=Paper= {{Paper |id=Vol-2713/paper03 |storemode=property |title=Complex networks theory and precursors of financial crashes |pdfUrl=https://ceur-ws.org/Vol-2713/paper03.pdf |volume=Vol-2713 |authors=Vladimir Soloviev,Victoria Solovieva,Anna Tuliakova,Alexey Hostryk,Lukáš Pichl |dblpUrl=https://dblp.org/rec/conf/m3e2/SolovievSTHP20 }} ==Complex networks theory and precursors of financial crashes== https://ceur-ws.org/Vol-2713/paper03.pdf

Complex networks theory and precursors of financial
crashes

Vladimir Soloviev1,2[0000-0002-4945-202X], Victoria Solovieva3[0000-0002-8090-9569],
Anna Tuliakova4, Alexey Hostryk5[0000-0001-6143-6797] and Lukáš Pichl6
1 Kryvyi Rih State Pedagogical University, 54 Gagarin Ave., Kryvyi Rih, 50086, Ukraine

2 The Bohdan Khmelnytsky National University of Cherkasy,

81 Shevchenka Blvd., 18031, Cherkasy, Ukraine
3 State University of Economics and Technology,

16 Medychna Str., Kryvyi Rih, 50005, Ukraine
4 Odessa I. I. Mechnikov National University, 2 Dvoryanskaya Str., Odessa, 65082, Ukraine

5 Odessa National Economic University, 8 Preobrazhenskaya Str., Odessa, 65082,

Ukraine
6 International Christian University, 3-10-2 Osawa, Mitaka-shi, Tokyo, 181-8585, Japan

vnsoloviev2016@gmail.com, vikasolovieva2027@gmail.com,
tuliakovaanna@gmail.com, alexeyGostrik@gmail.com,
lukas@icu.ac.jp

Abstract. Based on the network paradigm of complexity in the work, a
systematic analysis of the dynamics of the largest stock markets in the world and
cryptocurrency market has been carried out. According to the algorithms of the
visibility graph and recurrence plot, the daily values of stock and crypto indices
are converted into a networks and multiplex networks, the spectral and
topological properties of which are sensitive to the critical and crisis phenomena
of the studied complex systems. This work is the first to investigate the network
properties of the crypto index CCI30 and the multiplex network of key
cryptocurrencies. It is shown that some of the spectral and topological
characteristics can serve as measures of the complexity of the stock and crypto
market, and their specific behaviour in the pre-crisis period is used as indicators-
precursors of critical phenomena.

Keywords: crypto index, visibility graph, complexity measures of financial
crashes.

1 Introduction

The new interdisciplinary study of complex systems, known as the complex networks
theory, laid the foundation for a new network paradigm of synergetic [17]. The complex
networks theory studies the characteristics of networks, taking into account not only
their topology, but also statistical properties, the distribution of weights of individual
nodes and edges, the effects of information dissemination, robustness, etc. [1; 6; 10;

___________________
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
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20; 21; 22; 31]. Complex networks include electrical, transport, information, social,
economic, biological, neural and other networks [1; 6; 10; 20; 18; 22; 24; 31]. The
network paradigm has become dominant in the study of complex systems since it allows
you to enter new quantitative measures of complexity not existing for the time series
[3].
Previously, we introduced various quantitative measures of complexity for
individual time series [25; 26; 27; 28; 29]. Significant advantage of the introduced
measures is their dynamism, that is, the ability to monitor the time of change in the
chosen measure and compare with the corresponding dynamics of the output time
series. This allowed us to compare the critical changes in the dynamics of the system,
which is described by the time series, with the characteristic changes of concrete
measures of complexity. It turned out that quantitative measures of complexity respond
to critical changes in the dynamics of a complex system, which allows them to be used
in the diagnostic process and prediction of future changes. In [25], we introduced
network complexity measures and adapted them to study system dynamics. But
networks are rarely isolated. Therefore, it is necessary to take into account the
interconnection interaction, which can be realized within the framework of different
models [4]. In this paper we will consider it by simulating so-called multiplex networks,
the features of which are reduced to a fixed number of nodes in each layer, but they are
linked by different bonds.

