3.1. Materials

The data set consists of 465 stocks that were continuously traded on the NYSE or NASDAQ for the period from April 8, 2013 to March 10, 2023. The entire data set was divided into two data sets covering the period starting April 8, 2013 and ending March 8, 2018, with 1240 trading days, for data set 1, and from March 9, 2018 to March 10, 2023, with 1260 trading days (approximately 5 years), for data set 2. The 465 companies remaining in the dataset were included in the S&P500 index on the last day of the time span, and each stock has 2500 data points. The data set for a single observation of each stock contains: i) daily closing price adjusted for splits and dividends; ii) maximum price on the trading day; iii) minimum price on the trading day; and iv) daily trading volume. In addition, data were collected on: a) daily S&P500 index adjusted close prices; and b) interest rates on the 13-week Treasury Bill. All the networks 1) idiosyncratic return; 2) volatility; 3) trading volume; 4) multiplex stock market network, and 5) cross-correlation of stock return were constructed using data set 2. However, the idiosyncratic return network construction methodology requires stock prices, S&P500 indexes, and the 13-week Treasury Bill returns data to be obtained for an additional period encompassing data set 1. These historical data were collected from Yahoo Finance (https://finance.yahoo.com; accessed on 11.03.2023). The five networks mentioned above are binary and undirected. All statistical analyses were performed using the following programs: [87–89]. Additional analyzed data are provided in the S1 File supplementary Information files.

3.2. Methods

3.2.1. Construction of similarity-based networks.

Let be an undirected and unweighted network consisting of a set of vertices (stocks) , where , and a set of edges (relations) between the nodes . The adjacency matrix of a monoplex network is the N-dimensional, unweighted, and symmetric matrix with elements

(1)

The adoption of a binary representation of financial networks is justified by several advantages over weighted networks that retain the distance metric on the edges. First, unweighted networks derived from the distance matrix simplify the complexity of the network’s structure by reducing the granularity of edge weights. This approach focuses solely on the presence or absence of relationships between assets rather than their precise strength or magnitude. Furthermore, unweighted networks are inherently robust to minor fluctuations and noise in the underlying distance metrics. In financial datasets, the inherent noise and vulnerability to minor fluctuations in correlations can lead to potential overinterpretation of trivial variations in edge weights. By employing binarization, this risk is significantly reduced, thereby enhancing the reliability of the resulting network which effectively delineates only the most critical structural characteristics of the system, thus laying a more dependable foundation for subsequent analysis. Finally, unweighted networks present a uniform framework for comparative analyses across various layers of multiplex network. Weighted networks, by their very nature, complicate such comparisons because of the discrepancies in edge weights that do not have a direct comparability in different contexts. In other words, the added dimension of weighted networks brings with it more complexity, which may hide the basic topology and not allow direct comparison.

3.2.2. The cross-correlation of stock return network.

Denote by the adjusted closing price of stock at time . The daily logarithmic returns of the stock prices can be calculated as

(2)

The cross-correlation function based on the two log-return time series for each pair of stocks i and j can be computed using the Pearson correlation coefficient

(3)

where the notation  …  means the average value over a time period . The correlation coefficients establish a symmetric matrix C with elements and unity on the diagonal. The similarity measure between each pair of stocks requires the three axioms of Euclidean distance to be satisfied: i) positive definiteness; ii) symmetry; and iii) triangular inequality [90]. The correlation coefficients are converted into distance metrics using an appropriate transformation function

(4)

where denotes the distance metric between companies i and j, and ranges from 0, which corresponds to a strong positive correlation coefficient to 2, which corresponds to a strong negative correlation coefficient .

To filter out the huge amount of information from a fully-connected distance matrix of dimension, the MST method is applied. The MST is the spanning tree of the shortest length, effectively reducing the information space from edges to most important links, while connecting a set of N vertices without cycles or self-loops. The MST extraction problem is solved using Kruskal’s algorithm [91], which spans nodes using a subset of edges with a minimal sum of weights and forms an acyclic graph. The final binary network is obtained through its dichotomization. This network is referred to as the stock return network (SRN).

3.2.3. MST-based single-layer financial networks.

The following section presents the construction of three monoplex financial networks using three types of relations among assets.

