Standardized local assortativity in networks and systemic risk in financial markets

Introduction

Network analysis is a common technique used to detect and understand relationships among entities [1], and is employed in a variety of fields such as physics [2–4], social sciences [5, 6], biological, medical and epidemiological studies [7–13], business [14], and finance [15–19]. A network graph is made up of nodes that depict entities of a system and edges that indicate pairwise relationships. A graph is weighted if the edges between nodes have different strengths or attributes.

There are various measures for quantifying network structures, such as network degree and clustering coefficients to summarize the network size and the embeddedness of nodes in networks, respectively. Assortativity, or assortative mixing [4], is another measure used to quantify networks and tells us how likely nodes are to connect with other nodes that are similar to themselves. Sociological studies often describe assortative mixing as homophily, a phenomenon often found in a variety of social networks, among humans [20–22] and in nature [23]. In homophilous behavior, people tend to bond with others who are similar to themselves when they share common socio-demographic, behavioral, and intra-personal characteristics, such as gender, ethnicity, hobbies, and values, resulting in homogeneous personal networks for each individual [21]. Human networks are dynamic and individuals tend to form bonds with cooperators and break ties with defectors [22]. Constant formations and dissolutions of relationships form a continuously evolving human ecology. The evolution and ubiquity of homophily is affected by natural selection in that through collaborations and interactions with similar others, individuals create synergy and provide complementary skills to increase potential gains [24]. Over time, humans have learned to cooperate with assortative others and thereby achieve higher payoffs [22, 25].

Assortativity has implications for network resilience, in terms of how robust and reliable a network remains in the event of faults and operational challenges with some of its nodes [26]. In most networks, the connectivity can be destroyed by simply removing a few nodes with the highest number of edges. Consider technology hubs or service providers—they are usually the first targets of attacks because the failure of a few important “hubs” will create a domino effect that can bring down all services connected to them. Therefore, the study of network resilience is of high importance in the field of computer science, especially for highly critical systems in today’s digital age [26, 27].

Technological networks, along with biological and neural networks, are considered disassortative, suggesting that they are especially vulnerable and likely to be prone to targeted attacks [4]. Conversely, the assortative nature of social networks allows them to be more resilient, and their structure is therefore not easily disrupted. In the case of transmittable diseases, core groups in assortatively mixed social networks can form a “reservoir” for diseases that could sustain a pandemic [4]. As a result, pandemics have been found to be robust in face of simple targeted attacks on social networks, such as isolating a few infected individuals who have high degrees of connections in the midst of a pandemic [28]. Other studies have looked at regional networks to reveal the recent example of how the coronavirus disease 2019 (COVID-19) pandemic spread rapidly across the globe given the network’s connectedness [29, 30].

Global assortativity (GA) [3, 4, 31–33] measures the assortative mixing of networks and provides us with a glimpse of several aspects of those networks. In particular, the research provides significant insights into the relationship between global assortativity and network robustness, as well as the role of global assortativity in the evolution of cooperation in networks [31–33]. However, the coefficient is only representative if most of the nodes of a network have a degree of assortative mixing around the mean, and in some networks that is not the case [34]. When networks grow, thanks to technological advancement and easier data collection and storage, it is not hard to imagine that these complex networks could have more diverse local (dis)assortative mixing. As a measurement of averages, local assortative patterns are smoothed out and therefore might not present a full picture of the relationships in the networks.

A number of studies have proposed the use of local assortativity [34, 35] to overcome the shortcomings of a global measure, but there has not been a universal definition of how such a local coefficient should be calculated. With the lack of a local coefficient, it is difficult to compare and trace local assortativity features that might tell us more about the networks and reveal previously undiscovered patterns. In response, this study provides examples of when the global assortativity coefficient fails to address networks with local assortative hubs. We further extend the concept of local assortativity based on the existing literature, as well as proposing a standardized formula that we believe is an important add-on because it allows for comparisons on the same scale. We show in the Discussion section how this standardized local assortativity coefficient helps to explain the networks and the potential predictive power it provides.

