Network analysis is a powerful method to characterize the complexity and dynamics of socio-economic systems. However, traditional network analysis often ignores the higher-order dependencies that arise from the interactions of more than two nodes. In this paper, we propose to use high-order networks, which are generalized network structures that capture the higher-order dependencies, to study the temporal evolution of the Dow Jones Industrial Average (DJIA) index. We construct high-order networks from the DJIA time series using the visibility graph method, and we measure the topological complexity of the high-order networks using various metrics. We find that the complexity of the system changes drastically during crisis events, indicating that high-order network analysis can be used as an indicator (indicatorprecursor) of financial crashes. We also show that high-order network analysis and topology can provide more insights into the nonlinear and nonstationary behavior of the DJIA index than traditional tools of ifnancial time series analysis.
The proliferation of extensive and finely-grained data, often with temporal resolution, has
unlocked unprecedented opportunities to dissect the behaviors of complex systems spanning
diverse domains such as biology, technology, finance, and economics [
The notion of higher-order interactions finds historical roots in solid-state physics, where multiparticle potentials and quantum mechanical calculations supplanted paired interactions. Similarly, in thermodynamics and statistical physics, Tsallis introduced nonextensive interactions [11, 12]. However, in contrast to these simpler representations of higher-order interactions, complexity in complex systems demands more intricate mathematical structures like hypergraphs and simplicial complexes.
Diverse models of higher-order networks have surfaced [13], reflecting the growing importance of this domain. Here, we briefly highlight key models that have garnered attention [14, 15, 16].
Multiplex Networks: Multiplex networks, multilayer networks, and networks of networks
capture interactions between various entities and have found applicability in systems with
diverse interaction types [17]. However, most interactions remain dyadic and can be represented
through traditional networks [18]. Their application in financial analysis is well-documented
[
Hypergraphs and Simplicial Complexes: Algebraic topology’s computational techniques,
hypergraphs, and simplicial complexes encode units and hyperlinks, allowing explicit
consideration of systems beyond pairwise interactions [
Higher-Order Markov Models: First-order Markov models have gained traction in
describing flows of information, energy, money, etc. within networks [
Higher-Order Graphical Models and Markov Random Fields: Markov random fields,
including the Ising model, extended to higher-order models, capture interactions between
multiple objects [
Recently, Santoro et al. [
Visibility graph (VG), which was proposed by Lacasa et al. [
“Mutual visibility” can be understood by imagining two points at time and at time as two hills of a time series, which can be understood as a landscape, and these two points are “mutually visible” if has no any obstacles in the way on . Formally, two points are mutually visible if, all values of between and satisfy: < + − [ − ] , − ∀ ∶ < <
Horizontal visibility graph (HVG) [43] is a restriction of usual visibility graph, where two points and are connected if there can be drawn a horizontal path that does not intersect an intermediate point , < < . Equivalently, node at time and node at time are connected if the horizontal ordering criterion is fulfilled: < inf( , ),
∀ ∶ < < .
Multiplex network [45] is the representation of the system which consists of the variety of diferent subnetworks with inter-network connections. For working with multiplex financial networks, we set two tasks: • convert separated time series into network that represent a layer of a multiplex network.
The procedure of conversion is presented in section 2; • create intra-layer connection between each subnetwork. (3) (4) (5)
Multiplex network is the representation of a pair = (, ) , where { | ∈ 1, … , } is a set of graphs = ( , ) that called layers and = { ⊆ × | , ∈ 1, … , , ≠ } matrix as [] = ( ) ∈ Re × , where is a set of intra-links in layers and ( ≠ ). is intra-layer edge in , and each is denoted as inter-layer edge.
A set of nodes in a layer is denoted as = { 1 , … , }, and an intra-layer adjacency for 1 ≤ ≤ , 1 ≤ ≤ and 1 ≤ ≤ . For an inter-layer adjacency matrix, we have [, ] ( ) ∈ Re × , where = { 0. 1, ( ,
) ∈ , = { 0.
1, ( , ) ∈ ,
A multiplex network is a partial case of inter-layer networks, and it contains a fixed number of nodes connected by diferent types of links. Multiplex networks are characterized by correlations of diferent nature, which enable the introduction of additional multiplexes. while node :
[] is the element of the adjacency matrix of the layer . Specificity of the node degree in vector form allows describing additional quantities. One of them is the overlapping degree of
The next measure quantitatively describes the inter-layer information flow. For a given pair (, )
within layers and the degree distributions ( defined the so-called interlayer mutual information: [] ), ( [] ) of these layers, we can , = ∑ ∑ ( [] , [] )log
( [] , [] ) ( [] ), ( [] ) , where ( [] , [] ) is the joint probability of finding a node degree [] in a layer and a degree [] in a layer . The higher the value of , , the more correlated (or anti-correlated) is the degree distribution of the two layers and, consequently, the structure of a time series associated with them. We also find the mean value of , for all possible pairs of layers – the scalar ⟨ , ⟩ that quantifies the information flow in the system.
