# Persistent Homology of Fractional Gaussian Noise

H. Masoomy  Department of Physics, Shahid Beheshti University, 1983969411,
Tehran, Iran  B. Askari  Department of Physics, Shahid Beheshti University,
1983969411, Tehran, Iran  M. N. Najafi<EMAIL_ADDRESS>Department of
Physics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran  S. M.
S. Movahed<EMAIL_ADDRESS>Department of Physics, Shahid Beheshti
University, 1983969411, Tehran, Iran

###### Abstract

In this paper, we employ the persistent homology (PH) technique to examine the
topological properties of fractional Gaussian noise (fGn). We develop the
weighted natural visibility graph algorithm, and the associated simplicial
complexes through the filtration process are quantified by PH. The evolution
of the homology group dimension represented by Betti numbers demonstrates a
strong dependency on the Hurst exponent ($H$). The coefficients of the birth
and death curve of the $k$-dimensional topological holes ($k$-holes) at a
given threshold depend on $H$ which is almost not affected by finite sample
size. We show that the distribution function of a lifetime for $k$-holes
decays exponentially and the corresponding slope is an increasing function
versus $H$, and more interestingly, the sample size effect completely
disappears in this quantity. The persistence entropy logarithmically grows
with the size of the visibility graph of a system with almost $H$-dependent
prefactors. On the contrary, the local statistical features are not able to
determine the corresponding Hurst exponent of fGn data, while the moments of
eigenvalue distribution ($M_{n}$) for $n\geq 1$ reveal a dependency on $H$,
containing the sample size effect. Finally, the PH shows the correlated
behavior of electroencephalography for both healthy and schizophrenic samples.

Topological Data Analysis, Persistent Homology, Fractional Gaussian Noise,
Weighted Natural Visibility Graph, Topological Persistence, Persistence
Entropy

##  I Introduction

A powerful approach to study different types of data sets ranging from point
cloud data and scalar fields to a complex network (graph), particularly high-
dimensional data, is called topological data analysis (TDA) Edelsbrunner  _et
al._ (2001); Zomorodian and Carlsson (2005); Zomorodian (2005); Carlsson
(2009); Zomorodian (2012); Wasserman (2018). TDA as an application of
algebraic topology Nakahara (2003); Munkres (2018); Hatcher (2005) and a
branch of computational topology Edelsbrunner and Harer (2010); it analyzes
the shape of high-dimensional complex data in terms of global features (number
of connected components, loops, voids, etc.) of topological space underlying
the data set. In the persistent homology (PH) technique, as the main part of
TDA, the topological approximation of phase space of any type of data set
which is called the simplicial complex is assigned to the underlying data, and
then topological invariants are computed.

The PH aims to capture topological evolution of data set by varying scale
(parameter), and extracts topological invariants of the data set in each scale
summarizing them in different representations, e.g., persistence barcode (PB)
Ghrist (2008); Carlsson  _et al._ (2005), persistence diagram (PD) Patel
(2018), persistence landscape (PL) Bubenik and Dłotko (2017), persistence
image (PI) Adams  _et al._ (2017), persistence surface (PS), and
$\beta$-curve, which reveal topological information of the data set. Being
robust to noise, PH can clarify the essential features of the systems with
high internal degrees of freedom and is capable of classifying data sets
Chazal and Michel (2017); Munch (2017). The PH technique has attracted much
attention due to its vast applications in analyzing complex networks Serrano
_et al._ (2020); Saggar  _et al._ (2018); Horak  _et al._ (2009). Also it has
been used in various systems (see, e.g., Topaz  _et al._ (2015); Sizemore  _et
al._ (2019); Pranav  _et al._ (2017); Jaquette and Schweinhart (2020); Speidel
_et al._ (2018); Salnikov  _et al._ (2018); Bobrowski and Skraba (2020); Tran
_et al._ (2021); Olsthoorn  _et al._ (2020) and references therein).

A mathematical model containing correlation tuned by one parameter (the Hurst
exponent Hurst (1951)) is the fractional Brownian motion (fBm) and its
increment is known as fractional Gaussian noise (fGn) Mandelbrot and Van Ness
(1968). It is also used to model self-similar phenomena of various types
ranging from meteorology, engineering, econophysics, and astronomy to biology
(see, e.g., Eghdami  _et al._ (2018); Jiang  _et al._ (2019) and references
therein). To quantify the properties of a given self-similar data set whose
power spectrum behaves as a power-law in the frequency (wavelength) domain,
many methods have been proposed. A well-studied method is multi-fractal
detrended fluctuation analysis (MFDFA) Peng  _et al._ (1994, 1995);
Kantelhardt  _et al._ (2002). Taking into account the higher-order detrended
covariance, the multi-fractal detrended cross-correlation analysis (MFDXA) has
been introduced Zhou  _et al._ (2008). Despite many advantages brought by the
mentioned methods, the impact of more complicated trends and finite-size
effects have not been diminished completely in many previous approaches Hu
_et al._ (2001); Chen  _et al._ (2002); Kantelhardt  _et al._ (2001);
Nagarajan and Kavasseri (2005a). On the other hand, the finite size of time-
series affects the accurate estimation of the Hurst exponent by some of
previous methods Xu  _et al._ (2005).

Although different methods can be found in the literature to determine the
scaling exponent of fBm or fGn series, very little attention has been paid to
deal with the topological properties and the probable capability of TDA for
estimation of the Hurst exponent. In this regard, knowing a given time series
belongs to an fGn class, some relevant questions can be raised: (i) Does the
topological aspect of time series depend on the Hurst exponent? (ii) What are
the effects of sample size, trends, and irregularity of fGn signal on the PH
of topological motifs generated from the data set? (iii) How is the
multifractality expressed by PH? Motivated by the mentioned questions, in this
paper, we concentrate on the topological properties (homology group) of fGn.

A way to construct the higher dimensional manifold from a typical time-series
in order to evaluate the evolution of $k$-dimensional holes through the
filtration process is assigning a weighted graph to the underlying time-
series. There are various methods to assign a network to a typical time-
series, e.g., the proximity network, cycle network, visibility graph,
correlation network, recurrence network, and transition network (see, e.g.,
Zou  _et al._ (2019) and references therein). The idea of the visibility graph
(VG) is a complex network constructed by considering the visibility algorithm,
proposed by Lacasa et al as a novel way to analyze time-series in terms of
complex network language Lacasa  _et al._ (2008) proposed as an alternative
method to estimate $H$ for fBm and fGn series Lacasa  _et al._ (2009).

In this paper, we rely on statistical and topological properties (homology
group) of natural VGs (NVGs) associated with such signals. First, we propose a
weighted version of natural VGs (WNVGs) by defining a well-defined weight
function to quantify the quality of visibility between data points of the
signal for extracting nonsensitive (robust) features in the presence of
possible noises in the signal. Then, we apply the filtration process on the
weighted clique simplicial complex corresponding to WNVG by considering the
weight (visibility value) of links as a threshold parameter and higher-order
connections ($k$-cliques ; $k>2$) as building blocks of such higher-order
structure. Accordingly, we figure out the evolution of robust topological
features explaining the global structure of WNVG. Considering other techniques
for making networks from time-series and applying PH could provide interesting
results, but is time-consuming from computational points of view; the
possibility of distinguishing the non-stationary and stationary series, the
sensitivity of corresponding results to the value of the Hurst exponent, and
the robustness of results with sample size are some of our criteria for taking
VG in its weighted version.

Our research presented in this paper has the following advantages.

(1) Inspired by network science and a self-similar process characterized by a
scaling exponent called the ”Hurst exponent”, we implement the weighted
natural visibility graph (WNVG) to make a network from fractional Gaussian
noise (fGn). In this approach, the topological motifs survive; consequently,
their evolution concerning self-similar exponents can be examined. Our results
approve that such evolution is a robust feature for determining the Hurst
exponent.

(2) We will demonstrate that the statistical analysis of WNVGs such as
probability distribution functions of eigenvector, betweenness, and closeness
centralities are almost $H$-independent, and therefore, they can not recognize
the type of correlation of fGn series.

(3) We rely on TDA and implement the PH, which is a robust method to examine
the evolution of global properties during the filtration process. Almost all
relevant results are sensitive enough to the Hurst exponent of the underlying
data set and it is also possible to check whether the data are stationary or
not. More precisely, irrespective of either stationarity or non-stationarity
of underlying times series, to capture the homology generators, the
constructed network should be weighted. It is worth noting that the non-
stationary behavior may be produced due to the various types of trends
superimposed on the intrinsic fluctuation; therefore, in the presence of any
prior information regarding trends the pre-processing procedure including at
least one of the following algorithms must be employed to produce clean
series, e.g., the singular value decomposition algorithm (Nagarajan and
Kavasseri, 2005b), the adaptive detrending algorithm (Hu  _et al._ , 2009),
and empirical mode decomposition (Wu  _et al._ , 2007).

(4) Finally, we emphasize that the behavior of the local (statistical)
observables depend weakly on $H$, whereas the coefficients of global
(topological) observables are almost strongly $H$-dependent and even for a
part of measures, the size effect is almost diminished. We notify that TDA
provides a new type of measure to quantify the Hurst exponent, and therefore,
the scaling exponents of the correlation function, power spectrum, and fractal
dimension can be specified with reliable approaches.

The rest of paper is organized as follows: In the next section, our
methodology is introduced briefly. The construction of VGs for a time-series
and the key idea of topological network analysis are clarified in Sec. II. The
numerical results of synthetic fGn time series, which are covered in two
subsections, local statistical properties and topological properties, are
presented in Sec. III. The implementation of realistic data is described in
Sec. IV. We give some ideas for applying the proposed method to various
problems in Sec. V, and close the paper with a conclusion. More details on
synthetic data generation, statistical network analysis, algebraic topology,
and persistent homology are explained in appendices.