2 Review

Most complex systems inform their structural and dynamic nature by generating a
sequence of certain characteristics known as time series. In recent years, interesting
algorithms for the transformation of time series into a network have been developed,
which allows to extend the range of known characteristics of time series even to
network ones. Recently, several approaches have been proposed to transform time
sequences into complex network-like mappings. Three main classes can be
distinguished. The first is based on the study of the convexity of successive values of
the time series and is called visibility graph (VG) [12]. The second analyzes the mutual
approximation of different segments of the time sequence and uses the technique of
recurrent analysis [9]. The recurrent diagram reflects the existing repetition of phase
trajectories in the form of a binary matrix whose elements are units or zeros, depending
on whether they are close (recurrent) with given accuracy or not, the selected points of
the phase space of the dynamic system. The recurrence diagram is easily transformed
into adjacency matrix, on which the spectral and topological characteristics of the graph
are calculated [9]. Finally, if the basis of forming the links of the elements of the graph
is to put correlation relations between them, we obtain a correlation graph [9]. To
construct and analyze the properties of a correlation graph, we must form adjacency
matrix from the correlation matrix. To do this, you need to enter a value which, for the
correlation field, will serve as the distance between the correlated agents. So, if the
correlation coefficient between the two assets is significant, the distance between them
is small, and, starting from a certain critical value, assets can be considered bound on
55

the graph. For an adjacency matrix, this means that they are adjacent to the graph.
Otherwise, the assets are not contiguous. In this case, the binding condition of the graph
is a prerequisite.
The use of the complexity of recurrent and visibility graph networks to prevent
critical and crisis phenomena in stock markets has been considered by us in a recent
papers [27; 28]. In this paper, we consider multiplex implementations of these
techniques.
The recurrence diagrams for the visualization of phase space recurrences is based on
Henri Poincare’s idea of the phase space recurrence of dynamical systems. According
to Takens’s theorem [9], an equivalent phase trajectory that preserves the structure of
the original phase trajectory can be recovered from a single observation or time series
by the time delay method: ⃗( ) = , ,…, ( ) , where m – embedding
dimension, τ – time delay (real time delay is defined as τ·∆t). The recurrence plot shows
the existing repetitions in the form of a binary matrix R, where Ri, j = 1, if ⃗ it is adjacent

to the state ⃗ , and Ri,j = 0 otherwise. Neighboring (or recurrent) are states x j that fall
into a m-dimensional region with radius ε and center in ⃗ . It is clear that parameters m,
τ and ε are key when conducting recurrent analysis. The recurrence plot is easily
transformed into an adjacency matrix, by which the spectral and topological
characteristics of the graph are calculated [28].
The algorithm of the VG is realized as follows. Take a time series Y(t) = [y1, y2, …,
yn] of length N. Each point in the time series data can be considered as a vertex in an
associative network, and the edge connects two vertices if two corresponding data
points can “see” each other from the corresponding point of the time series (fig. 1).
Formally, two values ya of the series (at the time ta) and yb (at the time tb) are connected,
if for any other value (yc, tc), which is placed between them (that is, ta < tc < tb), the
condition is satisfied yc < ya + (yb – ya)((tc – ta)/(tb – ta)). Note that the visibility graph
is always connected by deﬁnition and also is invariant under affine transformations, due
to the mapping method.

Fig. 1. Illustration of constructing the visibility graph (red lines) and the horizontal visibility
graph (green lines) [16].
56

An alternative (and much simpler) algorithm is the horizontal visibility graph (HVG)
[16], in which a connection can be established between two data points a and b, if one
can draw a horizontal line in the time series joining them that does not intersect any
intermediate data by the following geometrical criterion: ya, yb > yc for all c such that
ta < tc < tb (fig. 1).
In multiplex networks, there are two tasks [13]: (1) turn separate time series on the
network for each layer; (2) connect the intra-loop networks to each other. The first
problem is solved within the framework of the standard algorithms described above.
For multiplex networks, the algorithm of the MVG for the three layers is presented in
fig. 2.