This paper makes the following contributions. First, we adopt a local assortativity measure from the literature and provide a standardized version of it in light of the limitations that this measure might have when the network size grows. We then compare the standardized local coefficient against the global one in several networks of different fields to highlight its significance. Last, we investigate how a standardized local assortativity could provide predictive power in a dynamic network setting, and shed light on tracking systemic risk in financial markets.

Assortative mixing

The commonly used approach to measuring the assortative mixing of a network is the framework proposed by Newman [3], which measures whether nodes tend to attach to each other based on the similarity in their degrees. We define pij as the joint probability distribution of the degrees between nodes vi and vj that satisfies the following sum rules:

where ai and bj represent the fraction of edges that start and end at nodes vi and vj, respectively. If no assortative mixing is found in the network, pij = aibj. Otherwise, the global assortativity, GA, is defined as:

where σa and σb denote the standard deviations of ai and bj, respectively [4]. GA has a range of −1 ≤ GA ≤ 1, with 1 indicating perfect assortativity and -1 is perfect disassortativity. An estimate of GA in Eq (2) can be found in [3].

GA describes the assortativity of a network on a global level and is essentially the Pearson’s correlation coefficient of the degree between nodes. However, in networks that show a mix of assortative and dis-assortative patterns, GA might not give an accurate picture. To look at local assortativity, LAj of a random node vj, we adopt the formula in [35]:

where dj is the degree of node vj, nodes vj(i) for i = 1, …, dj are the neighbour nodes linking to node vj, and dj(i) is the degree of the node vj(i).

This calculation is computationally less exhaustive, and assumes that all nodes in a network are, to a certain degree, disassortative. As local assortativity is a relative concept, the authors [35] argue that the average number of differences in node degrees between any node j and its neighbors can therefore be seen as the disassortativity, that is, how disassortative node j is when compared with its neighbouring nodes. Therefore, local assortativity, or LAj, can be expressed by Eq (3).

A shortcoming of LAj is that it is not scale-invariant, thus making the comparison of LAj among nodes dependent upon the values of dj. Consider a simple example having nodes j and j′ that dj′ = cdj, dj′(i) = cdj(i) and |dj − dj(i)| = M, where c and M are positive constants. It is obvious that nodes j and j′ have the same assortative properties, but LAj′=1dj′∑i=1dj′|dj′-dj′(i)|=cM=cLAj. In addition, if nodes have a huge difference in their degree, which is not uncommon in the case of complex networks, LAj can be heavily skewed given its unbounded range of [0, +∞). In view of this, we propose a standardized version of LAj, SLAj to account for the issue. Given |dj-dj(i)|2≤dj2+dj(i)2, we standardized the coefficient by dividing the summation of node degree differences in LAj by ∑i=1djdj2+dj(i)2,

Because of the normalization by the denominator ∑i=1djdj2+dj(i)2 instead of the denominator dj in LAj, SLAj is scale-invariant.

Fig 1 shows the effect of standardizing the coefficient in a dynamic financial network constructed using financial returns data. More details of the financial network construction are described in the section on “Data Collection”. The coefficient LAj is an unbounded coefficient, whose value is scale-dependent on the difference in degrees between node j and its neighboring nodes. Values of SLAj, on the other hand, lies within the range of 0 ≤ SLAj ≤ 1 as |dj-dj(i)|2≤dj2+dj(i)2. This provides a neater summary description of the network. According to the construction in Eq (4), when SLAj approaches one, node vj will be highly disassortative. When SLAj approaches zero, node vj will be highly assortative. To facilitate the comparison with the global assortativity, GA, we define the standardized local assortativity of node vj as

making -1≤Qj(s)≤1, with larger/smaller Qj(s) implying that node vj is more locally assortative/disassortative. In an extreme case where dj > 0 and all dj(i) are zeros implying that node j is highly dissortative, Qj(s)=-1. In another extreme case where dj > 0 and dj = dj(i) implying that node j is highly assortative, Qj(s)=1. In this paper, we will continue with this defined standardized local assortativity coefficient, Qj(s), to investigate network structures in the multiple examples, including the one shown in Fig 1.