##  II Methodology

Figure 1: The schematic representation of making a network for a typical data
set using VG algorithm. (a) The HVG and (b) NVG of a synthetic fGn with
$H=0.5$ (white-noise). To make more sense, we took the sample size equal to
$N=32$.

In this research, we aim to analyze the complex network of the visibility
graph constructed from a fractional Gaussian noise (fGn) (see Appendix A for
more details of generating a synthetic fGn series), with an emphasis on the
topological aspects. Besides this, we also compute some conventional
statistical properties of the mentioned signal.

###  II.1 Visibility Graph Method

Among growing applicability of complex networks in many fields and
interdisciplinary branches in science Costa  _et al._ (2011), a technique has
been suggested which converts a time-series to a network, the so-called
visibility graph method Lacasa  _et al._ (2008). Generally, suppose
$\\{x\\}:\\{x(t_{i}),i=1,...,N\\}$ represents a real-valued time-series. One
can construct a network, the so-called visibility graph, denoted by
$G=(V,E,w)$, $V\equiv\\{v_{i}\\}_{i=1}^{N}$ is the node (vertex) set, and $E$
is link (edge) set and $w$ is a map from $E$ to real numbers. The VG is
defined by using the bijection as follows,

$f:~{}V\equiv\Bigr{\\{}v_{i}\Bigr{\\}}_{i=1}^{N}\leftrightarrow\quad
T\equiv\Bigr{(}t_{i}\Bigr{)}_{i=1}^{N}\quad;\quad f(v_{i})=t_{i}$ (1)

and the connections are constructed according to the visibility condition
between the nodes, i.e., the nodes $v_{i}$ and $v_{j}$ are connected if the
node $v_{j}$ is visible from the node $v_{i}$ and vice versa, and therefore
the resulting graph is undirected (for more details on the properties of VGs,
see Lacasa  _et al._ (2008)). In general, there are two ways to construct a
network (graph) from a time-series: the horizontal visibility graph (HVG)
Gonçalves  _et al._ (2016); Gao  _et al._ (2016); Xie and Zhou (2011) and the
natural visibility graph (NVG) Zheng  _et al._ (2021); Yang  _et al._ (2009);
Ahmadlou  _et al._ (2010); the former is more sparse than the latter case and
in this work we focus on the NVG. In Fig. 1, we show how an HVG [panel (a)]
and an NVG [panel (b)] for a synthetic fGn series can be constructed. In a
binary setup, the corresponding visibility graph chooses the range of the
weights from a binary set,
$w^{(B)}:E\rightarrow\mathbb{Z}_{2}\equiv\\{0,1\\}$; e.g., for a binary NVG
(BNVG) the weight function can be written according to following relation,

$\displaystyle w_{ij}^{(BN)}\equiv\left\\{\begin{array}[]{l
l}1\qquad\qquad\qquad\qquad;\qquad\Bigr{|}f(v_{i})-f(v_{j})\Bigr{|}=1\\\
\displaystyle\prod_{k=i+1}^{j-1}\Theta\Bigr{(}s_{ij}-s_{ik}\Bigr{)}~{};\qquad\Bigr{|}f(v_{i})-f(v_{j})\Bigr{|}>1\end{array}\right.$
(4)

where $\Theta$ is the step function, and
$s_{ij}\equiv\frac{x(f(v_{j}))-x(f(v_{i}))}{f(v_{j})-f(v_{i})}$. The argument
of the $\Theta$ function being positive guarantees that the node $v_{j}$ is
visible from the node $v_{i}$ and vice versa. Since the edge in a BNVG has the
weight $0$ or $1$, it is unsuitable for continuous filtering. To take into
account the quality of visibility between nodes, we suggest the weighted
version of the natural visibility graph (WNVG), by considering the weight
function as follows,

$w_{ij}^{(WN)}\equiv\left\\{\begin{array}[]{l
l}\Bigr{|}s_{ij}\Bigr{|}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad;\qquad\Bigr{|}f(v_{i})-f(v_{j})\Bigr{|}=1\\\
\Bigr{(}\displaystyle\prod_{k=i+1}^{j-1}\Theta\Bigr{(}s_{ij}-s_{ik}\Bigr{)}\Bigr{[}s_{ij}-s_{ik}\Bigr{]}\Bigr{)}^{1/(j-i-1)}\quad;\qquad\Bigr{|}f(v_{i})-f(v_{j})\Bigr{|}>1\\\
\end{array}\right.$ (5)

There are two factors inside the product. In the second branch of Eq. (5), one
is the step function just like the binary graph, and the other is the weight
which is proportional to ”how visible is the site $j$ from $i$ and vice
versa”; i.e., the more distinguishable the data points are in the original
time-series, the higher the corresponding weight is in the constructed
network. The term $\frac{1}{j-i-1}$ is necessary to make the weights
reasonable numbers for comparison reasons. In the absence of this exponent,
the more the distance between the nodes is, the higher the corresponding
weights are. For both statistical and topological analysis, we use this weight
function which admits continuous filtering.

Figure 2: Probability distribution function of local features of WNVGs
constructed from fGns for various Hurst exponents. Panel (a) is the $p(c^{E})$
while panel (b) shows the probability distribution of betweenness centrality
for $N=2^{10}$ in log-log scale. Panels (c) and (d) respectively illustrate
the $p(c^{C})$ and corresponding moments, $M_{n}$, for $n=2$ (solid line) and
$n=3$ (dashed line) versus $H$. Different symbols are taken for various sizes.

###  II.2 Topological Network Analysis

A conventional approach to quantify the properties of a typical network is
determining degree, the number of nodes straightly connected with the
underlying node by non-zero weight, and centrality measures such as
betweenness centrality, the number of shortest paths between any pair of nodes
going through a distinct node, closeness centrality, eigenvector centrality,
etc. (see Appendix B for more details). Besides the mentioned approach, other
formalisms have been proposed in which not only the pairwise connections
(links) are examined but also a systematic pipeline is considered to
incorporate higher-order connections ($k$-cliques), including the simplicial
complex and hypergraph Bianconi and Rahmede (2016); Kovalenko  _et al._
(2021); Young  _et al._ (2020).

To analyze such higher-order structures, one can generalize the well-known
statistical quantities based on higher-order connections. On the other hand,
for global analysis of these structures newly born topological tools are
proposed. By the term ”global”, we mean the essential features of the object
which are not affected by geometric transformations. The well-known
topological quantities describing global properties of the space (or any
object mapped to a topological space) are Betti numbers. The $k$th Betti
number of an object ($\beta_{k}$) is a topological invariant. The dimension of
the $k$-homology group of the topological space corresponding to the object
counts the number of $k$-dimensional topological holes ($k$-holes) of the
object. Intuitively, $\beta_{0}$ indicates the number of connected components,
$\beta_{1}$ is the number of topological (unshrinkable) loops, and $\beta_{2}$
counts the number of topological (unshrinkable) voids (see the Appendix C for
more details). According to the mentioned properties of the topological
measures and weight-based analysis of weighted complex networks, the applied
algebraic topology tool, known as the persistent homology (PH) technique, is
useful to study global structures of the weighted complex network in higher-
dimensions Edelsbrunner and Harer (2010); Rote and Vegter (2006); Zomorodian
(2009).

In the PH-based analysis of a weighted complex network, the weighted network
is mapped to a weighted clique simplicial complex as a higher-order structure
containing any $(k+1)$-clique of the network as the $k$-simplex. Therefore, PH
creates a nested sequence of the clique simplicial complex, so-called
filtration, by considering the weight as the parameter, such that any link
(1-simplex) presents in the complex at a distinct weight if the weight of the
link is less than the distinct weight. Accordingly, the $k$-holes of the
complex appear and disappear when the weight (threshold) varies. To summarize
the evolution of these $k$-dimensional topological features of the system, PH
assigns an ordered tuple, the so-called persistence pair (PP), to this global
descriptor to visualize the topological variation of the system in the birth-
death space. This kind of visualization for the evolution of $k$-holes is
called the persistence diagram (PD) for the $k$-homology group. PD for the
$k$-homology group includes hidden topological information of the $k$-holes in
the weighted complex. For example, one can compute the population of birth,
death, and lifetime of $k$-holes using the distribution of PPs in PD for the
$k$-homology group. Persistence entropy (PE) for the $k$-homology group is
another quantity defined by the Shannon entropy of topological persistence
(lifetime) of PPs of the $k$-homology group (see Appendix D for more details).

##  III Results

To evaluate the statistical and topological features for the constructed VG
from a series, at first, we focus on the synthetic fractional Gaussian noise
(fGn) as a self-similar series. There are some exact and approximate
algorithms for generating a self-similar time-series. The Hosking Hosking
(1984), Cholesky Dieker and Mandjes (2003), and Davies-Harte methods Davies
and Harte (1987) are the examples of exact approach to generate fBm and fGn
series. Throughout this paper, we implement the Davies-Harte method to
generate fGn series with different self-similar exponents and sizes. To reduce
any bias in the nominal Hurst exponent as much as possible, all relevant
results are determined by doing an ensemble average over $10^{4}$ realizations
generated by Davies-Harte method for each $H$. To ensure the correctness of
the Hurst exponent value associated with each simulated time-series, we have
computed again the value of $H$ by our code based on the detrended fluctuation
analysis (DFA) method Kantelhardt  _et al._ (2002) and compare the expected
and computed Hurst exponents. Our results show that both Hurst exponents are
in agreement with each other.