Idiosyncratic return network. The premium of the firm-specific risk is utilized as a proxy for the idiosyncratic return of the company. To compute the idiosyncratic risk premium, the CAPM model is applied to filter out the common factor of the capital market

(5)

where represent the return of stock i at time t, the market return at time t, and the risk-free rate at time t; refers to the premium of the systematic risk in the capital market; denote the estimated regression coefficient of firm i (the intercept of the security market line) and the exogenous risk of stock i, respectively; and corresponds to the residual which represent the idiosyncratic component of the stock return dependent on firm-specific factors. The above are made for the combined time span of data sets 1 and 2. To separate the firm-specific return from the common factor property, the observed returns are regressed using the ordinary least squares (OLS) method. This approach has been applied in the work of numerous researchers in the field: [40,44,45,47,92,93].

Following [92], the daily log return of the S&P500 index is adopted as

(6)

where is the daily adjusted closing price of the S&P500 index at time t,and the risk-free rate is computed based on the interest rate on the 13-week Treasury Bill expressed in terms of one day

(7)

where denotes the annual interest rate of the 13-week Treasury Bill.

The residuals without the common factor property are utilized to construct the idiosyncratic return network for data set 2. The logarithmic returns of the residuals are calculated (Eq. (2)) and the rest of the MST-based procedure is followed as for the construction of the cross-correlation of stock return network (Eqs. (3), (4)).

Volatility network.

To produce the volatility network, the approach adopted in [69,74] is used, in which, for the daily stock price data, the volatility for each stock i and trading day t is calculated by utilizing the proxy

(8)

where and are the highest and lowest price of the trading day.

The selection of the above volatility measure employed in constructing the volatility network is grounded in its ability to accurately capture intraday price movements while remaining computationally efficient. Moreover, the methodology employed is supported by a number of considerations. Firstly, it directly measures the magnitude of price movements within a single trading day, thereby providing a robust proxy for realized volatility. Compared to methodologies based solely on returns or closing prices, this approach incorporates data from the entire trading day, enhancing its sensitivity to transient price dynamics. Secondly, the normalization by the sum of the highest and lowest prices ensures the scale independence of the measure, making it suitable for comparing stocks with varying price levels. This property is critical for constructing volatility networks, where relationships between stocks are assessed based on relative rather than absolute volatility levels. Finally, the simplicity of the formula makes it computationally efficient and more robust against the noise often present in high-frequency trading data. By using only the highest and lowest prices, the approach avoids potential biases introduced by closing prices, which may not fully exhibit intraday variability due to market microstructure effects or end-of-day trading behaviors.

Then the correlation of the volatility and the distance, applying Eqs. (3)-(4), are computed. Next, the Kruskal algorithm is used to build the MST-based volatility network. In the final step, the network is dichotomized.

Trading volume network.

This type of financial network is based on the cross-correlation of trading volumes between a pair of two stocks. Denote by the stock trading volume of company on trading day . The construction of this MST-based network is exactly the same as the construction of the stock return correlation network, where the correlation coefficients (Eq. (3)) are computed for the time series of the logarithmic expression of the trading volume .

3.2.4. Multiplex financial network.

The multiplex system of the financial network consists of N vertices and L binary layers . In our case there are three layers : 1) the idiosyncratic return network; 2) the volatility network, and 3) the trading volume network. The multiplex network can be expressed by the vector of the adjacency matrices of the L layers [94]

(9)(10)

where if nodes i and j are connected by an edge on layer l and otherwise.

In most interconnected complex systems, interactions occur not only among nodes in the same layer, but also between pairs of layers [95]. However, the multiplex network in the financial market consisting of the three network layers that is designed as described in Section 3.2.2, there are no edges between the layers. In a system that displays overlapping edges, the total number of links across all layers is meaningful [25]. Therefore, the multiplex financial network is created by joining the three single network layers. In other words, the multiplex network is obtained from the multilayer structure by combining all links that occur in at least one single layer. Note that possible multiple edges are omitted. The adjacency matrix of the multiplex financial network is defined as

(11)(12)

4.1. Statistical properties of financial networks

Fig 2 depicts a time-series representation of the residuals () derived from Eq. (5), which represent the firm-specific idiosyncratic component of stock returns.