Data collection

To demonstrate the significance of the standardized local assortativity coefficient, we made use of three networks from different fields of studies: (1) the Lancichinetti-Fortunato-Radicchi benchmark (LFR benchmark), (2) a Facebook ego, (3) a scale-free network, and (4) dynamic financial networks. The LFR benchmark network has 1000 nodes and was generated with the parameters shown in Table 1. We obtained the open Facebook ego network data from the Stanford Network Analysis Project, and it contains survey data from anonymized Facebook participants [36]. Each node represents an individual with their own profile attributes, and edges are the pairwise affiliations between individuals. To construct the dynamic financial networks, we used two data sources from Reuters. We collected the daily adjusted closing price of the constituents of five selected worldwide stock market indices, using the Reuters Terminal API: 1) Hong Kong’s Hang Seng Index (HSI50), 2) the United Kingdom’s Financial Times Stock Exchange 100 Index (FTSE100), 3) the United States’ Standard and Poor’s 500 (SP500), 4) France’s CAC40, and 5) Germany’s DAX30, from January 2018 to June 2023. Because the constituents of the indices might change over time, we manually collected a list of all historical constituents of the indices from the Reuters terminal. Only a few records could not be mapped due to unlisted stock information, and we were able to use the remaining 1158 constituents at a different time t for further analysis in our study.

Construction of financial networks

Using the daily closing price Pj,t of stock j on trading day t, we calculated the return of stock j at time t by

We defined dynamic undirected networks Gt = (Vt, Et) [30], where Vt is the set of nodes and Et is the set of edges at time t. We calculated the sample Pearson correlation between stocks i and j at time t, ρij,t, using a method often referred to as a “rolling-window” approach [37], and collecting the past 21 days of historical observations (Yi,t−k, Yj,t−k) for k = 0, …, 20. To construct a financial network at time t, an edge eij,t between two nodes vi and vj is created if ρij,t between stocks i and j at time t is greater than 0.5 [30].

The rolling-window approach enables the use of the most recent information to represent the network information at time t.

Network statistics

We made use of the degree of node vj at time t, dj,t, to investigate whether it could complement the local assortativity coefficient to predict future losses in dynamic financial networks. The degree dj,t describes the number of connections in the network Gt,

The network density in Table 3 was calculated as

which is defined as the total number of edges in the network Gt divided by the total number of possible edges with Vt nodes. We also followed [3, 30] to calculate the global assortativity coefficients listed in Table 3, as in Eq (10).

The global assortativity coefficient in Eq (10) can be interpreted as the degree correlation. It is calculated by using all pairs of degrees whose nodes are linked together to calculate the correlation coefficient. Therefore, it describes how strong degrees of linked nodes are related linearly with each other in the whole network (globally), rather than describing the local assortativity feature by Qj(s). In other words, the global assortativity coefficient is dominated by nodes with high degrees. In addition, if nodes that are highly assortative/disassortative have high degrees, the global assortativity tends to be positive/negative. On the other hand, the scale-invariant Qj(s) measure will not be affected by the scale of the node degrees, thus making the distribution of Qj(s) more objective in determining the assortativity of networks than the global assortativity coefficient is.

Correlation between standardized local assortativity and future volatility

With the daily closing price Pj,t of stock j at time t, we calculated the k-period return k at time t,

where k = 1, 2, 3, …. We correlated Qj,t(s) with the absolute k-period return, |Yj,t+k| of stock j at time t, with k = 1, …, 10 to investigate the predictability of Qj,t(s) for future volatility using |Yj,t+k| as a proxy:

For each combination of j, k and t, we estimated qjk,t using the sample correlation based on the paired data of the past 21 days, that is, (Qj,t-h(s),|Yj,t-h+k|), h = 0, …, 20, where h denotes a time lag within the past 21 days. Then, for each k, we produced the distribution of the correlation in Fig 5 using estimated qjk,t.

