ | TOKEN | LpoS
XBT | Bitcoin | https://bitcoin.org/ | COIN | Proof of Work
XDG | Dogecoin | http://dogecoin.com/ | COIN | Proof of Work
XLM | Stellar | https://www.stellar.org/ | COIN | Stellar Consensus Protocol
XMR | Monero | https://www.getmonero.org/ | COIN | Proof of Work
XRP | Ripple | https://ripple.com/ | COIN | Ripple consensus
XTZ | Tezos | https://www.tezos.com/ | COIN | Liquid Proof of Stake
YFI | yEarn | https://yearn.finance/ | TOKEN |
ZEC | Zcash | https://z.cash/ | COIN | Zk-SNARK
ZRX | 0x | https://0x.org/ | TOKEN |

Table 3: Tickers description. Each row provides some information on the
ticker, such as the name of the coin, the website of the project, its type
(COIN for cryptocoin which hold their own blockchain, TOKEN for cryptos built
on top of another cryptocoin, fiat for currencies and TOKEN - USD Stablecoin
for token hooked to the USD dollar price) and what algorithm it is used as
consensus mechanism (for COINS only).

|  | 2020 | 2021
---|---|---|---
crypto | firstday | EUR | USD | GBP | CAD | AUD | CHF | DAI | JPY | USDT | USDC | XBT | ETH | DOT | EUR | USD | GBP | CAD | AUD | CHF | DAI | JPY | USDT | USDC | XBT | ETH | DOT
1INCH | 20210810 |  |  |  |  |  |  |  |  |  |  |  |  |  | 144 | 144 |  |  |  |  |  |  |  |  |  |  |
AAVE | 20201215 | 17 | 17 | 17 |  |  |  |  |  |  |  | 17 | 17 |  | 365 | 365 | 365 |  | 345 |  |  |  |  |  | 365 | 365 |
ADA | 20191230 | 368 | 368 |  |  |  |  |  |  | 78 |  | 368 | 368 |  | 365 | 365 | 345 |  | 345 |  |  |  | 365 |  | 365 | 365 |
ALGO | 20200122 | 345 | 345 |  |  |  |  |  |  |  |  | 346 | 345 |  | 365 | 365 | 345 |  |  |  |  |  |  |  | 365 | 365 |
ANKR | 20210524 |  |  |  |  |  |  |  |  |  |  |  |  |  | 222 | 222 | 222 |  |  |  |  |  |  |  | 222 |  |
ANT | 20201124 | 38 | 38 |  |  |  |  |  |  |  |  | 38 | 38 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
ATOM | 20191230 | 368 | 368 |  |  |  |  |  |  |  |  | 368 | 368 |  | 365 | 365 | 345 |  | 345 |  |  |  |  |  | 365 | 365 |
AUD | 20200616 |  | 199 |  |  |  |  |  | 199 |  |  |  |  |  |  | 365 |  |  |  |  |  | 363 |  |  |  |  |
AVAX | 20211221 |  |  |  |  |  |  |  |  |  |  |  |  |  | 11 | 11 |  |  |  |  |  |  |  |  |  |  |
AXS | 20210713 |  |  |  |  |  |  |  |  |  |  |  |  |  | 172 | 172 |  |  |  |  |  |  |  |  |  |  |
BADGER | 20210803 |  |  |  |  |  |  |  |  |  |  |  |  |  | 151 | 151 |  |  |  |  |  |  |  |  |  |  |
BAL | 20200917 | 106 | 106 |  |  |  |  |  |  |  |  | 106 | 106 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
BAND | 20210810 |  |  |  |  |  |  |  |  |  |  |  |  |  | 144 | 144 |  |  |  |  |  |  |  |  |  |  |
BAT | 20191230 | 368 | 368 |  |  |  |  |  |  |  |  | 368 | 368 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
BCH | 20200616 | 368 | 368 | 246 |  | 199 |  |  | 71 | 246 |  | 368 | 246 |  | 365 | 365 | 365 |  | 365 |  |  | 365 | 365 |  | 365 | 365 |
BNC | 20211026 |  |  |  |  |  |  |  |  |  |  |  |  |  | 67 | 67 |  |  |  |  |  |  |  |  |  |  |
BNT | 20210524 |  |  |  |  |  |  |  |  |  |  |  |  |  | 222 | 222 | 222 |  |  |  |  |  |  |  | 222 |  |
CHZ | 20210713 |  |  |  |  |  |  |  |  |  |  |  |  |  | 172 | 172 |  |  |  |  |  |  |  |  |  |  |