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Fig 2. Time-series of idiosyncratic components of stock returns.

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The MST-based graphs of the systematic risk network, volatility network, trading volume network, and multiplex network are shown in Fig 3, and that of the stock return network in Fig 4. Heat maps of the network connections are presented in S1 File Appendix A (Supplementary files). All three layers and the MST-based stock return network demonstrate a typical tree-like structure with a few hub-nodes, while the multiplex network is a graph with a much higher density of connections. A wider assessment of the created networks is carried out on the basis of the global network indicators presented in Table 1.

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Table 1. Network indicators.

https://doi.org/10.1371/journal.pone.0320799.t001

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Fig 3. The MST networks of a) idiosyncratic return; b) volatility; c) trading volume; and d) multiplex network.

https://doi.org/10.1371/journal.pone.0320799.g003

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Fig 4. The MST network of stock returns (monoplex network).

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All networks have the same number of nodes . Since all three layers and the stock return network are MST-based networks, they all consist of the same number of edges , degree (928), and mean degree (1.996). In contrast, the multiplex network with the same number of vertices has more than twice the number of edges and has higher average degrees that exceed 4.5. There is a significantly higher number of maximum degrees compared to the other networks and a much smaller fraction of pendant nodes (0.03), the so-called leaf fraction, which means the fraction of vertices with no more than one link. For the remaining networks, the leaf fraction is approximately equal to half the number of nodes. This means that the multiplex network does not have any of the typical tree, chain, or star-like network structures, and the most connected vertices in the multiplex network has more connections than the hubs of other networks.

The stock return network and all three layers of the multiplex network are sparse graphs with a density of 0.004. This indicates that the probability of an edge between two randomly selected vertices is 0.4%. By contrast, the multiplex network is more dense and the probability is 1%.

The tree-like structure of the stock return network and the three layers is confirmed by the high value of the mean geodesic distance, which exceeds the value of 10. This means that between two selected nodes in these networks there are, on average, 9 intermediate nodes for the trading volume network and approximately 18–19 for the idiosyncratic risk network. The average shortest path length for the multiplex network is only 4.2 (approximately 3 intermediate nodes between two stocks), which indicates that the multiplex network is more compact than the other graphs. The compactness of the multiplex network is confirmed by the low values of the network diameter (9) and graph radius (6) compared to the other networks, for which the values range from 25 to 46 and from 13 to 23, respectively. It is worth noting that the diameter of the multiplex network (9) is lower than the minimal value of the graph radius of the remaining networks (13). This indicates that the FMN presents short cuts.

Since all the networks except the multiplex one are MST-based networks, which are graphs without loops, the global clustering coefficient and transitivity are equal to 0 (there are no triangles). By contrast, we observe positive values of the above indicators for the FMN. The likelihood of two adjacent neighbors of a vertex being clustered together is 15.2%. It should be emphasized that, based on the density, mean geodesic distance, network diameter and clustering coefficient, the properties of a small-world network may be exhibited by the multiplex network. The verification of this issue is the subject of Section 4.2.

The net tree length (NTL) is used to assess the length of the MST-based networks. For this reason, the multiplex network is not considered, because it does not meet the minimum spanning tree criteria, due to and the graph containing loops. We observe that the NTL is substantively shorter for the SRN and for one layer – the volatility network . As the NTL increases, the MST network structure becomes less tightly held. Comparing the shrunken trees of stock returns and volatility with those of idiosyncratic return and trading volume suggests that stock prices and volatility tend to move in the same direction more strongly than the firm-specific factor of the stock return and the turnover of assets. The SRN shows a higher level of correlation than the idiosyncratic return network layer because the construction of this layer is based on the common factor-free stock return.

The FMN has a significantly lower value for the mean occupation layer (MOL) compared to all three layers of the multiplex network and the SRN. This suggests that the FMN is more densely connected than the other networks, with each vertex located only approximately two nodes away from the center. A lower MOL value for the multiplex network indicates that the transmission of all information provided by each of the three layers from the central vertex to the other stocks requires fewer intermediate nodes than is required for the other networks. This is a consequence of the higher density of the multiplex network, in which local connectivity of the network as connected triples of nodes is allowed. Nonetheless, MOL values ranging from 7.9 to 13.9 for the SRN and all three layers confirm that these MST-based networks exhibit a tree-like structure.