Multiple correlations between standardized local assortativity and future losses

To investigate the predictability of future losses using the standardized local assortativity, we defined the loss of stock j at time t + 1 as

With that, we obtained the multiple correlation between Qj,t-l(s), l = 0, ⋯, N, and the loss Lj,t+1 by

where y denotes Lj,t+1, x denotes the predictor variables Qj,t-l(s) for l = 0, ⋯, N, depending on the number of lags N to include, Sxy denotes the sample covariance matrix between x and y, Syx denotes the transpose of Sxy, Sxx denotes the sample covariance matrix between the predictor variables, Sxx-1 denotes the inverse of Sxx, sy2 denotes the sample variance of y, Rxy denotes the sample correlation matrix between x and y, Ryx is the transpose of Rxy, Rxx denotes the sample correlation matrix of the predictor variables, and Rxx-1 denotes the inverse of Rxx. For each stock j and time t, the sample covariance and correlation matrices were calculated using data from the past 21 days, that is, (Qj,t-h-n(s),n=0,⋯,N) and |Lj,t−h+1|, for h = 0, ⋯, 20. We present the distribution of the multiple correlation for N = 0, ⋯, 5 in Fig 6.

With the additional inclusion of network statistics, dj,t, we calculated the multiple correlations between Lj,t+1 and Qj,t-l(s), dj,t−l, l = 0, ⋯, N, using Eq (14), except x here denotes variables Qj,t-l(s) and dj,t−l for l = 0, ⋯, N.

Conditional multiple correlations between standardized local assortativity and future losses given Qj,t(s)>γ

The conditional multiple correlations between Qj,t-l(s), l = 0, ⋯, N, and the loss Lj,t+1 given Qj,t(s)>γ were also calculated using Eq (14) but ignoring those days t in which the condition Qj,t(s)>γ was not satisfied. In this paper, we consider γ = 0.6 as the threshold of high standardized local assortativity.

Static networks

To compare the global and local assortativity coefficients, we made use of three static networks in this study: the Lancichinetti-Fortunato-Radicchi (LFR) benchmark, a Facebook ego network, and a scale-free network.

Lancichinetti-Fortunato-Radicchi (LFR) benchmark

The LFR benchmark is a computer-generated network with prior known communities for comparing community detection methods. It is considered a proxy of real networks as it is capable of generating heterogeneous distributions of node degrees that follow the power law and the community size [38]. As such, generated networks will have assortative groups, depending on parameter settings. Fig 2a shows an example of a generated LFR benchmark network with 1000 nodes and 4850 edges. Summarized in Table 2, this network had an assortativity coefficient of −0.0002, suggesting that the benchmark network was neither assortative nor disassortative, despite the visible observation of closely related clusters in the network at a high level.

Facebook ego network

The second network we considered was a network of social circles, or friend lists, from Facebook. It contains anonymized data from surveyed users [39]. We chose ego networks for our research because they are especially known for being assortative in some common social attributes, such as gender, age, and political views. Respondents, or egos, are asked to list the people that are directly linked to them, or alters, and to further describe the social relationships between them based on a variety of attributes. The extracted data set contained 10 combined ego networks, each of which if looked at separately showed a fairly strong assortative pattern, with GA between 0.084 and 0.503. This aligns with what the current literature suggests that social networks tend to be assortative in nature [21]. The combined network, which contained 4039 nodes and 88234 edges, as shown in Fig 2b, had a global assortativity of 0.0636 (Table 2), despite it containing multiple assortative sub-groups of individuals. This drastic change indicates that the global measure GA might not be an adequate coefficient for identifying local patterns in enormous networks.