Now we turn to the statistical and topological properties of the VGs
constructed from the fGn time-series and investigate their behavior concerning
the Hurst exponent. The networks of sizes $N=2^{7},2^{8},2^{9},2^{10},2^{11}$,
and $2^{12}$ (for which a desktop with $128$ GB memory is capable of
performing matrix operations) are considered. The Python toolbox
”NetworkX”111https://networkx.org/ is employed for the matrix operations on
the graphs. In the topological analysis, we especially focus on the Betti-$0$
(represented by the $\beta_{0}$ defined as the number of connected components
of the network) and Betti-$1$ (represented by $\beta_{1}$ defined as the
number of loops) features, which are extracted by using the ”Dionysus” Python
package Dmitriy (sus2). The persistence statistics, containing the lifetime
(the interval between birth and death) of the topological features, and its
Shannon entropy are also analyzed.

Each exponent has been estimated by Bayesian statistics accordingly; the
$\\{{\mathcal{D}}\\}$ and $\\{\Upsilon\\}$ reveal the data and model free
parameters, respectively. The posterior function is defined by

${\mathcal{P}}(\Upsilon|{\mathcal{D}})=\frac{{\mathcal{L}}({\mathcal{D}}|\Upsilon){\mathcal{P}}(\Upsilon)}{\int{\mathcal{L}}({\mathcal{D}}|\Upsilon){\mathcal{P}}(\Upsilon)d\Upsilon}$
(6)

where $\mathcal{L}$ is the likelihood and $\mathcal{P}(\Upsilon)$ is the prior
probability function containing all information concerning model parameters.
Here we adopt the top-hat function for the prior function whose window’s size
depends on the expected range of the corresponding exponent. Taking into
account the central limit theorem, the functional form of likelihood becomes
multivariate Gaussian, i.e.,
${\mathcal{L}}({\mathcal{D}}|\Upsilon)\sim\exp(-\chi^{2}/2)$. The $\chi^{2}$
for determining the best-fit value for the scaling exponent reads as

$\chi^{2}(\Upsilon)\equiv\Delta^{{\dagger}}.{C}^{-1}.\Delta$ (7)

where $\Delta$ is a column vector whose elements are determined by difference
between computed value and theoretical form for each measure, and ${C}$ is the
corresponding covariance matrix. Finally, the best fit value of the considered
exponent is computed by maximizing the likelihood probability distribution and
the associated errorbar is given by

$68.3\%=\int_{-\sigma_{\Upsilon}}^{+\sigma_{\Upsilon}}{\mathcal{L}}(\mathcal{D}|\Upsilon)d\Upsilon$
(8)

Subsequently, we report the best value of the scaling exponent at a $1\sigma$
confidence interval as $\Upsilon_{-\sigma_{\Upsilon}}^{+\sigma_{\Upsilon}}$.

###  III.1 Local Statistical Properties

By local properties, we mean the properties which are node-dependent and are
not necessarily globally defined. It has been confirmed that the distribution
function of the node degree of VGs of the fBms and fGns is power-law
$p(k)\propto k^{-\gamma}$ with the exponent $\gamma(H)=3-2H$ and
$\gamma(H)=5-2H$ , respectively Lacasa  _et al._ (2008, 2009). In this
subsection, we perform our computation for the WNVG, introduced in this paper.

Figure 3: (a) The probability distribution function of eigenvalue for
$N=2^{10}$ when the vertical axis is plotted in log-scale. (b) The
corresponding moments, $M_{n}$, for $n=2$ (solid line) and $n=3$ (dashed line)
versus $H$. Different symbols are taken for various sizes.

The probability distribution function of eigenvector ($c^{E}$) and betweenness
($c^{B}$) centralities are indicated in panels (a) and (b) of Fig. 2 in log-
log scale, in terms of $H$ for WNVGs. The scaling behavior of mentioned
distributions has been checked by the Kolmogorov-Smirnov test (KS test) and
the results confirm that the power-law behavior of the computed $p(c^{E})$ and
$p(c^{B})$ is highly $H$\- and size-dependent [see panels (a) and (b) of Fig.
2]. The probability distribution of closeness centrality and corresponding
moments $M_{n}$ for $n=2$ and $n=3$ are illustrated in panels (c) and (d) of
Fig. 2, respectively. This part depicts that not only the Hurst dependency is
not confirmed but also the overall shape of probability distribution functions
of statistical measures depends on the size of the underlying fGn series.
Subsequently, such distributions can not discriminate between different fGn
series.

The full spectrum of the eigenvalues [Eq. (12)] is illustrated in panel (a) of
Fig. 3. We see that the impact of $H$ is changing the range of the spectrum,
and by increasing the Hurst exponent, the range of the spectrum for WNVGs
becomes tight. This phenomenon can be understood by recalling that
correlations (obtained by increasing $H$) smooth the underlying time-series,
causing the corresponding network to have more links at low weight. For more
smoothed time-series, the typical slopes for the associated WNVG become down,
leading to lower weights according to Eq. (5), and equivalently making a
shorter range for the distribution of $\lambda$s. Panel (b) of Fig. 3
indicates different moments for $p(\lambda)$. The solid and dashed lines
correspond to $M_{2}$ and $M_{3}$, respectively. The $n$th moment of this
distribution behaves as an $H$-dependent quantity. The higher the order of
moments, the stronger the dependency on sample size is.

###  III.2 Topological Properties

Figure 4: (a) The normalized $\beta_{0}$-curve in log-scale for clique
complexes of WNVGs associated with fGns of various Hurst exponent versus
threshold. (b) The $w_{0}$ as a function of $H$. (c) The normalized
$\beta_{0}^{(\rm death)}$ in log-scale as a function of $w^{2}$ (see the text)
for various Hurst exponents as a function of threshold. (d) The value of
$\alpha_{0}^{\rm(death)}$ as a function of Hurst exponent.

Figure 5: (a) The normalized $\beta_{1}$-curve for clique complexes of WNVGs
associated with fGns of various Hurst exponents versus threshold. (b) The
$w_{1}$ as a function of $H$ for different lengths of data set. (c) The
distribution of the persistence diagram in log-scale versus the birth-axis as
a function of threshold. (d) The corresponding coefficient of
$\beta_{1}^{\rm(birth)}$ in terms of threshold known as
$\alpha_{1}^{\rm(birth)}$ as a function of Hurst exponent. Panels (e) and (f)
are the same as panels (c) and (d) respectively just for dying 1-hole
statistics. In this plot for computing $\beta_{1}$, we took $N=2^{10}$.

Figure 6:  (a) The probability distribution of lifetime for 1-dimensional
holes. Different symbols correspond to various $H$. This diagram has been
obtained by doing an ensemble average. We set the vertical axis in log-scale.
(b) The $\alpha_{1}^{({\rm lifetime})}$ coefficient for different sizes
represented by different symbols.

For any given value of Hurst exponent, the associated BNVG of fGn is a
topological tree; therefore, the BNVGs are topologically equivalent to each
other. In other words, various BNVGs are homeomorphic for different $H$
according to the homeomorphism theorem. The looplessness of BNVGs indicates
that these networks do not contain some classes of network motifs
corresponding to topological loops, so-called topological motifs Xie and Zhou
(2011). To make more sense, suppose 4 successive data points as
$\\{x_{j}\\}_{j=i}^{i+3},(j=1,...,N-4)$ from a given time-series
$\\{x_{i}\\}_{i=1}^{N}$ to make a minimal topological loop. From a network
perspective, the necessary (not sufficient) condition to construct a
topological loop is that $w_{i,i+1}=w_{i,i+3}=w_{i+1,i+2}=w_{i+2,i+3}=1$ and
the sufficient condition is $w_{i,i+2}=w_{i+1,i+3}=0$. The weights here are
determined by Eq. (4). Geometrically, the necessary condition is equivalent to
the case where data points have upward concavity. The visibility condition
does not satisfy the sufficient condition, such that either $w_{i,i+2}=1$
(creating two 2-simplices $[v_{i},v_{i+1},v_{i+2}]$ and
$[v_{i},v_{i+2},v_{i+3}]$) or $w_{i+1,i+3}=1$ (making two 2-simplices
$[v_{i},v_{i+1},v_{i+3}]$ and $[v_{i},v_{i+2},v_{i+3}]$) or even
$w_{i,i+2}=w_{i+1,i+3}=1$ (constructing a 3-simplex
$[v_{i},v_{i+1},v_{i+2},v_{i+3}]$) killing the topological loop and leads the
binary network to be topologically tree. Therefore, the BNVG does not contain
a certain class of network motifs equivalent to topological holes, whereas, in
the WNVG case these local patterns can be captured as topological loops. Also
mentioned loops are extracted through the filtration of the corresponding
weighted simplicial complex. According to the PH language, the sufficient
condition can be treated as
$w_{i,i+1},w_{i+1,i+2},w_{i+2,i+3},w_{i,i+3}<w_{i,i+2},w_{i+1,i+3}$;
therefore, the associated topological motifs now remain. Subsequently, we only
focus on the topological aspects of WNVGs. Using this terminology, we can
analyze local behaviors of a generic fGn using the statistics of corresponding
patterns (topological motifs) in WNVGs which are presented as PPs in PD.