Fig. 2. Scheme for forming bonds between three layers of the multiplex network [13].

The recurrence multiplex network (MCRP) is formed from recurrence plots of
individual layers.

2.1 Spectral and topological graph properties
Spectral theory of graphs is based on algebraic invariants of a graph – its spectra. The
spectrum of graph G is the set of eigenvalues of a matrix Sp(G) corresponding to a given
graph. For an adjacency matrix A of a graph, there exists a characteristic polynomial
|λI – A|, which is called the characteristic polynomial of a graph PG(λ). The eigenvalues
of the matrix A (the zeros of the polynomial |λI – A|) and the spectrum of the matrix A
(the set of eigenvalues) are called respectively their eigenvalues and the spectrum of
graph G. The eigenvalues of the matrix A satisfy the equality A ̅ = ̅ ( ̅ – non-zero
vector). Vectors ̅ satisfying this equality are called eigenvectors of matrix A (or graph
G) corresponding to their eigenvalues.
Another common type of graph spectrum is the spectrum of the Laplace matrix L.
The Laplace matrix is used to calculate the tree graphs, as well as to obtain some
important spectral characteristics of the graph. In particular, the positive eigenvalues λ2
is called the index of algebraic connectivity of the graph. This value represents the
“force” of the connectivity of the graph component and is used in the analysis of
reliability and synchronization of the graph.
57

Important derivative characteristics are spectral gap, graph energy, spectral moments
and spectral radius. The spectral gap is the difference between the largest and the next
eigenvalues of the adjacency matrix and characterizes the rate of return of the system
to the equilibrium state. The graph energy is the sum of the modules of the eigenvalues
of the graph adjacency matrix. The spectral radius is the largest modulus of the
eigenvalue of the adjacency matrix. Denote by Nc the value which corresponds to an
“average eigenvalue” of the graph adjacency matrix =− ∑ and is
called natural connectivity.
The k-th spectral moment of the adjacency matrix is determined by the expression
n
mk ( A)  1 / n ik , where i is the eigenvalues of the adjacency matrix, n is the vertex
i 1

of G.
Among the topological measures one of the most important is the node degree k –
the number of links attached to this node. For non-directed networks, the node’s degree
ki is determined by the sum ki   aij , where aij – the elements of the adjacency matrix.
j

To characterize the “linear size” of the network, useful concepts of mean and
maximum lmax shortest paths. For a connected network of N nodes, the average path
length (ApLen) is equal to  l  2 / ((n( N  1)) lij , where lij – the length of the
i j

shortest path between the nodes. The diameter of the connected graph is the maximum
possible distance between its two vertices, while the minimum possible is the radius of
the graph.
If the average length of the shortest path gives an idea of the whole network and is a
global characteristic, the next parameter – the clustering coefficient – is a local value
and characterizes a separate node. For a given node m, the clustering coefficient Cm is
defined as the ratio of the existing number of links between its closest neighbors to the
maximum possible number of such relationships Cm=2Em/(km(km–1)). Here km(km–1)/2
is the maximum number of links between the closest neighbors. The clustering
coefficient of the entire network is defined as the average value Cm of all its nodes. The
clustering coefficient shows how many of the nearest neighbors of the given node are
also the closest neighbors to each other. He characterizes the tendency to form groups
of interconnected nodes – clusters. For real-life networks, the high values of the
clustering coefficient are high.
Another feature of the node is the between’s. It reflects the role of the node in
establishing network connections and shows how many shortest paths pass through this
node. Node betweenness is defined as =∑ ( , , )/ ( , ), where B(i, j) –
the total number of shortest paths between nodes i and j, B(i, m, j) – the number of
shortest paths between i, j those passing through the node m.
One of the main characteristics of the network is the distribution of nodes P(k),
which is defined as the probability that the node i has a degree ki=k. For most natural
and actual artificial networks there is a power distribution P(k) ~ 1/kγ, k≠0, γ > 0.
Also, important topological characteristics are the vertex eccentricity – the largest
distance between m and any other vertex, that is, how far the vertex is far from the other
58

vertices of the graph. The centrality of the vertex measures its relative importance in
the graph. At the same time, the farness of a node is defined as the sum of its distances
to all other nodes, and its closeness is defined as the backward distance. Thus, the
centrality of the node is lower than its total distance to all other nodes.
Another important measure is the link density in the graph, which is defined as the
number of links ne, divided by the expression nn(nn–1)/2, where nn is the number of
nodes of the graph.