Scale-free network

We constructed a scale-free network using a standard preferential attachment (the Barabási-Albert model) to examine its assortativity properties. We set the number of edges that were added from a new node to existing nodes in each step of the network’s growth to be 5. Fig 2c shows the scale-free network with the global assortativity in Table 2 of -0.0022, which is close to zero. Here, therefore, the global measure GA shows no obvious assortative or disassortative trend globally.

Application of Qj(s) on the three networks

With the limitations of GA in mind, we calculated Qj(s) for each node vj for the LFR benchmark, the Facebook ego network, and the scale-free networks. The second column of Table 2 shows the median values of Qj(s) for the three networks. The median Qj(s) values were 0.3718, -0.2110 and -0.1382 for the LFR benchmark, the Facebook ego network, and the scale-free network, respectively, while the Facebook ego network showed slightly more non-assortative mixing among its nodes. If we look at the Qj(s) distributions in Fig 2d, we can observe that the LFR benchmark network is comparatively more skewed to the left, with more nodes showing local assortative patterns, and the proportion of positive values of Qj(s) was 85%. On the other hand, the Qj(s) distribution of the Facebook ego network revealed a small portion of nodes with an assortative pattern, such that the proportion of positive Qj(s) was approximately 38%. Other nodes appear to have fairly uniform distributions except for the nodes that had a Qj(s) value of nearly −1. With Qj(s)=-1 representing high disassortativity, the nodes included in the leftmost bar in the Qj(s) distribution of the Facebook ego networks were highly unlikely to connect with other nodes with similar degrees. These nodes could potentially be the egos and mutual friends that acted as “hubs” linking diverse groups of friends in the network. In the scale-free network, the distribution of Qj(s) is more symmetrical than it was in the LFR benchmark and Facebook ego networks. The median Qj(s) was -0.1382, suggesting a locally disassortative mixing pattern. In other words, there was small tendency for nodes to connect to others with a different degree from them though GA is close to zero. Table 2 also presents the mean values of Qj(s) which are consistent with what we observe from the median Qj(s).

Dynamic financial networks

The same limitation of a global coefficient is believed to apply to financial networks. One interesting feature of financial networks being added to the picture is that these networks are dynamic, that is, changes in the stock prices and indices occur within days, if not hours [17, 19]. The highly varying nature of the dynamic in financial markets leads to constant changes to the network structures and it is more challenging, but still important, to capture them.

As an illustration, Fig 3 shows networks on several different trading days t. Table 3 presents the network density, global assortativity and the median Qj,t(s) of the financial networks at the five selected time points. The standardized local assortativity Qj,t(s) was obtained from the dynamic financial networks, as detailed in the Materials and Methods section. In Fig 3a, the network on 13 March 2020 exhibited an unusually high network density with the lowest global assortativity among the selected dates. The financial network became less dense as time went by, and eventually separated into two big clusters at the end of 2020, as shown in Fig 3c. The network at that time had the lowest network density, but the highest global assortative mixing compared with the other networks. The first selected date of 13 March 2020 was two days after the World Health Organization (WHO) had declared COVID-19 to be a global pandemic. On that day, with the financial market greatly affected by the COVID-19 pandemic, the global assortativity (0.14) was very different from the median standardized local assortativity, Qj,t(s) (0.86). This demonstrates that our standardized local assortativity contains supplementary information on network assortative mixing. On the other two selected dates in 2020 when the financial market was relatively stable, the global assortativity and the median standardized local assortativity are relatively more consistent with each other. Table 3 also shows the network statistics for two selected dates in 2022, when we again note a large discrepancy between the global assortativity and the median Qj,t(s). Possibly due to the tensions in Ukraine, the worry of economic recession and interest rate increases, the global financial markets dropped to their lowest points in the year around these two dates. In the three dates when the GA and the median Qj,t(s) were very different, the mean Qj,t(s) has value smaller than the corresponding median because the distribution of Qj,t(s) was skewed to the left. Fig 3 and Table 3 thus demonstrate that when the financial market is very volatile due to an unexpected shock, global assortativity and Qj,t(s) can be largely inconsistent, and the global assortativity may indicate spuriously low assortative mixing among stocks. Fig 4 presents the MSCI World Index, the proportion of Qj,t(s)>0.6 at each time point, the time series of the global assortativity, and the median and interquartile range of Qj,t(s). There was severe downside movement in the MSCI in January, December 2018, March 2020 and June, October 2022, triggering potentially high systemic risk. In the five downside movements, the median and lower and upper quartiles of Qj,t(s) increased sharply. The proportion of Qj,t(s)>0.6 also increased quickly when there was an extreme market adjustment. Even more striking observations were in March 2020 during the outbreak of COVID-19, when we note that the median and both quartiles of Qj,t(s) were above 0.8. There were also obvious peaks in the proportion of Qj,t(s)>0.6 in June and October 2022. Together, these findings motivate us to anticipate that Qj,t(s) could provide useful information for tracing systemic risk in financial markets, and thus to consider the empirical study we discuss next.