Now, we are going to evaluate topological measures for WNVG constructed from a
generic time series. Since in this paper, we are interested in examining the
capability of the PH technique in determining the Hurst exponent of a typical
self-similar process, therefore we take fGn which is an increment of fBm.
Classifying the stationary and non-stationary series is in principle possible
when the weighted network is constructed from time series in our approach.
Intuitively, by increasing the value of the Hurst exponent, the underlying
series becomes more smooth and the statistics of weights grow for lower values
of weight. For an fBm data set which is the profile set of fGn Taqqu  _et al._
(1995); Kantelhardt  _et al._ (2002), the maximum value of distribution
function of weights occurs at lower weight values. Subsequently, the PH of
WNVG which is represented by topological curves can recognize the stationary
and non-stationary series. The constructed networks according to the
visibility graph possess the footprint of Hurst exponent empirically
demonstrated in Lacasa  _et al._ (2009). The higher value of $H$ causes making
the denser network. Also, the number of $k$-simplices at lower value of
threshold increase in the weighted version of the visibility graph; on the
other hand, such simplices influence the evolution of $k$-holes in associated
simplicial complexes. Therefore, the coefficients of topological curves depend
on Hurst exponent. Hereafter, we consider fGn as our case study. Panel (a) of
Fig. 4 indicates the $\beta_{0}$ as a function of filtration parameter
(threshold) for clique complexes of WNVG produced for various time-series with
different values of Hurst exponent. To reduce the effect of sample size, we
divide the Betti numbers by the sample size and call these normalized Betti
numbers. By increasing the value of the Hurst exponent, the normalized
$\beta_{0}$ versus threshold decreases. Namely, the number of connected
components in the network decreases with increasing $H$ (the graphs become
steeper). Interestingly, the slope of normalized $\beta_{0}$ as a function of
$w$ is different for different $H$; consequently, the WNVG of fGn sets reaches
to the connected regime (path-connected), $\beta_{0}=1$, [in panel (a) of Fig.
4, we take $N=2^{10}$] by different rates and at different thresholds,
$w_{0}$. Panel (b) of Fig. 4 represents the value of $w_{0}$ as a function of
Hurst exponent for different network sizes. As depicted in the mentioned
panel, the value of $w_{0}$ behaves as a sample size-dependent quantity and by
increasing $H$, such dependency becomes negligible.

We define the $\beta_{k}^{\text{(birth)}}(w)$ and
$\beta_{k}^{\text{(death)}}(w)$ as the number of $k$-dimensional holes that
are born and die at a given threshold, $w$, respectively. It turns out that
$\beta_{0}^{\text{(birth)}}=0$ for $w>0$, since at $w=0$ the underlying
network has $N$ connected components; therefore, all connected components are
born at $w=0$. Panel (c) of Fig. 4 illustrates the
$\beta_{0}^{\text{(death)}}/N$ as a function of $w^{2}$. As shown in this
plot, one of the proper fitting functions to describe the normalized
$\beta_{0}^{\text{(death)}}$ is given by
$\beta_{0}^{\text{(death)}}(w)\sim\exp\left[-\alpha_{0}^{\text{(death)}}w^{2}\right]$
for $2\lesssim w^{2}\lesssim w^{2}_{\rm max}$, where the value of $w_{\rm
max}$ depends on the $H$ value. The $\alpha_{0}^{\rm(death)}$ depends on the
Hurst exponent as increasing function and it behaves as an almost independent
function on size of series [panel (d) of Fig. 4].

Panel (a) of Fig. 5 is devoted to $\beta_{1}/N$ for various Hurst exponents.
As we expect, for a trivial threshold, $w=0$, we have $N$ connected components
($\beta_{0}=N$) and therefore the number of loops is identically zero
($\beta_{1}=0$). By increasing the threshold, the higher value of the Hurst
exponent leads to a more rapidly increasing rate of $\beta_{1}$. On the other
hand, for a high enough value of the threshold, again the underlying data set
behaves like a topological tree without any topological loops. Therefore, the
normalized $\beta_{1}$ goes asymptotically to zero, and such descending is
more rapid for higher $H$. We also determine the lowest non-trivial threshold
for which there is no loop in the underlying network denoted by $w_{1}$, and
depict this threshold versus $H$ for different samples size in panel (b) of
Fig. 5. The $w_{1}$ is also a size-dependent quantity. Comparing the
$\beta_{0}/N$ and $\beta_{1}/N$ demonstrate that by increasing threshold
value, the WNVGs of the fGn series reach the loopless regime ($\beta_{1}=0$)
before appearing in the connected regime (for which $\beta_{0}=1$),
irrespective of the Hurst exponent, i.e., $w_{0}(H)>w_{1}(H)$. The quantities
$\beta_{1}^{\text{(birth)}}\propto\exp\left[-\alpha_{1}^{\text{(birth)}}w\right]$,
$\beta_{1}^{\text{(death)}}\propto\exp\left[-\alpha_{1}^{\text{(death)}}w\right]$
versus thresholds and corresponding coefficients are illustrated in panels
(c)-(f) of Fig. 5, respectively. The $\alpha_{1}^{\rm(birth)}$ and
$\alpha_{1}^{\rm(death)}$ are almost size-independent and they grow by
increasing $H$. The local upward concavity behavior of time series satisfying
the loop condition and associated topological motifs (1-holes) appears and
disappears earlier as the Hurst exponent of the time-series increases.

Another interesting property to assess is the probability distribution of
lifetime for 1-holes which is the difference between the death and birth
thresholds of a typical measure in a 1-homology class. Panel (a) of Fig. 6
shows the probability distribution of topological 1-dimensional hole lifetime,
$\ell_{1}$, for various synthetic data sets with different values for the
Hurst exponent when the vertical axis is plotted in log-scale. Our results
confirm that
$p(\ell_{1})\propto\exp\left[-\alpha_{1}^{\text{(lifetime)}}\ell_{1}\right]$.
The $H$-dependency of $\alpha_{1}^{\rm(lifetime)}$ for various systems sizes
is depicted in panel (b) of Fig. 6. This result confirms that
$\alpha_{1}^{\rm(lifetime)}$ can be considered as a robust measure for
determining the Hurst exponent of the fGn signal which is not affected by
sample size even compared to $\alpha_{0}^{\rm(death)}$,
$\alpha_{1}^{\rm(birth)}$, and $\alpha_{1}^{\rm(death)}$. The increasing
behavior of the lifetime coefficient $\alpha_{1}^{\rm(lifetime)}$ versus the
Hurst exponent also indicates that the anti-correlated signals ($H<0.5$)
contain more persistent topological motifs (1-holes). This behavior also
demonstrates a considerable difference between the amount of visibility
between the inner and the outer nodes which are corresponding to the local
upward concavity behaving data points in these signals.

Figure 7: The persistence diagram and persistence barcode (inset) of weighted
clique complex of WNVG corresponding to fGn with (a) $H=0.2$ (anticorrelated
noise), (b) $H=0.5$ (uncorrelated noise) and (c) $H=0.8$ (correlated noise).
The blue circle and orange triangle symbols are considered for the
0-dimensional and 1-dimensional homology generators, respectively.

Figure 7 indicates the persistence diagram (PD) and persistence barcode (PB)
for three types of fGn signals for (a) $H=0.2$ (anticorrelated), (b) $H=0.5$
(uncorrelated) and (c) $H=0.8$ (correlated). The open circle and triangle
symbols correspond, respectively to the $0$\- and $1$-homology groups in a
persistence diagram for WNVGs of fGn. Each symbol depicts a pair $(w_{\rm
birth},w_{\rm death})$ of a $k$-dimensional hole. As expected, all 0-holes are
born in $w_{\rm birth}=0$, and also always $w_{\rm death}>w_{\rm birth}$. In
the barcode representation (inset plot), the horizontal lines (blue lines for
$k=0$ and orange lines for $k=1$) start and end on the threshold values on
which $k$-dimensional holes are born and die, respectively.

The associated persistence entropies (PEs) defined by Eq. (30) are obtained
using the persistence diagram. Panels (a) and (b) of Fig. 8 depict the
${PE}_{0}$ and ${PE}_{1}$, as a function of sample size, respectively. We
compute persistence entropy for all available Hurst exponent values
represented by different symbols. Our results demonstrate that
${PE}_{k}=\mathcal{A}_{k}(H)\log_{10}N$ for $k=0,1$. The behavior of prefactor
$\mathcal{A}_{k}$ versus $H$ is represented in panel (c) of Fig. 8. The
$\mathcal{A}_{0}$ is almost an increasing function versus Hurst exponent,
while the $\mathcal{A}_{1}$ becomes constant.

Figure 8: (a) The persistence entropy for the 0-homology group, $PE_{0}$. (b)
The $PE_{1}$ for 1-hole as a function of sample size in the log-scale.
Different symbols correspond to various values of the Hurst exponent. (c) The
prefactor of persistence entropy as a function of $H$ computed by the ensemble
average.

##  IV Implementation on Realistic Data

To illustrate the applicability of PH to characterize the self-similar nature
of real data, in this section, we use the physiological data set studied in
Olejarczyk and Jernajczyk (2017). We implement the DFA algorithm to verify the
scaling behavior of their correlation functions. Then, we apply our approach
to the channel F8 of the standard 10–20 electroencephalography (EEG) montage
recorded from 14 patients (7 males: 27.9 ± 3.3 years, 7 females: 28.3 ± 4.1
years) with paranoid schizophrenia and 14 healthy controls (7 males: 26.8 ±
2.9, 7 females: 28.7 ± 3.4 years). The EEG data were recorded in all subjects
during an eyes-closed resting-state condition for 15 minutes with the sampling
frequency of 250 Hz. We construct the WNVG from the EEG signals and the
evolution of topological motifs through the filtration process is examined as
depicted in Fig. 9. To infer the statistical significance of the results, we
carry out the KS test and we obtain the exponential behavior of results
indicated in Fig. 9. Then, according to the likelihood approach, we estimate
the topological coefficients and corresponding errorbars. The value of
coefficients $\alpha_{1}^{(\rm birth)}$, $\alpha_{1}^{(\rm death)}$,
$\alpha_{1}^{(\rm lifetime)}$ and associated Hurst exponents for both healthy
and schizophrenic cases are reported in Table 1. Comparing the topological
coefficients with our results for estimation of the Hurst exponent (Figs. 5
and 6), enables us to compute the associated Hurst exponent. The value of the
Hurst exponent for each coefficient is reported in Table 1. Averaging on three
kinds of Hurst exponents for EEG signals leads to $H=0.86\pm 0.02$ and
$H=0.69\pm 0.03$ at the $1\sigma$ confidence interval for healthy and
schizophrenic cases, respectively. Our results also reveal a correlated
behavior in both data sets. It is worth noting that, the PH can classify
different types of data.