The multiplex degree entropy is another multiplex measure which quantitatively describes the distribution of a node degree between diferent layers. It can be defined as = − ∑ =1 [] log [] .
Entropy is close to zero if th node degree is within one special layer of a multiplex network, and it has the maximum value when th node degree is uniformly distributed between diferent layers.
For a multiplex network, the node degree is already a vector
with the degree
[] of the node in the layer , namely
= ( [
Financial networks are strongly influenced by the ordering and timing of links. In their context of their temporality, we must consider time-respecting paths, an extension of the concept of paths in static network topologies which additionally respects the timing and ordering of timestamped links [47, 48, 49]. For a source node and a target node , a time-respecting path can be presented by any sequence of time-stamped links where 0 = , = and 1 < 2 < ... < . Time ordering of temporal financial networks is important since it implies causality, i.e. a node is able to influence node relying on two time-stamped links (, ) and (, ) only if edge (, ) has occurred before edge (, ) .
Apart the restriction on networks to have the correct ordering, it is common to impose a maximum time diference between consecutive edges [ 50], i.e. there is a maximum time diference and, example, two time-stamped edges (, ; ) and (, ; ′) that contribute to a time-respecting path if 0 ≤ ′ − ≤ . If = 1 , we are usually interested in paths with short time scales. For = ∞ , we impose no restrictions on time-range and consider a path definition where links can be weeks or years apart.
The key idea behind this abstraction is that the commonly used time-aggregated network is the simplest possible time-aggregated representation, whose weighted links capture the frequencies of time-stamped links. Considering that each time-stamped link is a time-respecting path of length one, it is easy to generalize this abstraction to higher-order time-aggregate networks in which weighted links capture the frequencies of longer time-respecting paths.
There are several variants for encoding high-order interactions [
Simplex is another mathematical abstraction to accomplish high-order interaction. Formally, a -simplex is a set of + 1 fully interacting nodes = [ 0, 1, ..., ]. Essentially, a node is 0-simplex, a link is 1-simplex, a triangle is 2-simplex, a tetrahedron is 3-simplex, etc. Since a standard graph is a collection of edges, simplicial complexes are collections of simplices = { 0, 1, ..., }.
Figure 3 demonstrates examples of simplices and hyperlinks of orders 1, 2, and 3.
For a temporal network = ( , ) we thus formally define a kth order time-aggregated (or simply aggregate) network as a tuple () = ( () , () ) where () ⊆ is a set of node k-tuples and () ⊆ () × () is a set of links. For simplicity, we call each of the k-tuples = 1 − 2 − ... − ( ∈ () , ∈ ) a kth order node, while each link ∈ () is called a kth order link. Between two kth order nodes and exists kth order edge ( , ) if they overlap in exactly − 1 elements. Resembling so-called De Bruijn graphs [51], the basic idea behind this construction is that each kth order link represents a possible time-respecting path of length in the underlying temporal network, which connects node 1 to node via time-stamped links ( 1, 2 = 1; 1), ..., ( = −1 , ; ). (12)
Importantly, and diferent from a first-order representation, kth order aggregate networks allow to capture non-Markovian characteristics of temporal networks. In particular, they allow to represent temporal networks in which the kth time-stamped link ( = −1 , ) on a timerespecting path depends on the − 1 previous time-stamped links on this path. With this, we obtain a simple static network topology that contains information both on the presence of time-stamped links in the underlying temporal network, as well as on the ordering in which sequences of of these time-stamped links occur.
Network centralities are node-related measures that quantify how “central’’ a node is in a network. There are many ways in which a node can be considered so: for example, it can be central if it is connected to many other nodes (degree centrality), or relatively to its connectivity to the rest of the network (path based centralities, eigenvector centrality). One of the simplest centrality measure is the degree of a node, which counts the number of edges incident to an ith node.
For any adjacency matrix the degree of a node can be defined as = ∑ .
= 1
=1
∑ .
High-order degree centrality counts the number of kth-order edges incident to the kth-order node . To get a scalar value which will serve as an indicator of high-order dynamics, we obtain mean degree
: Except this measure, we can calculate nth moment of the degree distribution, which can be (13) (14)
In this study we will present the dynamics of the first moment, which is the mean weighted degree of a network, and its high-order behavior.
Assortativity is a property of network nodes that characterizes the degree of connectivity between them. Many networks demonstrate “assortative mixing” on their nodes, when highdegree nodes tend to be connected to other high-degree nodes. Other networks demonstrate disassortative mixing when their high-degree nodes tend to be connected to low-degree nodes. Assortativity of a network can be defined via the Pearson correlation coeficient of the degrees at either ends of an edge. For an observed network, we can write it as (15) (16)
). (17) defined as ∞ min ⟨ ⟩ = ∑ ≈ ∫
. ∞ 1 ( 2 + 2) − [ −1 ∑ 1 (
+ )] 2 2 2 2 , where −1 ≤ ≤ 1 ; , are the degrees of the nodes at the ends of the ith edge, with = 1, ..., , where is the number of edges of a network.