Based on the network degree centrality index (NCDI), it can be concluded that the FMN network is the most centralized of all the networks considered, and that the idiosyncratic return network is the least centralized . In other words, the most connected nodes in the FMN network have a higher degree than in the other networks. This confirms the finding that the FMN network has a significantly higher maximum degree than the other networks.

The compactness of the FMN is confirmed by the network closeness centrality index (NCCI), which is approximately 2–4 times larger than the NCCI of the remaining networks. This implies that there are several vertices in the FMN that are very close to other nodes, the so-called short cuts. On the other hand, the FMN shows a much lower level of betweenness than the other networks. The network betweenness centralization index (NCBI) is 18.5%, while it ranges from 62.5% to 70.3% for the other networks. The observed higher centralization of the SRN and all three layers of the multiplex network in terms of betweenness is due to their tree-like structures, where several vertices act as central intermediate nodes. In contrast, the FMN network exhibits a much more evenly distributed betweenness centrality. It should be stressed that the NCCI is larger than the NCBI only for the multiplex network, which clearly indicates a different shape and structure of this network.

Modularity (Q) values for all four networks – SRN, idiosyncratic return, volatility, and trading volume – range from 0.901 to 0.904, indicating that the networks have a strong community structure. The modularity result obtained is significantly high. However, the multiplex network has a lower modularity value of 0.555, but still shows significant community partitioning . There is also a difference in the number of communities. The Louvain Fast Unfolding algorithm [96] identifies fewer communities (14) for the multiplex network than for the other networks (from 19 to 20) for the same number of nodes (). This is the result of the clustering tendency observed in the multilayer network but not in the other networks.

To assess whether the analyzed networks are scale-free, 930,000 iterations (2,000 iterations per vertex) a powerful statistical approach developed in [97] are performed. The power-law exponents of all the networks range from 2.534 (volatility layer) to 4.921 (idiosyncratic return layer) and the corresponding p-values are larger than 0.1, indicating that both networks, the SRN and the FMN, and each layer of the multiplex network obey a power-law vertex degree distribution. This implies that these networks have a scale-free structure, which means that they are composed of self-similar structures at different scales. The power-law degree distribution indicates that the network is highly heterogeneous, meaning that there are a few highly connected hub-nodes that play a crucial role in the overall connectivity of the network, and many vertices with low degree. This observed nature, which is typical of a scale-free network, distinguishes the analyzed financial networks from random networks in the sense of E–R random graphs, whose degree distribution follows the Poisson distribution. S1 File Appendix B (Supplementary files) provides more information on the analysis of the power-law degree distributions, such as the KS test statistics, and the lower bound (), and a graphical presentation of the degree distributions using the complementary cumulative distribution function (CDF). It should be noted that a power-law degree distribution is also observed for the overlapping degree of the multiplex network

(13)

where the exponent is 2.972 (p-value = 0.144).

Since the constructed financial multiplex network is a fully multiplexed system in which every node is presented in each of the three layers, a suitable measure of the distribution of the degree of the vertex i among the separate layers is the multiplex participation coefficient (MPC) [54,94]

(14)

(15)

The MPC measures the distribution of the edges connected to node i across multiple layers. takes values in the range , where 1 indicates that node i has the same number of edges on each of the L layers, and 0 means that vertex i has connections in only one layer. The average multiplex participation coefficient (MPC) is equal to P=0.933, indicating that the participation of most vertices by degree is uniformly distributed among the three layers of the multiplex network. The MPC distribution is depicted in Fig 5.

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Fig 5. Distribution of the multiplex participation coefficient.

https://doi.org/10.1371/journal.pone.0320799.g005

Table 2 reports the degree–degree correlation analysis with the Quadratic Assignment Procedure (QAP) results. The results show substantial negative assortativity coefficients for all networks. This means that the indicated networks display disassortative behavior, where stocks tend to be adjacent to assets with dissimilar degrees. This can be interpreted as a tendency towards heterogeneity in the connections between vertices. Although the multiplex network has a negative assortativity coefficient, it is closest to 0 with a value of -0.072, indicating that there is a poor tendency to connect vertices with other nodes of dissimilar degrees.