Measure | Healthy | Schizophrenic
---|---|---
$\alpha_{1}^{(\rm birth)}$ | $4.85\pm 0.04$ | $3.52\pm 0.05$
$H_{(\rm birth)}$ | $0.84\pm 0.02$ | $0.66\pm 0.03$
$\alpha_{1}^{(\rm death)}$ | $3.96\pm 0.04$ | $2.83\pm 0.04$
$H_{(\rm death)}$ | $0.85\pm 0.02$ | $0.68\pm 0.03$
$\alpha_{1}^{(\rm lifetime)}$ | $7.03\pm 0.03$ | $5.00\pm 0.03$
$H_{(\rm lifetime)}$ | $0.89\pm 0.02$ | $0.73\pm 0.03$
Table 1: The value of topological coefficients and corresponding Hurst
exponents for healthy and schizophrenic cases at $1\sigma$ level of
confidence.

Figure 9: (a) The distribution of persistence diagram versus birth-axis
($\beta_{1}^{(\rm birth)}$) in log-scale for clique complexes of WNVGs
associated with EEG signals for healthy (blue circle symbols) and
schizophrenic cases (orange triangle symbols) as a function of threshold. (b)
The $\beta_{1}^{(\rm death)}$ in log-scale as a function of threshold. (c) The
lifetime distribution in log-scale for 1-homology generator statistics.

##  V Summary and Concluding Remarks

Applying the various tools developed in network science on a network
constructed from a typical time series reveals non-trivial information.
Particularly, some of the features of the network for a generic fractional
Gaussian noise (fGn), characterized by Hurst exponent $H$, exhibit a promising
relation to the corresponding Hurst exponent. In this paper, we developed a
method to generate a WNVG from the fGn (an increment of fBm) series which is a
more general network compared to a binary network class. According to the
filtration process, the evolution of topological holes in the associated WNVG
can be captured. To this end, we used the PH technique, to examine the
topological motifs.

The statistical properties of the WNVG were analyzed using the standard
network measures. Our main results are as follows:

(1) The probability distribution functions of eigenvector, betweenness and
closeness centralities computed for WNVGs constructed from fGn series [panel
(b) of Fig. 1] do not depend on $H$ and even the overall shape of mentioned
distributions is highly size-dependent. The main message is that such kinds of
statistical measures are not proper measures for determining the Hurst
exponent of the fGn series (Fig. 2).

(2) Increasing the amount of correlation results in obtaining more dense
networks and shifts the weights of the links to the lower values and,
consequently, the width of the distribution function of the eigenvalues to be
tighter. The $n$th moment of this distribution behaves as an $H$-dependent
quantity. The higher the order of moments, the stronger the dependency on
sample size (Fig. 3).

In the second part, we considered the topological properties of synthetic fGn.
Our main results are summarized below:

(3) The corresponding BNVG of a typical fGn is a topological tree which means
that all topological motifs have identically vanished while taking into
account a weight function to construct the so-called weighted NVG (WNVG) leads
to the survival of topological motifs and therefore their evolution with
respect to self-similar exponents can be examined. Such evolution reveals a
robust feature for determining the Hurst exponent reliably.

(4) The Betti-0 is a representative of the number of connected components of
the weighted clique simplicial complex associated with the WNVG with respect
to the threshold (weight). The decreasing rate of Betti-0 increases by
increasing $H$. The threshold value for which the underlying WNVG reaches the
connected regime (path-connected) indicates an $H$-dependency [panels (a) and
(b) of Fig. 4].

(5) Our result also confirms that the coefficient corresponding to the
exponentially decaying function of the number of connected components dying
(merging), $\alpha_{0}^{(\rm death)}$, as a function of threshold, depends on
the Hurst exponent and interestingly it is almost size-independent [panels (c)
and (d) of Fig. 4].

(6) The statistics of 1-holes (topological loops) show that the vanishing
threshold for the number of topological loops, $w_{1}$, is $H$-dependent and
it contains the sample size effect. The coefficients corresponding to the
exponentially decaying function of the number of topological loops appearing
($\alpha_{1}^{({\rm birth})}$) and disappearing ($\alpha_{1}^{({\rm death})}$)
as a function of threshold reveal the proper criteria for measuring the Hurst
exponent (Fig. 5). These coefficients also indicate that the topological
motifs (upward concavity behavior data points of time-series satisfying loop
condition) evolve (appear and disappear) in low weights for correlated
signals.

(7) The probability distribution of the lifetime of 1-holes ($\ell_{1}$)
confirms that the corresponding coefficient ($\alpha_{1}^{({\rm lifetime})}$)
is an increasing function versus $H$ emphasizing that the sample size effect
is completely diminished in this quantity. Consequently, for a self-similar
time-series in the absence of trends, this coefficient can be a reliable
measure for estimating the Hurst exponent even for a small sample size
irrespective of the value of the $H$ exponent (Fig. 6). This coefficient also
indicates that the fGn signals living in negatively correlated regimes
incorporate more robust topological motifs (1-holes) which shows a significant
difference between the quality of visibility of inner and outer nodes
corresponding to the local upward concavity behaving data points in these
signals.

(8) The persistence pairs (PPs) in the persistence diagram (PD) and the
persistence barcode (PB) for the weighted clique complex of WNVGs which are
indicators of persistent homology have been computed and with increasing the
value of the Hurst exponent shrink to the origin of the coordinate (Fig. 7).
We also computed persistence entropies (PEs) for 0-homology and 1-homology
groups. Both quantities depend on sample size as expected and the
corresponding slopes in semi-log scale were almost $H$-dependent (Fig. 8).
Implementing the PH method on real data and computing the topological
parameters, we could determine the corresponding Hurst exponent.
Interestingly, PH admitted its capability for the classification of various
data sets.

(9) Employing PH on the constructed weighted network from the realistic series
confirmed the correlated behavior of electroencephalography for both healthy
and schizophrenic samples.

Finally, we emphasize that the behavior of the local (statistical) observables
depends weakly on $H$, whereas the exponents of global (topological)
observables are almost strongly $H$-dependent and even for the
$\alpha_{1}^{({\rm lifetime})}$, the size effect is completely diminished. A
take-home message is that the TDA provides a new type of measure to quantify
the Hurst exponent and, therefore, the scaling exponents of the correlation
function, power spectrum, and fractal dimension can be specified with reliable
approaches.

In this paper, we have not verified whether the persistent homology technique
is capable of recognizing that the underlying time-series is a self-similar
set or not. Indeed, this purpose is beyond the scope of this paper. To address
the capability of our method to discriminate between stationary and non-
stationary times series, more simulations for different types of non-
stationary cases should be performed; this is left for our future study. Also,
it would be interesting to examine the effect of trends and irregularity which
may occur in a wide range of events in nature in the context of the TDA and
more precisely via the PH approach, and we leave this too for future research.
The above analysis can be done on different phenomena ranging from cosmology,
astrophysics, and economics to biology Akrami  _et al._ (2020); Eghdami  _et
al._ (2018); Vafaei Sadr  _et al._ (2018) with different approaches to make
graphs from time-series.

## Acknowledgment

The authors are very grateful to Marco Piangerelli for his notice on TDA as a
starting point in this research. In addition, we thank to the anonymous
referees, who helped us to improve the paper, considerably. The numerical
simulations were carried out on the computing cluster of the University of
Mohaghegh Ardabili.

## Appendices

### Appendix A: Synthetic Fractional Gaussian Noise

To model the stochastic fractal processes, Mandelbrot and Van Ness introduced
the fractional Brownian motion (fBm) and fractional Gaussian noise (fGn)
Mandelbrot and Van Ness (1968). The theory of fBm is a mathematical
generalization of the classical random walk and Brownian motion Mandelbrot and
Van Ness (1968); Kahane and Kahane (1993). A 1-dimensional fBm is represented
by $B\equiv\\{B(t):t\geq 0\\}$, with power-law variance, for which
$Var(B(at))\triangleq Var(a^{H}B(t))=a^{2H}Var(B(t))$, where $H\in(0,1)$ is
called the Hurst exponent. For this random force, the Markov property and the
stationarity are violated (note that when we have domain Markov property,
stationarity, and continuity for a time series, then it should be proportional
to a 1-dimensional Brownian motion). A model for generating fBm (denoted by
$B_{H}$ to emphasize on its Hurst exponent $H$) is a generalization of the
Brownian motion which is non-stationary and non-Markovian Reed  _et al._
(1995), and is given by the Holmgren-Riemann-Liouville fractional integral,

$B_{H}(t)=\frac{1}{{\Gamma(H+\frac{1}{2})}}\int\limits_{0}^{t}{{{(t-s)}^{H-\frac{1}{2}}}}\text{d}B(s)$
(9)

where $\Gamma$ is the Gamma function, $\text{d}B(s)\equiv B(s+\text{d}s)-B(s)$
is the increment of 1-dimensional Brownian motion, and it has the following
covariance:

$\langle{{B_{H}}(t){B_{H}}(s)}\rangle=\frac{\sigma^{2}}{2}\left({{{\left|t\right|}^{2H}}+{{\left|s\right|}^{2H}}-{{\left|{t-s}\right|}^{2H}}}\right)$
(10)

where $\sigma^{2}\equiv\langle B(0)\rangle$ and also $\langle B_{H}\rangle=0$.
The increments, $x_{H}(t)\equiv\delta_{t}\left({B_{H}}(t+\delta
t)-{B_{H}}(t)\right)$, are known as fractional Gaussian noise (fGn). The power
spectrum of fBm and fGn behaves as $S(f)\sim f^{-\xi}$, where $\xi(H)=2H+1$
and $\xi(H)=2H-1$ for fBm and fGn, respectively. For $H>\frac{1}{2}$
($H<\frac{1}{2}$) the corresponding fGn is positively (negatively) correlated.
According to the results provided by detrended fluctuation analysis (DFA), the
relation between the scaling exponent derived by the DFA method and associated
Hurst exponent of fBm is $H+1$, while constructing the fGn series by making
the increment of fBm confirmed that the scaling exponent of fluctuation
functions computed by DFA is directly related to the Hurst exponent Taqqu  _et
al._ (1995); Kantelhardt  _et al._ (2002).