COMP | 20200715 | 170 | 170 |  |  |  |  |  |  |  |  | 170 | 170 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
CQT | 20210706 |  |  |  |  |  |  |  |  |  |  |  |  |  | 179 | 179 |  |  |  |  |  |  |  |  |  |  |
CRV | 20200917 | 106 | 106 |  |  |  |  |  |  |  |  | 106 | 106 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
CTSI | 20210803 |  |  |  |  |  |  |  |  |  |  |  |  |  | 151 | 151 |  |  |  |  |  |  |  |  |  |  |
DAI | 20191230 | 368 | 368 |  |  |  |  |  |  | 368 |  |  |  |  | 365 | 365 |  |  |  |  |  |  | 365 |  |  |  |
DASH | 20191230 | 368 | 368 |  |  |  |  |  |  |  |  | 368 |  |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 |  |
DOT | 20201015 | 136 | 136 |  |  |  |  |  |  | 78 |  | 136 | 136 |  | 365 | 365 | 345 |  | 345 |  |  |  | 365 |  | 365 | 365 |
DYDX | 20210914 |  |  |  |  |  |  |  |  |  |  |  |  |  | 109 | 109 |  |  |  |  |  |  |  |  |  |  |
ENJ | 20210520 |  |  |  |  |  |  |  |  |  |  |  |  |  | 226 | 226 | 225 |  |  |  |  |  |  |  | 226 |  |
EOS | 20201015 | 368 | 368 |  |  |  |  |  |  | 78 |  | 368 | 368 |  | 365 | 365 |  |  |  |  |  |  | 365 |  | 365 | 365 |
ETC | 20191230 | 368 | 368 |  |  |  |  |  |  |  |  | 368 | 368 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
ETH | 20191230 | 368 | 368 | 368 | 368 | 198 | 368 | 368 | 368 | 368 | 359 | 368 |  |  | 365 | 365 | 365 | 365 | 365 | 365 | 365 | 365 | 365 | 365 | 365 |  |
ETH2.S | 20211001 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 92 |
EUR | 20200617 |  | 295 | 295 | 295 | 198 | 295 |  | 295 |  |  |  |  |  |  | 365 | 365 | 365 | 365 | 365 |  | 365 |  |  |  |  |
EWT | 20210401 |  |  |  |  |  |  |  |  |  |  |  |  |  | 275 | 275 | 275 |  |  |  |  |  |  |  | 275 |  |
FIL | 20201015 | 78 | 78 |  |  |  |  |  |  |  |  | 78 | 78 |  | 365 | 365 | 345 |  | 345 |  |  |  |  |  | 365 | 365 |
FLOW | 20210127 |  |  |  |  |  |  |  |  |  |  |  |  |  | 339 | 339 | 339 |  |  |  |  |  |  |  | 339 | 339 |
GBP | 20200312 |  | 295 |  |  |  |  |  |  |  |  |  |  |  |  | 365 |  |  |  |  |  |  |  |  |  |  |
GHST | 20210517 |  |  |  |  |  |  |  |  |  |  |  |  |  | 229 | 229 | 229 |  |  |  |  |  |  |  | 229 |  |
GNO | 20191230 | 368 | 368 |  |  |  |  |  |  |  |  | 368 | 368 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
GRT | 20201218 | 14 | 14 |  |  |  |  |  |  |  |  | 14 | 14 |  | 365 | 365 | 345 |  | 345 |  |  |  |  |  | 365 | 365 |
ICX | 20191230 | 368 | 368 |  |  |  |  |  |  |  |  | 368 | 368 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
INJ | 20210810 |  |  |  |  |  |  |  |  |  |  |  |  |  | 144 | 144 |  |  |  |  |  |  |  |  |  |  |
KAR | 20210720 |  |  |  |  |  |  |  |  |  |  |  |  |  | 165 | 165 |  |  |  |  |  |  |  |  |  |  |
KAVA | 20200715 | 170 | 170 |  |  |  |  |  |  |  |  | 170 | 170 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
KEEP | 20201124 | 38 | 38 |  |  |  |  |  |  |  |  | 38 | 38 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
KILT | 20211130 |  |  |  |  |  |  |  |  |  |  |  |  |  | 32 | 32 |  |  |  |  |  |  |  |  |  |  |
KNC | 20200715 | 170 | 170 |  |  |  |  |  |  |  |  | 170 | 170 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