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Table 2. Results of degree assortativity mixing.

https://doi.org/10.1371/journal.pone.0320799.t002

It should be emphasized that the QAP analysis [98], performed with 930,000 iterations, demonstrates a high probability (equal to or close to 1) that the observed assortativity coefficient values are significantly different from the expected value close to 0, which characterizes a non-randomly created network.

A preliminary analysis of the networks confirms the presence of sectoral assortativity (see Table 3). A positive assortativity coefficient indicates that networks exhibit assortative behavior, characterized by a tendency to form connections between assets belonging to the same sector. This property is most prominent in the idiosyncratic return layer. Previous research has demonstrated that stocks in correlation-based networks consistently form clusters closely aligned with economic sector classifications [6,28,46,72,90,92,99–103]. Likewise, the results are statistically significant when employing the QAP approach.

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Table 3. Results of sector assortativity mixing.

https://doi.org/10.1371/journal.pone.0320799.t003

4.2. The small-world structure of the multiplex network

The small-world characteristics of a corporate network [104] are widely acknowledged as a stylized fact [105] and refers to the idea that any two nodes in a large complex network can be linked to each other through a relatively short chain of intermediaries. The multiplex network is the only one considered here because the other networks do not exhibit the property of cliquishness. In other words, the global clustering coefficient and transitivity are only greater than zero for the multiplex network (see Table 2), which means that the other networks do not show the small-world phenomenon. The indicators utilized to evaluate the small-world character of the network are presented in Table 4.

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Table 4. Small-world quantities for the multiplex network.

https://doi.org/10.1371/journal.pone.0320799.t004

The multiplex network is numerically large and sparsely connected, since , where and . We compare the values of and with the asymptotic approximation of the clustering coefficient and the mean shortest path length for the equivalent E–R random network with the same number of nodes and edges. Since the relations and are satisfied, indicating that the multiplex network exhibits the small-world property. Additional evidence of the small-world phenomenon leads to a similar conclusion. Since and are observed, the ratio of to λ is much larger than one (). S1 File Fig. C1 presented in Appendix C (Supplementary files) depicts the relationship between and λ.

The small-world property of the multiplex network means that there are short cuts that reduce the distance between nodes that are not directly connected. These long-range connections allow for closer relationships between stocks and shorter paths between vertices that are otherwise separated by many intermediate nodes. Short cuts play a crucial role in creating a more compact and interconnected network. The presence of short cuts in a multiplex network results from the compactness created by several nodes that have a high degree of connectivity and act as hubs within the graph. Although the existence of hub-nodes has been detected for all the networks (the power-law property), only the FMN is a scale-free and small-world network.

4.3. Similarity between layers and networks

Similarity refers to the degree of association between two or more networks, which is measure of how the networks are related. The similarity coefficient can be used to evaluate the similarity between two different networks by comparing the sets of edges that they have in common. The Pearson correlation coefficient, the Jaccard index and the Czekanowski–Sørensen–Dice similarity coefficient, are utilized.

Table 5 reports the correlation coefficients between the individual networks. The correlation between each layer and the multiplex network is moderate , which suggests that the different layers contribute distinct information towards the final form of the multiplex network. In comparison, the correlation between the FMN and the SRN is lower, at 0.514. This is due to the varied relationship between the different layers of the multiplex network and the stock return network, where the layer of the network based on the firm-specific factor of the stock return is the most similar to the SRN, while the layer of the graph based on the trading volume is the least similar. Moreover, the similarity between the three layers is moderate (ρ is approximately 0.4) or weak (below 0.3). It should be pointed out that, based on the QAP analysis, all the correlation coefficients are statistically significant at the level of 0.1%. A better measure of similarity between networks is the matching measure. Table 6 presents the Jaccard similarity measure (lower triangle) and the Czekanowski–Sørensen–Dice similarity coefficient (upper triangle).

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Table 5. Pearson correlation coefficients.

https://doi.org/10.1371/journal.pone.0320799.t005

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Table 6. Jaccard and Czekanowski–Sørensen–Dice matching coefficients.