2.2 Multiplex complexity measures
A multilayer/multiplex network is a pair M=(G, C) where = { ; ∈ {1, . . . , }}
there is a family of graphs (whether directed or not, weighed or not) = ( , ),
called layers = { ⊆ × ; , ∈ {1,...,M}, ≠ }. The latter is a set of links
between nodes of different layers and at ≠ . Each element is intralayer
bonds M in contrast to the elements of each ( ≠ ), called interlayer bonds.
A set of nodes of a layer is denoted = { ,..., }, and a adjacency matrix
1, ( , ) ∈ ,
as [ ] = ( ) ∈ ℝ × , = for 1 ≤ , ≤ and 1 ≤ ≤ .
0
For an interlayer adjacency matrix we have [ , ] = ( ) ∈ ℝ × , where

= 1, ( , ) ∈ ,.
0
A multiplex network is a partial interlayer case and contains a fixed number of nodes
connected by different types of links. Multiplex networks are characterized by
correlations of different nature [27], which enable the introduction of additional
multiplexes.
Let's evaluate the quantitative overlap between the various layers. The average edge
[ ]
overlap obviously equal = ∑ ∑ ∑ / ∑ ∑ (1 − ,∑ [ ] ) and determines
the number of layers in which this bond is present. Its value lies on the interval [1/ , 1]
and equals 1/M if the connection (i, j) exists only in one layer, that is, if there is a layer
[ ] [ ]
such that = 1, =0 ∀ ≠ . If all layers are identical, then = 1.
Consequently, this measure can serve as a measure of the coherence of the output time
series: high values indicate a noticeable correlation in the structure of time series.
The total overlap between the two layers and is defined as the total number
of bonds that are shared between the layers and  : =∑ , where ≠ .
For a multiplex network, the vertex degree k is already a vector
[ ] [ ] [ ]
=( ,..., ), where is the degree of the node in the layer, that is, while the
elements of the matrix of adjacency for the layer. Specificity of the vector character of
the degree of the peak in multiplex networks allows for the introduction of additional
interlayer characteristics. One of these is the overlap of the node’s degree
[ ]
=∑ .
The next measure quantitatively describes the interlayer correlations between the
degrees of the selected node in two different layers. If, chosen from M the layers of the
59

pair ( , ) characterized by the distribution of degrees ( [ ] ), ( [ ] ), the so-called
interlayer mutual information is determined by the formula
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]
, = ∑∑ ( , ) ( ( , ))/( ( ) ( )) , where ( , ) is
the probability of finding a node degree [ ] in a layer and degree [ ] in a layer .
The higher the , value, the more correlated are the distributions of the levels of the
two layers, and, consequently, the structure of the time series associated with them. We
also find the mean value , for all possible pairs of layers – the scalar value < , >
that quantifies the information flow in the system.
The quantity that quantitatively describes the distribution of the node degree i
between different layers is the entropy of the multiplexed degree
[ ] [ ]
= −∑ / ( / ). Entropy is zero if all nodes are in the same layer and
vice versa, has the maximum value when they are evenly distributed between different
layers. That is, the higher the value Si, the even more links evenly the nodes’
connections are distributed between the layers.
A similar magnitude is the multiplex participation coefficient
[ ]
= /( − 1) 1 − ∑ / . Pi takes values on the interval [0, 1] and
determines that homogeneous links of node i are distributed among M the layers. If all
links of the node i lie in one layer, Pi = 0 and Pi =1 if the node has a precisely defined
number of links in each of the M layers. Consequently, the larger the coefficient Pi is,
the more evenly distributed the participation of the node in the multiplex.
Obviously, the magnitudes Si and Pi are very similar. We will show that some of
these spectral and topological measures serve as measures of complexity of the system,
and the dynamics of their changes allows us to build predictors of crisis situations on
financial markets.