### Appendix B: Statistical Network Analysis

Suppose that a network (graph) is represented by $G=(V,E,w)$. Here,
$V\equiv\\{v_{i}\\}_{i=1}^{N}$ is a node (vertex) set, $E=V\times
V\equiv\Bigr{\\{}e_{ij}=(v_{i},v_{j})\Bigr{|}v_{i},v_{j}\in
V\Bigr{\\}}_{i,j=1}^{N}$ is a link (edge) set, and $w:E\rightarrow\mathbb{R}$
is a weight function (threshold). Subsequently, the degree of $i$th node
($v_{i}$) is the number of nodes straightly connected with the underlying node
by non-zero weight, and it is denoted by
$k_{i}\equiv\sum_{j}(1-\delta_{0,w_{ij}})$. The degree centrality
($c_{i}^{D}$) is defined by $\frac{k_{i}}{N-1}$, which is apparently related
directly to how important the underlying node is, since it is the number of
agents that have a connection with it. An important function concerning this
quantity is the degree distribution showing the probability distribution
function for degree of all nodes in the network:

$p(k)=\displaystyle\frac{1}{N}\displaystyle\sum_{i=1}^{N}\delta_{k,k_{i}}$
(11)

The eigenvector centrality ($c_{i}^{E}$) is also defined via the eigenvalue
equation

$\lambda c_{i}^{E}=\sum_{j=1}^{N}w_{ij}c_{j}^{E}$ (12)

where $\lambda$ is the eigenvalue. The maximum value of the $\lambda$
spectrum, i.e., $\lambda_{\rm max}$ plays the dominant role in the network
properties, the corresponding eigenvectors of which are denoted by $c_{i,{\rm
max}}^{E}$ revealing the importance of the nodes. Let us denote the shortest
distance between nodes $v_{i}$ and $v_{j}$ by $d_{ij}$ which is assumed to be
$N$ when there is no path connecting them (disconnected graphs). Then the
closeness centrality is defined by

$c_{i}^{C}=\frac{N-1}{\sum_{j=1}^{N}d_{ij}},$ (13)

and the betweenness centrality is as follows:

$c_{i}^{B}=\frac{2}{(N-1)(N-2)}{\sum_{j=1,j\neq i}^{N}}{\sum^{N}_{k=1,k\neq
i,j}}\frac{n_{jk}(i)}{n_{jk}},$ (14)

where $n_{jk}$ is the number of geodesics from $v_{j}$ to $v_{k}$, and
$n_{jk}(i)$ is the number of geodesics from $v_{j}$ to $v_{k}$ which pass
through node $v_{i}$.

### Appendix C: Algebraic Topology

Topology generally refers to the global features in contrast to the
geometrical invariants of underlying objects or sets. We have two spaces
represented by $X$ and $Y$, and they have the same local (geometric) features
if any relevant features are invariant under congruence, while the mentioned
spaces are topologically equivalent if the associated features are invariant
under homeomorphisms. In other words, they are homeomorphic. Homology theory
plays a crucial role in the mathematical description of the relevant building
block of a typical topological space and reveals the connectedness of
underlying space Edelsbrunner and Harer (2010); Rote and Vegter (2006);
Zomorodian (2009). Based on such properties, for the sake of clarity, we will
give a brief review on the building blocks of algebraic topology which are
useful to set up homology groups.

Simplex: A $k$-simplex ($\sigma_{k}$) is a convex-hull of any geometrically
independent subset, accordingly
$\sigma_{k}\equiv[v_{0},v_{1},...,v_{k}]\subseteq\mathbb{R}^{D}$. By this
definition, a 0-simplex is a point, a 1-simplex is a segment of a line, a
2-simplex is a filled triangle, a 3-simplex is a filled tetrahedron, and so on
(Fig. 10).

Figure 10: Low dimensional simplices : (a) 0-simplex (point), (b) 1-simplex
(line segment), (c) 2-simplex (filled triangle), and (d) 3-simplex (filled
tetrahedron).

Face: An $l$-simplex which is denoted by $\sigma_{l}$ is a subset of the
$k$-simplex ($\sigma_{l}\subseteq\sigma_{k}$) and it is called the $l$-face of
the $k$-simplex.

Simplicial complex: A simplicial complex ($\psi$) is a collection of simplices
such that: any $l$-face of any $k$-simplex of a typical complex ($0<l<k$), is
a member of the complex. In addition, the non-empty intersection of any two
simplices, $\sigma_{k}$, and $\sigma_{m}$, from the complex is an $l$-face of
both simplices. The dimension of a complex is the maximum dimension of all
simplices of the complex. According to the definition of the complex, one can
define its $k$-ordered subcollection of complexes $\psi$ as follows:

$\Sigma_{k}(\psi)\equiv\Bigr{\\{}\sigma\in\psi~{}\Bigr{|}~{}dim(\sigma)=k\Bigr{\\}}$
(15)

Chain: For a given simplicial complex, a $k$-chain ($k$-dimensional chain) is
a linear combination of $k$-simplices of $\psi$, defined by

$c_{k}\equiv\displaystyle\sum_{i=1}^{|\Sigma_{k}(\psi)|}a_{i}~{}\sigma_{k}^{(i)}\quad;\quad\sigma_{k}^{(i)}\in\Sigma_{k}(\psi)$
(16)

where $|\Sigma_{k}(\psi)|$ corresponds to the cardinality of the $k$-ordered
subcollection of complex and the coefficients, $a_{i}$s, belong to a field,
which is usually considered as $\mathbb{F}=\mathbb{Z}_{2}\equiv\\{0,1\\}$. The
collection of all possible $k$-chains in the simplicial complex is called the
$k$-chain group as

$C_{k}(\psi)\equiv\Bigr{\\{}c_{k}~{}\Bigr{|}~{}c_{k}=\displaystyle\sum_{i=1}^{|\Sigma_{k}(\psi)|}a_{i}\sigma_{k}^{(i)};\sigma_{k}^{(i)}\in\Sigma_{k}(\psi),a_{i}\in\mathbb{F}\Bigr{\\}}$
(17)

Boundary operator: For the simplices in any dimension, the boundary operator
$\partial_{k}$ is an operator mapping $\sigma_{k}$ to its boundary according
to

$\partial_{k}(\sigma_{k})\equiv\displaystyle\sum_{j=0}^{k}(-1)^{j}~{}[x_{0},x_{1},...,x_{j-1},x_{j+1},...,x_{k}]~{}~{}\subseteq~{}\sigma_{k}$
(18)

Boundary: A $k$-chain which is the boundary of a $(k+1)$-chain, is called the
$k$-boundary, denoted by $b_{k}$. The $k$-boundary group is the collection of
all $k$-boundaries in complex $\psi$,

$\displaystyle B_{k}(\psi)$ $\displaystyle\equiv$
$\displaystyle\Bigr{\\{}c_{k}\in C_{k}(\psi)\Bigr{|}\exists c_{k+1}\in
C_{k+1}(\psi);\partial_{k+1}(c_{k+1})=c_{k}\Bigr{\\}}$ (19)
$\displaystyle\equiv$
$\displaystyle\Bigr{\\{}b_{k}^{(i)}\Bigr{\\}}_{i}^{|B_{k}(\psi)|}\subseteq
C_{k}(\psi)$

Cycle: A $k$-chain that has no boundary, is called a $k$-cycle denoted by
$z_{k}$ as

$\partial_{k}(z_{k})=\oslash$ (20)

The $k$-cycle group is defined as the collection of all $k$-cycles in complex
$\psi$,

$\displaystyle Z_{k}(\psi)$ $\displaystyle\equiv$
$\displaystyle\Bigr{\\{}c_{k}\in
C_{k}(\psi)~{}\Bigr{|}~{}\partial_{k}(c_{k})=\oslash\Bigr{\\}}$ (21)
$\displaystyle\equiv$
$\displaystyle\Bigr{\\{}z_{k}^{(i)}\Bigr{\\}}_{i}^{|Z_{k}(\psi)|}\subseteq
C_{k}(\psi)$

Since ”boundaries have no boundary”, therefore, we have

$\partial_{k}(b_{k})=\partial_{k}(\partial_{k+1}(c_{k+1}))=\oslash$ (22)

hence

$B_{k}(\psi)\subseteq Z_{k}(\psi)\subseteq C_{k}(\psi)$ (23)

Homology group: The $k$-homology group is defined by the quotient group of the
$k$-cycles group by the $k$-boundary group,

$H_{k}(\psi)\equiv Z_{k}(\psi)/B_{k}(\psi).$ (24)

The $k$th Betti number of a simplicial complex, denoted by $\beta_{k}(\psi)$ ,
is a topological invariant which counts the number of $k$-homology classes
corresponding to the number of $k$-dimensional holes of complex $\psi$ (Fig.
11),

$\beta_{k}(\psi)\equiv dim(H_{k}(\psi))$ (25)

Clique (flag) simplicial complex of an unweighted (binary) network,
$G=(V,E,w\in\\{0,1\\}$), is a simplicial complex, denoted by $\psi^{(G)}$,
such that any $k$-simplex of each dimension in the complex corresponds to a
$(k+1)$-clique in the network and vice versa (Fig. 12).

Figure 11:  Betti numbers of some topological spaces: point, line, circle,
sphere, torus, 2-torus.

Figure 12: (a) Clique complex of (b) a typical network.