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The Jaccard similarity between each layer and the FMN is 0.439, which means that 43.9% of the edges overlap, while the remaining 56.1% of the links are derived from the other two layers. The edge overlapping between the FMN and the SRN is less than one third (362/1,158 = 0.313). As we saw in the analysis based on the correlation coefficient, the smallest similarity to the SRN is shown by the layer of the trading volume network (12.9%), and the greatest similarity is shown by the layer of the multiplex network based on the firm-specific factor of stock return (53.9%). When comparing the separate layers of the multiplex network, the edge overlap is relatively low, indicating their mutual diversity. The values of the Czekanowski–Sørensen–Dice coefficients confirm the above observations. The Sørensen–Dice coefficient tends to give slightly higher similarity values than the Jaccard similarity index because the Sørensen–Dice measure gives more weight to the common elements and less weight to the non-common elements. After conducting 9,300 QAP iterations, it was determined that all the similarity measures in Table 6 exhibit statistical significance. Specifically, the observed matching coefficients are larger than would be expected by random chance with a probability of 1.0. It is noteworthy that the similarity measures, including Pearson correlation coefficients, the Jaccard index, and Czekanowski-Sørensen-Dice matching coefficients, for each layer with the FMN, yield identical values. This phenomenon results from aggregating three layers with varying similarities into a multiplex form, where some edges overlap across layers, and each layer contains the same number of edges due to the MST properties.

Computing the Jaccard similarity coefficient for the four networks is recommended, considering all three layers and the multiplex network. The formula in the following can be used to extend the Jaccard coefficient to more than two sets

(16)

where represent the edges of each network, and s is the number of sets. Table 7 shows the Jaccard index for all three layers.

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Table 7. Jaccard index for two or more sets.

https://doi.org/10.1371/journal.pone.0320799.t007

The Jaccard index for all three layers and the multiplex network is 7.95%, which means that, of the 1,056 edges contained in all four networks together, only 84 overlap (84/1,056 = 0.0795). Fig 6 depicts a graphical illustration of the edge overlapping for the multiplex network and its three layers, as well as for the FMN with the SRN. The Jaccard index for three layers and for each pair of layers ranges from 10.2% to 11.5%. This indicates a higher proportion of overlapping edges within each pair of layers compared to the multiplexed representation.

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Fig 6. Edge overlapping for

a) FMN with all three layers (red edges; 7.95%); b) FMN with the SRN (green edges; 31.3%).

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Another approach is the edge overlapping ratio in the compared networks, which is defined as the ratio of the common edges found in s consecutive networks and the maximum number of edges in the network

(17)

where represent the sets of links of networks ; are the number of edges in each network. The EOR index is an original proposal defined in this study.

It should be noted that the edge overlapping ratio (EOR) proposed in this study for the FMN and its three layers corresponds exactly to the value of the Jaccard index (EOR = 7.95%). This convergence is because the denominator of the EOR – the maximum number of edges of one network – corresponds to the denominator of the Jaccard index, i.e., the union of the sets of edges of all the networks considered . On the other hand, the numerators of both indexes are the same by definition.

However, the EOR for the FMN with the SRN is different from Jaccard’s ratio and is 34.4%. In other words, the overlapping edges cover 34.3% of the maximum number of links in one of these networks (362/1056). The edge overlapping ratio for comparisons between individual layers of the multiplex network and the SRN is included in S1 File Appendix D (Supplementary files).

Another quantity used to analyze multiplex networks is the correlation of degrees across layers. In order to determine the inter-layer degree correlations, the mutual information of the degree sequences can be used. The inter-layer mutual information between the degree distributions of two layers is defined as follows [67]

(18)

where is the joint probability that a node has degree k in both layers

(19)

where denotes the number of vertices with degrees and in layers α and β, respectively. The higher the value of the inter-layer mutual information, the more correlated are the degree distributions of the two layers.

The inter-layer mutual information I for individual pairs of the three layers is as follows: 0.419 for the idiosyncratic return–volatility pair; 0.449 for volatility–trading volume; and 0.359 for idiosyncratic return–trading volume. The average quantity among all possible pairs is 0.409, which means that the degree distribution is not strongly correlated between the multiplex layers.