KSM | 20210121 | 106 | 106 |  |  |  |  |  |  |  |  | 106 | 106 |  | 365 | 365 | 345 |  | 344 |  |  |  |  |  | 365 | 365 | 267
LINK | 20210121 | 368 | 368 |  |  |  |  |  |  | 78 |  | 368 | 368 |  | 365 | 365 | 345 |  | 345 |  |  |  | 365 |  | 365 | 365 |
LPT | 20210520 |  |  |  |  |  |  |  |  |  |  |  |  |  | 226 | 226 | 226 |  |  |  |  |  |  |  | 226 |  |
LRC | 20210803 |  |  |  |  |  |  |  |  |  |  |  |  |  | 151 | 151 |  |  |  |  |  |  |  |  |  |  |
LSK | 20191230 | 368 | 368 |  |  |  |  |  |  |  |  | 368 | 368 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
LTC | 20191230 | 368 | 368 | 245 |  | 198 |  |  | 71 | 246 |  | 368 | 246 |  | 365 | 365 | 365 |  | 365 |  |  | 365 | 365 |  | 365 | 365 |
LUNA | 20211216 |  |  |  |  |  |  |  |  |  |  |  |  |  | 16 | 16 |  |  |  |  |  |  |  |  |  |  |
MANA | 20201215 | 17 | 17 |  |  |  |  |  |  |  |  | 17 | 17 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
MATIC | 20210517 |  |  |  |  |  |  |  |  |  |  |  |  |  | 229 | 229 | 229 |  |  |  |  |  |  |  | 229 |  |
MINA | 20210601 |  |  |  |  |  |  |  |  |  |  |  |  |  | 214 | 214 | 214 |  |  |  |  |  |  |  | 214 |  |
MIR | 20210810 |  |  |  |  |  |  |  |  |  |  |  |  |  | 144 | 144 |  |  |  |  |  |  |  |  |  |  |
MKR | 20210514 |  |  |  |  |  |  |  |  |  |  |  |  |  | 232 | 232 | 232 |  |  |  |  |  |  |  | 232 |  |
MLN | 20191230 | 368 | 368 |  |  |  |  |  |  |  |  | 368 | 368 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
MOVR | 20210827 |  |  |  |  |  |  |  |  |  |  |  |  |  | 127 | 127 |  |  |  |  |  |  |  |  |  |  |
NANO | 20191230 | 368 | 368 |  |  |  |  |  |  |  |  | 368 | 368 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
OCEAN | 20210401 |  |  |  |  |  |  |  |  |  |  |  |  |  | 275 | 275 | 275 |  |  |  |  |  |  |  | 275 |  |
OGN | 20210713 |  |  |  |  |  |  |  |  |  |  |  |  |  | 172 | 172 |  |  |  |  |  |  |  |  |  |  |
OMG | 20191230 | 368 | 368 |  |  |  |  |  |  |  |  | 368 | 368 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
OXT | 20200602 | 213 | 213 |  |  |  |  |  |  |  |  | 213 | 213 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
OXY | 20210928 |  |  |  |  |  |  |  |  |  |  |  |  |  | 95 | 95 |  |  |  |  |  |  |  |  |  |  |
PAXG | 20191230 | 368 | 368 |  |  |  |  |  |  |  |  | 368 | 368 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
PERP | 20210713 |  |  |  |  |  |  |  |  |  |  |  |  |  | 172 | 172 |  |  |  |  |  |  |  |  |  |  |
PHA | 20211008 |  |  |  |  |  |  |  |  |  |  |  |  |  | 85 | 85 |  |  |  |  |  |  |  |  |  |  |
QTUM | 20191230 | 368 | 368 |  |  |  |  |  |  |  |  | 368 | 368 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
RARI | 20210517 |  |  |  |  |  |  |  |  |  |  |  |  |  | 229 | 229 | 229 |  |  |  |  |  |  |  | 229 |  |
RAY | 20210928 |  |  |  |  |  |  |  |  |  |  |  |  |  | 95 | 95 |  |  |  |  |  |  |  |  |  |  |
REN | 20210514 |  |  |  |  |  |  |  |  |  |  |  |  |  | 232 | 232 | 232 |  |  |  |  |  |  |  | 232 |  |
REP | 20191230 | 368 | 368 |  |  |  |  |  |  |  |  | 368 | 368 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