With the daily closing price Pj,t of stock j on trading day t, we defined the return k periods ahead as Yj,t+k = log(Pj,t+k) − log(Pj,t) as described in Eq (11). Then, we correlated Qj,t(s) with the absolute k-period return, |Yj,t+k|, where k = 1, …, 10 to investigate the predictability of the volatility and future losses using the local assortativity feature. We were interested here in the volatility of the stocks, and we correlated Qj,t(s) with |Yj,t+k| because in statistical terms, volatility refers to the standard deviation of financial returns. Unfortunately, the standard deviation of a random variable (in our case the financial returns) is not observable. Therefore, to quantify the market fluctuation, we used the absolute returns here which can be regarded as proxies of the fluctuations [40]. Correlating Qj,t(s) with |Yj,t+k| for stock j would provide evidence of how the local feature is related to volatility of a stock [41]. It can be observed in Fig 5 that in our case Qj,t(s) and |Yj,t+k| were not linearly correlated. Regardless of the lag k under observation, Qj,t(s) does not seem to provide significant predictive power for future absolute returns and thus for volatility.

Qj,t(s) and future losses

Instead of correlating Qj,t(s) with the absolute return |Yj,t+k|, we next focused only on trading days t + 1 when a loss occurred. In other words, we explored the multiple correlations between the local coefficient Qj,t-l(s) for l = {0, ⋯, 5} on trading day t − l, and the loss on trading day t + 1, denoted as Lj,t+1 in Eq (13). In this investigation, “l + 1” represents the number of lagged variables in the vector (Qj,t-l(s),…,Qj,t(s)) to correlate with Lj,t+1. For example, if l = 2, we study the multiple correlation of the previous three standardized local assortativity (Qj,t-2(s), Qj,t-1(s), Qj,t(s)) to correlate with Lj,t+1. The network degree of node vj at time t − l, dj,t−l was added alongside Qj,t-l(s) to investigate whether the network statistics could complement each other and provide additional predictive power to future losses.

The density ridge plot in Fig 6 suggests a moderately positive correlation between Qj,t-l(s) and Lj,t+1. A stronger correlation with Lj,t+1 emerged when more lags were included, in contrast to the previous case when we correlated Qj,t(s) with |Yj,t+k| in Fig 5. The standardized local assortativity seemed to be more related to actual financial losses than to market volatility.

When dj,t−l was added into the equation, the strength in correlation increased further. For example, the medians of the multiple correlations were 0.130 and 0.078 for l = 0, and 0.221 and 0.147 for l = 5, with and without dj,t−l, respectively. Combining the degree of nodes and the standardized local assortativity can give a median multiple correlation as high as 0.265, thus indicating mild predictability of the network statistics on financial losses.