4.4. Multiplex network regression analysis

Another method of testing the importance of different layers is to perform the Multiple Regression Quadratic Assignment Procedure (MR-QAP) network regressions [106]. In this study, three multiple regression models are proposed to determine whether the structure of each layer explains i) the financial multiplex network and ii) the stock return network

(20)

(21)

(22)

where is the adjacency matrix of the financial multiplex network; the adjacency matrix of the stock return network; α indicates a constant; mean regression coefficients; and denote the adjacency matrices of the idiosyncratic return network, the volatility network, and the trading volume network, respectively.

These regression models estimate how the layers of the multiplex network affect the two networks – the multiplex financial network and the stock return network. The objective of Model 1, defined in Eq. (20), is to assess the effect of its individual components (layers) on the generation of the FMN. Model 1 serves as the initial framework for Model 2, which is described by Eq. (21). Both Model 1 and Model 2 are used to evaluate the differences in the strength of the impact that each layer of the multiplex network has on the FMN and SRN, respectively. To evaluate the statistical significance of the specified models, the MR-QAP test is used, which, as an extension of the bivariate QAP model, is a permutation test for multiple linear regression model coefficients for network data organized in square matrices [107]. The Double-Semi-Partialing (DSP) approach introduced by Dekker et al. [107], which is a residual permutation method, is utilized. Under permutations, the DSP method reduces the potential effects of multicollinearity between the focal variable and the control variables.

Table 8 reports the results of the estimation of the three cross-network regression models formulated by Eqs. (20)-(22). Based on the regression coefficients of Model 1, the positive effect of all three layers on the multiplex network can be observed, with the trading volume network layer having the strongest effect on the FMN. Positive regression coefficients are reported for Model 2. The strongest effect on the SRN is the multiplex network layer concerning idiosyncratic return. The explanation for this is that the rate of return is the sum of the systematic and the idiosyncratic risk premiums. It is important to emphasize that the proportion of variance in the dependent variable that can be explained by the independent variables in Model 2 is . This means that about half of the information contained in the SRN is determined by the transfer of information embedded in the three layers of the financial multiplex network. The remaining variability is explained by other factors not included in Model 2. It should be pointed out that, in contrast to Model 1, where the trading volume layer has the strongest influence on the formation of the FMN, this layer in Model 2 exerts the weakest influence on the SRN. This change underscores the differences between the FMN and SRN.

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Table 8. Network regression results.

https://doi.org/10.1371/journal.pone.0320799.t008

The third model (Model 3) deals with the impact of the SRN on the FMN. The modeled reverse relationship exhibits a much lower overall measure of goodness-of-fit, expressed by the coefficient of determination , than Model 2. However, Models 2 and 3 are not directly comparable, as one contains a multiplex network and the other the three layers of the multiplex network.

In all the regression models, the F-test value and the regression coefficients for all variables are statistically significant, while, by applying 46,500 iterations of Double-Semi-Partialing of MR-QAP, the observed quantities are statistically significantly larger than expected under the random chance assumption.

4.5. Robustness to network failure

Since all networks – the FMN, its layers, and the SRN – are scale-free graphs, we can determine the network’s robustness to random vertex failures. Evaluation in this regard has been carried out using the critical threshold of network failure, which, by applying the Molloy-Reed criterion to form a giant component in a network, can be defined as follows [108]

(23)

where and denote the second and first moments of the degree distribution, respectively.

This critical threshold indicates a finite fraction of random node removal to collapse the largest component structure. In other words, the random removal of a fraction of vertices will result in the fragmentation of the network.

Table 9 reveals that the FMN exhibits the most robust structure against random failures. In order for the financial multiplex network to fall apart, one would have to remove 84% of its vertices, while to damage the giant component of the stock return network, it is enough to randomly remove 58.6% of the nodes. It should be emphasized that the result robustness of the SRN is consistent with those obtained by [37], where for the MST-based cross-correlation network the critical robustness coefficient is and , respectively, depending on the period. This suggests that the financial market demonstrates resilience to failures in the aftermath of random stock removal.