This correlation function is zero for no assortative mixing. If = 1 , then we have perfect assortative mixing pattern. For = −1 , we can observe perfect disassortativity.
Studying financial networks, with time-respecting paths, we can consider four type of assortativity: (, ), (, ), (, ), (, ) , which will correspond to tendencies to have similar in and out degrees. We can denote one of the studied in/out pairs as (, ) . Suppose, for a given ith edge, we have got the source (i.e. tail) node of the edge and target (i.e. head) node Assortativity coeficient for degrees of a specific type can be defined as of the edge. We can denote them as -degree of the source ( ) and -degree of the target ( (, ) = ∑(
− ) ( − ) √ ∑( − )2 ∑( − )2
√ where and are the average -degree of sources and -degree of targets.
following is done:
To build indicators (indicators-precursors) based on multiplex and high-order networks, the
• databases of 6 most influential stock market indices for the period from 02.01.2004 to
18.10.2022 were selected for multiplex analysis (see figure 4). The data were extracted
using Yahoo! Finance API based on Python programming language [52];
• the indicators described in the previous sections were calculated using the sliding window
procedure [
In figure 5 presented the dynamics of inter-layer mutual information ( ) and multiplex degree entropy ( ) along with the DJIA index.
From figure 5 we can see that multiplex mutual information increases before the crisis of 2008. Also, it noticeably becomes higher before COVID-19 crash. For the last months, it demonstrates decreasing pattern, which indicates that the economies of diferent countries may be experiencing diferent evolutions now. Nevertheless, it can be seen that, as a rule, this indicator is characterized by growth, indicating an increase in the interconnection of the economies of diferent countries. In a crisis, this indicator usually declines, demonstrating diferent resistance to the collapse events of the stock markets of countries and the diference in the actions that they take. Entropy indicator shows asymmetric behavior.
Next, we compare one of the multiplex measure, overlapping degree ( ), with the mean degree of a network ( ). Figure 6 represents this result.
In figure 6 we can see that both and are characterized by similar dynamics. These indicators increase near the crash, which indicates an increase in the concentration of connections for some network nodes, and further, based on the indicators during the crisis, there is a decline in concentration both in the dynamics of the DJIA and the inter-layer connectedness of stock indices. We may see that the multiplex approach does not significantly change the dynamics of the concentration degree indicator in comparison with the indicator based on the classical univariate graph.
Figure 7 demonstrates the dynamics of mean weighted degree (equation (15)) for order 1 and 2 along with the DJIA index.
In figure 7 we can see that the second-order is slightly diferent from the first-order one. The second-order starts to increase a slightly earlier before the crisis of 2008. We can see that before crisis of 2020 second-order declines more noticeably comparing to the ifrst-order one. However, this diference between the first and second order is still insignificant, what can we say about the fact that the classical visibility graph can reflect all the information that the series under study can represent.
Next, let us present high-order dynamics of the assortativity coeficient for the DJIA index (see figure 8).
Figure 8 presents the assortativity coeficient for first, second, and third orders. Assortativity declines before crashes and increases during them. We see that high-orderness does not change radically change the dynamics of this indicator. Third-order assortativity responds better for the crash of 2008, but worse for the COVID-19 crisis, comparing to first- and second-order assortativity.
In this article, we have introduced methods to measure and model systems with causal, multiplex,
and high-order interactions. We have shown that these methods can capture the long-range
spatio-temporal correlations that characterize non-Markovian, non-stationary, non-linear
systems, which are better described by the high-order paradigm. We have used hypergraphs
[
We have presented indicators (indicators-precursors) based on classic visibility graphs,
multiplex networks, and high-order networks. We have applied these indicators to the time series of
the Dow Jones Industrial Average (DJIA) index and a database of six stock indices from diferent
countries and sectors. We have used the sliding window algorithm to calculate various network
measures, such as the mean degree of a node ( ), the first-moment degree of a network, the
assortativity coeficient, the inter-layer mutual information ( ), the multiplex degree entropy
( ), and the mean overlapping degree of a network ( ). We have found that multiplex and
high-order networks do not difer significantly from the traditional pairwise visibility model in
terms of their dynamics. This may suggest that the classical visibility graph reflects all possible
short-term and long-term dependencies in the DJIA index. We have also found that all the
presented measures work similarly as indicators (indicators-precursors) of critical financial
events, increasing or decreasing before and during them. However, multiplex and high-order
network indicators still need further development and improvement for studying complex
ifnancial time series. A possible solution may be to combine Markov chains of multiple, higher
orders into a multi-layer graphical model that captures temporal correlations in pathways at
multiple length scales simultaneously [
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