2.3 Experimental results and their discussion

In our recent work [26] we investigated the multiplex properties of the most capitalized
stock indices for the period from 01/01/1983 to 10/01/2019. Here we will add an
analysis of their complex network properties, while also preserving the analysis of
multiplex properties for a whiter period of time, but including the COVID-19 crisis
[23]: 02/01/2004 – 17/08/2020 [34]. In addition, for the first time, we are examining
the network complex properties of the CCI30 crypto index and comparing them with
the properties of the multiplex network of the 3 most capitalized cryptocurrencies
included in the CCI30 index basket. For the crypto market, the period under
consideration corresponds to the time the index was observed: from 07/08/2015 to
17/08/2020 [7]. The time series of daily values of stock market indices were selected
as databases, which contained significant changes in the indices, and were identified as
crisis phenomena [15]. Daily values of cryptocurrency prices for a period comparable
to the CCI30 index were borrowed from [33]. Among the set of stock indexes are the
following: SP (S&P500) – USA; DAX (DAX PERFORMANCE-INDEX) – Germany;
N225 (Nikkei 225) – Japan; HSI (HANG SENG INDEX) – China; GSPTSE (S&P/TSX
Composite index) – Canada.
60

Up to three selected cryptocurrencies include the following: Bitcoin – BTC;
Ethereum – ETH; Litecoin – LTC.
Their comparative dynamics on a relative scale is shown in fig. 3.

Fig. 3. The dynamics of the daily values of the selected stock market indices (a) and
cryptocurrencies and CCI30 (b).

Despite a slight difference in the time series dynamics, the main trends (both rising and
falling – actually crises) are observed for all time series.
Crypto index CCI30 refers to a number of emerging crypto indices [32] similar to
many stock market indices. CCI30 is an index that is designed to objectively measure
the overall growth, daily and long-term movement of the blockchain sector. This allows
you to track 30 cryptocurrencies by market cap. This index serves as a tool for passive
investors to participate in this asset class and an industry benchmark for investment
managers. The main characteristics of the CCI30 index are: 1) versatility;
2) reproducibility; 3) transparent; 4) provides comprehensive coverage of the entire
sector; 5) represents the best possible risk-adjusted performance profile. The CCI30
was launched on Jan 1st, 2017. Its starting value is arbitrarily set at 100 on Jan 1st,
2015. Members are automatically selected from the top 30 cryptocurrencies by adjusted
market cap and included in the index. The minimum number of constituents required
to obtain statistical significance was set at 30. Because using more components will
lead to higher fees without significantly improving performance, and using fewer than
thirty components will run the risk of reduced performance, lack of diversification,
decreased statistical significance, and wasted opportunities to pick the next leader. To
date, having hit the top 30 cryptocurrencies, CCi30 captures approximately 90% of the
cryptocurrency market capitalization. With this scope, the index statistically represents
the entire cryptocurrency market with a confidence level of 99% and a confidence
interval of 1.11. The margin of error of the index value as an indicator of the market is
just 1.11%. To calculate the weights for each cryptocurrency, you first need to calculate
the adjusted market cap. Market capitalization is not calculated as some instantaneous
number – the volatility in the cryptocurrency market is such that it too destabilizes the
composition of the index. Instead, the CCI30 uses an exponentially weighted moving
average of the market cap. Weighted average market cap helps smooth out volatility to
give the most accurate portrait of market cap at any given moment.
61