REPV2 | 20200804 | 150 | 150 |  |  |  |  |  |  |  |  | 150 | 150 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
SAND | 20210520 |  |  |  |  |  |  |  |  |  |  |  |  |  | 226 | 226 | 226 |  |  |  |  |  |  |  | 226 |  |
SC | 20191230 | 368 | 368 |  |  |  |  |  |  |  |  | 368 | 368 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
SDN | 20210902 |  |  |  |  |  |  |  |  |  |  |  |  |  | 121 | 121 |  |  |  |  |  |  |  |  |  |  |
SHIB | 20211130 |  |  |  |  |  |  |  |  |  |  |  |  |  | 32 | 32 |  |  |  |  |  |  |  |  |  |  |
SNX | 20200917 | 106 | 106 |  |  |  |  |  |  |  |  | 106 | 106 |  | 365 | 365 | 345 |  | 345 |  |  |  |  |  | 365 | 365 |
SOL | 20210617 |  |  |  |  |  |  |  |  |  |  |  |  |  | 198 | 198 | 198 |  |  |  |  |  |  |  | 198 |  |
SRM | 20210617 |  |  |  |  |  |  |  |  |  |  |  |  |  | 198 | 198 | 198 |  |  |  |  |  |  |  | 198 |  |
STORJ | 20200715 | 170 | 170 |  |  |  |  |  |  |  |  | 170 | 170 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
SUSHI | 20210524 |  |  |  |  |  |  |  |  |  |  |  |  |  | 222 | 222 | 222 |  |  |  |  |  |  |  | 222 |  |
TBTC | 20201125 | 38 | 37 |  |  |  |  |  |  |  |  | 37 | 38 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
TRX | 20200305 | 302 | 302 |  |  |  |  |  |  |  |  | 302 | 302 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
UNI | 20201015 | 78 | 78 |  |  |  |  |  |  |  |  | 78 | 78 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
USD | 20200312 |  |  |  | 295 |  | 295 |  | 295 |  |  |  |  |  |  |  |  | 365 |  | 365 |  | 365 |  |  |  |  |
USDC | 20200108 | 359 | 359 |  |  |  |  |  |  | 359 |  |  |  |  | 365 | 365 | 345 |  | 345 |  |  |  | 365 |  |  |  |
USDT | 20191230 | 368 | 368 | 368 | 368 | 198 | 245 |  | 244 |  |  |  |  |  | 365 | 365 | 365 | 365 | 365 | 365 |  | 365 |  |  |  |  |
WAVES | 20191230 | 368 | 368 |  |  |  |  |  |  |  |  | 368 | 368 |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 | 365 |
WBTC | 20210825 |  |  |  |  |  |  |  |  |  |  |  |  |  | 151 | 151 |  |  |  |  |  |  |  |  | 129 |  |
XBT | 20191230 | 368 | 368 | 368 | 368 | 198 | 368 | 368 | 368 | 368 | 359 |  |  |  | 365 | 365 | 365 | 365 | 365 | 365 | 365 | 365 | 365 | 365 |  |  |
XDG | 20191230 | 368 | 368 |  |  |  |  |  |  |  |  | 368 |  |  | 365 | 365 |  |  |  |  |  |  | 196 |  | 365 |  |
XLM | 20191230 | 368 | 368 |  |  |  |  |  |  |  |  | 368 |  |  | 365 | 365 | 345 |  | 345 |  |  |  |  |  | 365 |  |
XMR | 20191230 | 368 | 368 |  |  |  |  |  |  |  |  | 368 |  |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 |  |
XRP | 20191230 | 368 | 368 | 246 | 368 | 198 |  |  | 368 | 246 |  | 368 | 246 |  | 365 | 365 | 365 | 365 | 365 |  |  | 365 | 365 |  | 365 | 365 |
XTZ | 20191230 | 368 | 368 |  |  |  |  |  |  |  |  | 368 | 368 |  | 365 | 365 | 345 |  | 345 |  |  |  |  |  | 365 | 365 |
YFI | 20201015 | 78 | 78 |  |  |  |  |  |  |  |  | 78 | 78 |  | 365 | 365 | 345 |  | 345 |  |  |  |  |  | 365 | 365 |
ZEC | 20191230 | 368 | 368 |  |  |  |  |  |  |  |  | 368 |  |  | 365 | 365 |  |  |  |  |  |  |  |  | 365 |  |
ZRX | 20210514 |  |  |  |  |  |  |  |  |  |  |  |  |  | 232 | 232 | 232 |  |  |  |  |  |  |  | 232 |  |