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Table 9. Critical threshold for robustness to failure.

https://doi.org/10.1371/journal.pone.0320799.t009

4.6. Robustness analysis

The fundamental analysis was conducted through the simplification of a network approach, which entailed the dichotomization of the MSTs obtained using the distance metric and the subsequent construction of an unweighted multiplex network. The robustness test is applied to the relaxation of the network binarization assumption, resulting in the formation of a multiplex network, its corresponding individual layers, and the SRN as a weighted network. Fig 7 illustrates the visualization of the multiplex graph (FMN) using a force-directed layout, which was generated with the ForceAtlas2 algorithm (node colors correspond to sector affiliation). In the presented graph, four companies (PRU, MSFT, AMP, ETN) have been highlighted, as their degree centrality in the weighted multiplex network exceeds the threshold value of 0.05 (degree centrality > 0.05). Two of these firms belong to the Financial Services sector, while one represents the Technology sector and another the Industrials sector. Notably, the AMP node stands out due to both its high degree of connectivity and its considerable distance from other vertices.

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Fig 7. Weighted multiplex network.

https://doi.org/10.1371/journal.pone.0320799.g007

It should be stressed that comparisons of the statistical properties of the networks were omitted, as weighted and unweighted networks inherently differ. The robustness test is applied to the analyses presented in Sections 4.3, 4.4, and 4.5.

Table 10 shows the results of the correlations between the weighted networks. The results do not reveal significant differences compared to the basic analyses. It should be pointed out that the assessment of network similarity, as measured by Jaccard similarity, Czekanowski-Sørensen-Dice matching coefficients, and EOR, yields identical values. This is because the evaluation considers only the presence of edges between nodes in the network.

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Table 10. Pearson correlation coefficients (weighted networks).

https://doi.org/10.1371/journal.pone.0320799.t010

The results obtained for network regression (Table 11) and the threshold for robustness to failure (Table 12) in weighted networks are similar to those obtained for unweighted networks. This indicates the robustness of the results with respect to the assumption of dichotomization of the networks at the final stage of their construction.

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Table 11. Network regression results (weighted networks).

https://doi.org/10.1371/journal.pone.0320799.t011

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Table 12. Critical threshold for robustness to failure (weighted networks).

https://doi.org/10.1371/journal.pone.0320799.t012

4.7. Out-of-sample assessment of the dependency structure representation

In order to assess which financial network has a structure that better represents stock price movements, a regression analysis has been performed employing the ordinary least squares method. The dependent variable is the Sharpe ratio , as proxy for stock performance [34], using out-of-sample data of the stock returns over the 252 days, with a one-year lag from the last data point used to construct the financial networks. The time frame for the out-of-sample data is March 13, 2023, to March 12, 2024. The explanatory variables employed are two centrality measures calculated for the two networks, respectively FMN and SRN (for the in-sample data): 1) degree centrality and 2) eigenvector centrality. Degree centrality is a simple measure that captures connections in the nearest vertex neighborhood in the network. In contrast, eigenvector centrality is a more complex measure that considers the wider context of the structure of the connections in the network. A similar approach was applied by [101], investigating the correlation between the in-sample centrality and the out-of-sample Sharpe ratio. Due to changes in listed shares over the out-of-sample period, the number of observations is 456. Table 13 reports the results of a regression analysis of the in-sample centrality measures and their relationship with the out-of-sample Sharpe ratio.

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Table 13. Regression analysis results for out-of-sample Sharpe ratio.

https://doi.org/10.1371/journal.pone.0320799.t013

The first two models are directly comparable and evaluate the impact of the eigenvector centrality of the SRN (Model A) and the FMN (Model B) on the Sharpe ratio realized in a later time period. The results of the standardized regression coefficients indicate a more pronounced effect of the FMN than of the SRN. Furthermore, the value for Model B is greater than that of Model A. Model C includes the eigenvector centrality of both networks and confirms a larger regression coefficient for the FMN. In addition, the regression coefficient for the SRN is not statistically significant. The last model (D) includes the degree and eigenvector centrality derived from both networks. The effect of the FMN centrality measures on the Sharpe ratio is more pronounced than that of the SRN centrality measures, as evidenced by the absolute values of the standardized regression coefficients. Additionally, the regression coefficient for the eigenvector centrality_SRN is not statistically significant. Note that , indicating that there is no problem with multicollinearity.