The weight of each constituent cryptocurrency is measured by the square root of its
adjusted market capitalization. The square root function was chosen as a hybrid that
most accurately weighs the constituents based on the current cryptocurrency market
conditions. A simple weighted market cap index would be dominated by the two
leading cryptocurrencies, while a slower declining weight, or in a pinch, equal
weighting, would give too much weight to the tiny illiquid cryptocurrencies at the
bottom of the range. In order to accurately capture market movements, restrictions or
minimum values for the weight of cryptocurrencies are not applied. The index is a better
investment vehicle than Bitcoin itself, and a much safer approach than trying to pick
individual coins. Investing in an index allows you to profit from the unpredictable
growth of some cryptocurrencies, while limiting losses associated with the fall of
others.
The spectral and topological properties of both stock market indices and the CCI30
index were calculated. The calculations were carried out using the moving window
algorithm. In the case of stock indices, the window size was chosen equal to 250 days,
and the window offset was equal to 5 days. In the case of a more volatile crypto market,
the window size was chosen equal to 100 days.
Figure 4, using only two measures as an example, shows their typical dynamics. It
follows from the figure 4 that in the pre-crisis period the system has a higher
complexity. In this case, node degree takes maximum values, while the average path
length, on the contrary, is minimized. After the crisis, the opposite picture is observed.
Calculations of spectral and topological measures by methods of MVG, MCRP were
carried out in the following way. For the selected window, the corresponding graphs
were constructed and the spectral, topological and multiplex properties were calculated.
Next, the window was shifted step by step, for example, one week (5 trading days) and
the procedure repeated until the time series were exhausted. Knowing the time of the
onset of the crisis and comparing the time series with the dynamics of a certain
indicator, it is possible to investigate its dependence on certain the characteristic
changes in the stock market: pre-crisis, crisis and post-crisis periods.
The results of calculations for revived time series of graphs are shown in figures 5-
8.

a)
62

Fig. 4. Window dynamics of spectral kmax and topological ApLen complexity measures for the
S&P500 and CCI30 indices.
63

Fig. 5. The entropy of the multiplexed degree and the multiplex participation coefficient
calculated for a sliding window (w) of 250 days in increments (Δ ) of 5 days. Moments of
major stock crises are indicated.

Fig. 6. Nodes degree overlap and the total overlap calculated for a sliding window (w) of 250
days in increments (Δ ) of 5 days.

Figures 5-6 show that the pair of parameters , in figure 5 is antisymmetric to the pair
parameters , of figure 6. However, all of them in a characteristic way (decreasing or
increasing before the crisis) signal about its possible onset.
Similar calculations were carried out for the CCI30 index (figures 7-8). As in the
case of a multiplex visibility graph, multiplex recurrence measures are also informative
indicators-predictors of crisis phenomena.
Parameters such as the width of the window w and the step ∆t of its displacement
along the time series are important. When w is small, the degree of complexity
fluctuates noticeably, reacting not only to crises, but also to more or less noticeable
fluctuations of the index. On the contrary, with too much window width there is a
noticeable smoothing of the appropriate measure and if two crises are at a distance that
64

is smaller w, the indicators of both crises are averaged and less informative. If you
choose an oversized parameter ∆t, you might miss the actual crisis that distorts the
indicator.

Fig. 7. MVG complexity measures: the entropy of the multiplexed degree and the multiplex
participation coefficient.

Fig. 8. MVG complexity measures: nodes degree overlaps and the total overlap.

Multiplex networks are actively used to simulate complex networks of different nature:
from financial (banks [2], stock market [19], guarantee market [14]) to social [30].
Particular attention should be paid to the work [19], in which the above multiplex
measures are analyzed for the subject of correlations with known stock markets crises.
Yet there is no systematic analysis of network and multiplex measures and the
construction of indicators-predictors of the crisis phenomena in the stock market.
Therefore, our studies to some extent fill this gap.
65

3 Conclusions

We have demonstrated the possibility of studying complex socio-economic systems as
part of a network paradigm of complexity. A time series can be represented in an
equivalent way – a complex/multiplex network, which has a wide range of
characteristics; both spectral and topological, and multiplexed. Examples of known
financial crises have shown that most of the network measures can serve as indicators-
precursors of crisis phenomena and can be used for possible early prevention of
unwanted crises in the financial markets. They are an extension of the already proposed
by us and “working” indicators, which use other measures of complexity.
It should be noted that the proposed indicators-precursors do not solve the more
general problem of forecasting future values or trends of the stock market. In this way,
it is possible to use new approaches or alternative methods based on algorithms of
(deep) machine learning [5; 8; 11].

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