Table 4: Data coverage on 2020 (starting on Monday 30/12/2019) and 2021. Rows
are cryptocoins, columns correspond to fiat coin used to trade. The crypto
column reports the Ticker used to list the cryptocoin on the Kraken Exchange.
Firstday column show when the cryptopair appear for the first time (on the
2020 or 2021) using the format yyymmdd. For each fiat, are shown the number of
days available for the crypto-fiat pair in 2020 and in 2021.

##  Appendix B Results for other fiat

For completeness, in Figure 9 we show the time correlation matrices obtained
by using also the other fiat currencies of the dataset. Concerning the
stability of the network, we see that results for EUR 9(b) is quite similar to
USD 9(a). On the other hand, in Figures 9(c) and 9(d) it can be seen that the
matrix of correlation for XBT and ETH has low and often non-significant
values, indicating that the network is quite unstable.

|
---|---
|

(a) USD networks, 99 nodes  (b) EUR networks, 96 nodes

(c) XBT networks, 68 nodes  (d) ETH networks, 44 nodes

Figure 9:  The Pearson correlation between adjacency matrices at different
time windows. The element $c_{ij}=\rho(A^{i},A^{j})$ of each of these matrices
is the Pearson correlation between the vectorized adjacency matrices at
windows $i$ and $j$. High values of $c_{ij}$ indicate that the structure of
the network at windows $i$ and $j$ is very similar. 9(a) and 9(b) are the
correlation between cryptos networks traded using respectively USD and EUR. As
it is possible to see they are quite similar, except for some window where
correlation is discontinued on USD. Panels 9(c) and 9(d) show the correlation
between networks using XBT and ETH as fiat. Despite the number of nodes being
similar to USD and EUR networks, the correlation profile is quite low compared
to them. Correlations that are not significant ($p>0.01$), are indicated in
gray.

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