A binary network is a topological tree, if and only if, it is topologically
holeless in all dimensions. Namely, the associated clique simplicial complex
has the following property:

$\beta_{k}(\psi^{(G)})=min(\beta_{k})=\left\\{\begin{array}[]{l
l}1\qquad;\qquad k=0\\\ 0\qquad;\qquad k>0\\\ \end{array}\right.$ (26)

### Appendix D: Persistent Homology

In the context of topological data analysis (TDA), we are interested in
statistical analysis of the structure in data. Generally, in TDA-based
analysis, data of any type (point cloud data, scalar field, time-series,
network, etc.) when mapped to a weighted simplicial complex are worked out
topologically in terms of the parameters present inherently in the original
data, e.g. the weight of links in a weighted network, or the pairwise distance
between data points in point cloud data. More precisely, TDA maps parameter-
dependent data, $X(w)$, (where $w$ is a typical parameter, like the threshold
value for any weighted network) to a weighted simplicial complex,
$\psi^{X}(w)$. Such approach produces the chain group, the cycle group, the
boundary group, the homology group and particularly, the $k$th Betti number
Nakahara (2003); Edelsbrunner and Harer (2010).

Persistent homology (PH) as a powerful tool of TDA examines the creation
(birth) and destruction (death) of topological invariants associated with
homology classes during a mathematical process called filtration Edelsbrunner
and Harer (2010). Filtration, $\phi$, is a nested sequence of weighted complex
$\psi(w)$ in which any complex with a distinct weight is a subcollection of
any complex with higher weight,

$\phi(\psi(w))\equiv\left(\psi(w)~{}\Bigr{|}~{}\forall
w^{\prime}<w^{\prime\prime}:\psi(w^{\prime})\subseteq\psi(w^{\prime\prime})\right)_{w_{\rm
min}}^{w_{\rm max}}$ (27)

More precisely, the PH technique enumerates the $k$th Betti number of any
subcomplex in $\phi$ and assigns an ordered tuple $w^{(h_{k})}\equiv(w_{{\rm
birth}}^{(h_{k})},w_{{\rm death}}^{(h_{k})})$ to the existing $k$-dimensional
topological hole. Here $w_{{\rm birth}}^{(h_{k})}$ and $w_{{\rm
death}}^{(h_{k})}$ are the thresholds for which $h_{k}$ appears (birth) and
disappears (death), respectively. Since $w_{{\rm birth}}^{(h_{k})}<w_{{\rm
death}}^{(h_{k})}$, we can define the positive-value quantity
$\ell^{(h_{k})}\equiv w_{{\rm death}}^{(h_{k})}-w_{{\rm birth}}^{(h_{k})}$ as
persistency (lifetime) of a $k$-dimensional hole. The persistence barcode (PB)
and equivalently the persistence diagram (PD) are the famous representations
of PH. As an illustration, the $k$-dimensional persistence diagram of weighted
complex $\psi(w)$ is a multiset
$PD_{k}\Bigr{(}\phi(\psi(w))\Bigr{)}\equiv(\mathcal{M},\mathcal{N})$, where

$\mathcal{M}~{}\equiv~{}\Bigr{\\{}w^{(h_{k})}~{}\Bigr{|}~{}h_{k}\in
H_{k}(\psi(w))~{},~{}\psi(w)\in\phi\Bigr{\\}}$ (28)

and $\mathcal{N}:\mathcal{M}\rightarrow\mathbb{N}$ is the count function.
Inspired by Shannon entropy for a typical state probability, one can define
the persistence entropy (PE) of the $k$th PD (PB). To this end, we construct
the probability for lifetime of homology classes as

$\displaystyle
p(\ell^{(h_{k})})\equiv\frac{\ell^{(h_{k})}}{\mathcal{L}}\quad;\quad\mathcal{L}~{}\equiv\displaystyle\sum_{w^{(h_{k})}\in\mathcal{M}({PD}_{k})}\ell^{(h_{k})}$
(29)

Therefore, the PE for the $k$-dimensional topological hole is defined by
Merelli  _et al._ (2015); Atienza  _et al._ (2019)

$\displaystyle
PE_{k}=-\displaystyle\sum_{w^{(h_{k})}\in\mathcal{M}({PD}_{k})}p(\ell^{(h_{k})})~{}\log_{10}p(\ell^{(h_{k})})$
(30)

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October 1, 2022

# Embedding and correlation tensor for XRP transaction networks

Abhijit Chakraborty1,2,∗ Tetsuo Hatsuda2 and Yuichi Ikeda1 1 Kyoto University1
Kyoto University Graduate School of Advanced Integrated Studies in Human
Survivability Graduate School of Advanced Integrated Studies in Human
Survivability Kyoto Kyoto 606-8306 606-8306 Japan
2RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program Japan
2RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program Saitama
Saitama 351-0198 351-0198 Japan Japan<EMAIL_ADDRESS>
###### Abstract

Cryptoassets are growing rapidly worldwide. One of the large cap cryptoassets
is XRP. In this article, we focus on analyzing transaction data for the
2017–2018 period that consist one of the significant XRP market price bursts.
We construct weekly weighted directed networks of XRP transactions. These
weekly networks are embedded on continuous vector space using a network
embedding technique that encodes structural regularities present in the
network structure in terms of node vectors. Using a suitable time window we
calculate a correlation tensor. A double singular value decomposition of the
correlation tensor provides key insights about the system. The significance of
the correlation tensor is captured using a randomized correlation tensor. We
present a detailed dependence of correlation tensor on model parameters.

cryptoaasets, XRP, network embedding, correlation tensor, double singular
value decomposition

##  1 Introduction

Cryptoassets are digital assets, which use cryptography and depend on
distributed ledger technology, also known as blockchain. The cryptomarket
consists of many cryptoassets, namely, Bitcoin, Etherium, and XRP etc.
Recently, we have witnessed increase in investor attention on cryptoassets.
However, the risk for investing in cryptoassets is generally high due to the
high volatility of their market price. Moreover, as the growing cryptoassets
market can disrupt the financial market and the possibility of money
laundering, policymakers in different countries have already started enacting
regulations for its use. The cryptoassets have also attracted attention from
researchers of different fields to study the fluctuation in price, transaction
patterns of users, and efficiency, robustness of the underlying blockchain
technology. While there are many studies on Bitcoin and Etherium [1, 2, 3],
there are very few studies on XRP [4, 5]. In this work, we focus on analyzing
XRP transactions.

Time series analysis comprises different methods to characterize and extract
crucial insights for various time series data in the stock markets [6, 7, 8],
foreign exchange markets [9, 10] or even in medical recordings [11]. The cross
correlation is one of the popular methods to analyze time series data [6, 7,
8, 11]. The simplest way to measure correlation is the Pearson correlation,
which is defined for a pair of variables $(x,y)$ as
$r_{x,y}=\sum\limits_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y})/((n-1)\sigma_{x}\sigma_{y})$.
Here $n$ is the no of observations or the length of the time series. The mean
and standard deviations of $x$ and $y$ are represented as
$\overline{x},\overline{y}$, and $\sigma_{x},\sigma_{y}$. The applications of
cross correlation method with random matrix theory on time series data have
provided valuable insights for different systems.

Recently, inspired by the cross correlation analysis of time series, A.
Chakraborty et al. have developed a method of correlation tensor spectra for
dynamical XRP transaction networks [12] to capture price burst. Following this
article, we show the details of the embedding method for weekly directed
weighted XRP networks. Using the embedded node vectors, we calculate the
correlation tensors for XRP transaction netwokrs. The significance of the
elements of the correlation tensor and its spectra is shown by comparing with
randomized correlation tensor. Furthermore, We uncover the dependence of the
correlation tensor on various model parameters, such as the embedding
dimension, and the time window in detail.

##  2 Data description

Our data consist of the direct transactions between different XRP wallets from
October 2, 2017 to March 4, 2018. The dataset was recorded as ledger data
using the Ripple Transaction Protocol. We divided this dataset into $22$
groups based on different weeks. We constructed a weekly directed weighted
network of XRP transactions by aggregating all the transactions for a week. In
this network, XRP wallets are the nodes and a link represents the flow of XRP
from a source wallet to a destination wallet. The total amount of weekly XRP
flow between a pair of wallets represents the link weight. The XRP
transactions data were collected from the ripple data API at
https://xrpl.org/data-api.html.

##  3 Embedding of XRP transaction networks

Network embedding is a technique for representation of a network in low
dimensional vector space by preserving key network features, which is useful
for downstream tasks, such as network visualization, network analysis, link
predication etc. The DeepWalk [13] and node2vec [14] are the two well-known
methods for network embedding with structural information. We briefly describe
these two methods below.

###  3.1 DeepWalk

Following the natural language models [15], B. Perozzi et al. developed the
DeepWalk algorithm [13], which maps each node $v_{i}\in V$ of a network as a
vector in a $D$ dimensional continuous space $R^{{\mid V\mid}\times D}$ by
modeling a set of truncated random walks. The algorithm encodes the network
community structure into a vector representation of the nodes. Many short
random walk are used to compute local community structure information
efficiently. These walks correspond to short sentences in language modeling.
More specifically, it uses many random walks $R_{w}$ of length $l$ from each
node. For each node, the algorithm generates a random walk of length $l$ and
uses it to update the SkipGram algorithm in accordance with an objective
function. The objective function maximizes the co-occurrence probability with
other nodes present in the short random walk sequence. For a mapping function
$\phi:v\in V\to R^{{\mid V\mid}\times D}$, the objective function can be
written as

$\min_{\phi}-\log Pr\biggl{(}\\{v_{i-w},...,v_{i+w}\\}\backslash
v_{i}|\phi(v_{i})\biggr{)},$ (1)

where $w$ is the window size, $\\{v_{i-w},...,v_{i+w}\\}$ is the node set
generated by the random walks from $v_{i}$ and $\phi(v_{i})$ represents the
$v_{i}$ on the vector space. The algorithm uses the hierarchical softmax [16]
to speed up the optimization procedure.

###  3.2 node2vec

A. Grover et al. further generalized the Deepwalk method by modifying the
random walks as second order biased random walks. This method can encode more
complex regularities of the network, such as functional relationships into the
vector representation of nodes, depending on the tuning parameters $p$ and $q$
for the walks . It becomes equivalent to DeepWalk method when $p=q=1$.

Here we have used node2vec with $p=q=1$ to embed weighted directed weekly XRP
transactions networks.

##  4 Correlation tensor from embedded XRP transaction networks

Figure 1: The daily XRP/USD close price from May 05, 2017 to October 13, 2022.
The dotted blue lines indicate the period that we consider for our analysis.
This figure is adapted from [12].

We embed each of these $22$ weeekly weighted directed networks using node2vec
with $p=q=1$. It gives vector representations for each node
$V_{i}^{\alpha}(t)$, where $t=1,2,3,\cdots,22$; $\alpha=1,2,3,\cdots,D$;
$i=1,2,3,\cdots,N_{t}$. Here $D$ represents the dimension of the embedding
space and $N_{t}$ is the total number of nodes in the $t$-th week network. We
identify that there are $N=71$ nodes that carry out at least one transaction
each week. We call these $N=71$ nodes, regular nodes. The regular nodes play
key role in the XRP trading market as they carry out frequent and consistent
transactions. Considering the embedded vectors of regular nodes
$V_{i}^{\alpha}(t)$, where $i=1,2,3,\cdots,N$, we calculate the correlation
tensor as follows

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

where $\sum$ is taken over $(2\Delta T+1)$ weekly $\\{t-\Delta T,t-(\Delta
T-1),\cdots,t,\cdots,t+(\Delta T-1),t+\Delta T\\}$ networks. The
$\overline{V_{i}^{\alpha}}$ and $\sigma_{V_{i}^{\alpha}}$ represents mean and
standard deviation of $V_{i}^{\alpha}$ over a time window of $(2\Delta T+1)$
networks.

The correlation tensor has $(N\times N\times D\times D)$ elements. To uncover
the crucial insights from the correlation tensor, we use a double singular
value decomposition (SVD) technique as follows:

As a first step, we conduct the diagonalization of $M_{ij}^{\alpha\beta}$ in
terms of $(ij)$-index

$M_{ij}^{\alpha\beta}=\sum\limits_{k=1}^{N}L_{ik}\sigma_{k}^{\alpha\beta}R_{kj},$
(3)

and as a second step, we perform the digonalization in terms of
$(\alpha\beta)$-index successively

$\sigma_{k}^{\alpha\beta}=\sum\limits_{\gamma=1}^{D}\mathcal{L}^{\alpha\gamma}\rho_{k}^{\gamma}\mathcal{R}^{\gamma\beta}.$
(4)

Then, using Eq. 3 and Eq. 4 we have

$M_{ij}^{\alpha\beta}=\sum\limits_{k=1}^{N}\sum\limits_{\gamma=1}^{D}\rho_{k}^{\gamma}(L_{ik}R_{kj})(\mathcal{L}^{\alpha\gamma}\mathcal{R}^{\gamma\beta}).$
(5)

Here $\rho_{k}^{\gamma}$ represents the $N\times D$ generalized real and
positive singular values as the correlation tensor $M$ is real.

##  5 Results

Figure 2:  The temporal variation in the properties of directed weighted
weekly XRP networks. Temporal variation of (a) node numbers, (b) number of
links per node and (c) total transaction volume. This figure is adapted from
[12].  Figure 3: Distributions for three different components $(\alpha=1,2,32$
are indicated as legends$)$ of all regular nodes during the week, 2017,
October 02 to October 08.

We show the daily XRP/USD price between 2017, May 18 and 2022 October 13 in
Fig. 1. The period consists several peaks of XRP/USD price. The most
extraordinary rise and fall of XRP/USD price is observed around January 2018.
So we focus our study between October 2, 2017 and March 4, 2018, which covers
22 weeks.

We construct a weekly network of XRP transactions by aggregating all the
transactions among the wallets. Wallets are the node and a directed link is
formed between a source wallet and a destination wallet. The total flow of XRP
between two wallets represents the link-weight. We show the temporal variation
of number of nodes, links and value of total link-weight in Fig. 2. We observe
that the number of nodes in the weekly networks increases rapidly around
December 2017 and started decreasing again from the second week of January
2018, as shown in Fig. 2 (a). Fig. 2 (b) shows that the number of links per
node, also known as average in-degree or out-degree decreases from $2$ to
$1.2$ during October, 2017-March 04, 2018. The variation of the total
transaction volume which represents total link-weight in weekly networks, is
shown in Fig. 2 (c). We observe three sudden peaks in the transactions volume,
which may be associated with the XRP price bubble during January 2018.

The distributions for the components of the node vector $V_{i}^{\alpha}$ for
all regular nodes for three different values of $\alpha$ are shown in Fig. 3
for the week 2017, October 02 to October 08. The distributions have a peak
indicating that the nodes are not embedded randomly on the vector space but
capture the regularities of the network.

Figure 4: The auto-correlation function acfi for three different components
$(\alpha=1,2,32$ are indicated as legends$)$ of three nodes
$i={\rm(a)}~{}1,{\rm(b)}~{}35,{\rm and~{}(c)}~{}71$.  Figure 5: (a)
Distribution for the elements of correlation tensors $M(t)$ with different
time windows $2\Delta T+1=5,11$ and $21$, respectively. (b) Sorted singular
values for the corresponding correlation tensors. The correlation tensors are
calculated for the week, $t$ = 2017, October 16 - October 22, 2017, November
06 - November 12, and 2017, December 11 - December 17 respectively.  Figure 6:
Comparison between empirical correlation tensor and randomize correlation
tensor with time window $2\Delta T+1=21$. (a) Distribution for the elements of
correlation tensors. (b) Sorted singular values for the corresponding
correlation tensors. The correlation tensor is calculated for the week, $t$ =
2017, December 11 - December 17.

We calculate auto-correlation for each component of the node vectors
$V_{i}^{\alpha}(t)$. The auto-correlation is defined as the Pearson
correlation for a time series as a function of the time lag. The auto-
correlation function for the different components for different values of
$\alpha$, and $i$ is shown in Fig 4. It is evident that the components do not
have any significant auto correlation.

We show the distributions for the elements of correlation tensors with
different time windows $(2\Delta T+1)$ in Fig. 5 (a). It is observed that the
peak of the distribution becomes sharper as we increase the time window and
approaches a saturation form. We also notice that if we consider the time
window $(2\Delta T+1)<5$, the distribution appears with two more peaks at $\pm
1$ due to high noise. The correlation tensor contains more noise as we
decrease the values of the time window $(2\Delta T+1)$. The singular values
$\rho_{k}^{\gamma}$ of correlation tensors with different time windows
$(2\Delta T+1)$ is shown in Fig. 5 (b). We observe that the largest singular
value decreases as the time window increases.

To understand the significance of the correlation tensor, we need a reference
correlation tensor. We consider the following null hypothesis for the
reference correlation tensor. We assign uniform random numbers between
$[-1,1]$ to the components of the embedding node vectors. We then calculate
the randomized correlation tensor from these components of the embedding node
vectors. We compare the distributions for the elements of the empirical
correlation tensor and randomized correlation tensor in Fig. 6 (a). We observe
that the distribution for the randomized correlation tensor has a higher peak
and it is symmetric around zero. The distribution for the empirical
correlation tensor is asymmetric and has a positive mean.

We further compare the singular values $\rho_{k}^{\gamma}$ of the empirical
and randomized correlation tensors in Fig. 6 (b). We observe that the largest
singular value of the empirical correlation tensor lies significantly beyond
the largest singular value of the randomized correlation tensor.

We show the dependence of the correlation tensor on the dimension $D$ of
embedding vector space in Fig. 7. The distribution of the elements of the
correlation tensor becomes narrower as we increase the dimension, as shown in
Fig. 7 (a). The corresponding singular values $\rho_{k}^{\gamma}$ of the
correlation tensors are shown in Fig. 7 (b). The largest singular value
$\rho_{1}^{1}$ increases as we embed the weekly networks in a higher
dimension.

Figure 7:  Correlation tensor with time window $2\Delta T+1=21$ for embedding
dimension $D=16,$ and $32$, respectively (a) Distribution for the elements of
correlation tensors. (b) Sorted singular values for the corresponding
correlation tensors.

##  6 Conclusion

We have analyzed the XRP transactions focusing during the period between
October, 02, 2017 and March 04, 2018. During this period, a significant price
burst of XRP was observed. We have constructed weekly directed networks from
XRP transactions. We have embedded these weekly networks in vector space,
which capture the community structure information of the weekly networks in
the embedded node vectors. The correlation tensor and its spectra are
calculated using different time windows. It is observed that the the
correlation tensor is sensitive to the time window. Lower the values of time
window, it contains more noise. As we increase the time window, the
distribution of the elements of correlation tensor approaches to the fixed
form. The results were compared with randomized counterpart. It is observed
that the distribution for the elements of empirical correlation tensor differs
significantly. The largest singular value of the empirical correlation tensor
appears well above the largest singular value of the randomized correlation
tensor. Furthermore, we have presented the dependence of the correlation
tensor on model parameters in detail.

This method is very useful and can capture valuable insights about the
dynamical properties of the XRP transaction networks [12]. The embedding of
the nodes encodes the community structure in the node vectors. The correlation
is calculated between the components of the regular node vectors. Therefore,
the spectra of the correlation tensors can capture whether there is a change
in the community structure of the XRP transaction networks.

## Acknowledgements

We thank the members of the Kyoto Univ.- RIKEN blockchain study group for
discussions. YI acknowledges the grant, ”University Blockchain Research
Initiative”, provided by Ripple, Inc. to Kyoto University for partial support
to this work.

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