Conditioned Qj,t(s) and future losses

The correlation between Qj,t(s) and Lj,t+1 has suggested that the standardized local assortativity of financial networks tends to correlate with future loss. In Fig 4, the top panel gives the proportion of stocks with Qj,t(s)>0.6 on each day. When this proportion of locally assortative stocks with Qj,t(s)>0.6 is high, the financial market tends to adjust downward. Taking this one step further, we calculated the multiple correlations of Qj,t(s) and Lj,t+1 on days t, conditioned on Qj,t(s) being greater than a threshold of 0.6. The main purpose was to pick out days when stock j exhibited a highly assortative pattern locally. When this occurred, we wanted to know whether the standardized local assortativity would help in detecting future losses more accurately.

Fig 7 displays a view similar to Fig 6, except the correlation was conditioned to when Qj,t(s) was greater than 0.6. The results show a drastic increase in the multiple correlations between future loss and the respective lagged Qj,t-l(s). As with the previous graph, we observe that the more lags we included, the stronger the positive correlation was. However, we also see that the distributions were heavily skewed when compared with those in Fig 6. On the left panel, when only Qj,t-l(s) with five lags were included, the median of multiple correlations had increased to approximately 0.32. On the other hand, when dj,t−l was added to the picture, in the right panel the median of multiple correlations was already close to 0.4 when five lags were included, suggesting that Qj,t-l(s) and dj,t−l for l = {0, ⋯, 5} may help predict future losses when the node at time t exhibits assortative behaviors. From the risk management perspective, the inclusion of a large number of lags is generally not recommended; however, the experiments show the potential predictive power for future losses by using the standardized local assortativity coefficient in dynamic financial networks.

Discussion

With increasingly complex networks and the presence of clusters with their own degree of assortativity, we see the inadequacy of using a single global measure to quantify the assortative mixing of networks. We have shown a few networks with social and financial data in this research, all of which have indicated that local patterns are smoothed out if we simply use a global assortativity coefficient to summarize the networks.

A few research studies have suggested the use of local assortativity [34, 35]. However, there has not been a universal definition of how local assortativity should be calculated. Furthermore, the downside of the definition proposed by [35] faces the issue of unbounded upper ranges as the network size grows. To handle that shortcoming, we propose a standardized version of the previously defined local assortativity, with the coefficient lying within the range of -1 and 1, thus providing a neater summary representation of networks.

In this paper we tested the standardized local assortativity coefficient on several networks, from a generated community detection benchmark, to a real-world Facebook ego network, and to dynamic financial networks. In our LFR benchmark network, the global coefficient suggested neutral assortative mixing despite the presence of communities in the network. Facebook’s social network also showed that the global coefficient failed to capture the assortative nature of an ego network. The application of a standardized local measure, as is shown in Fig 2d, displays a more granular distribution for these networks. The comparison in Table 3 demonstrates a substantial discrepancy between the median Qj,t(s) and the global assortativity in March 2020, signifying incredibly high systemic risk when the COVID-19 pandemic began to have a great impact on the world economy and financial markets. As is more evidence in Fig 4, the proportion of stocks having high values of Qj,t(s) greater than 0.6 in our case might provide good early warning signals on severe market downturns.

Standardized local assortativity was also seen to provide predictive power in the chosen financial networks. Whereas the global assortativity at time t was unable to predict the future absolute returns, a proxy as future volatility, Qj,t-l(s) did provide a higher predictive power in the case of a loss situation in the future. The standardized local assortativity coefficient Qj,t-l(s), when in combination with other network statistics (i.e., network degree in our case), showed even higher predictive power for the future performance in terms of loss. The correlation became stronger when more lags of Qj,t-l(s) were included. To further test out scenarios when networks were showing highly assortative patterns, we selected networks with Qj,t(s) above the threshold of 0.6, and the predictability became even more obvious. When we included only the one lag of Qj,t-l(s), the median correlation was already as strong as when we included five lags where no conditioning of Qj,t(s) was applied, and the correlation continued to increase up to a median value of approximately 0.3 when five lags were included. The network degree provided an extra boost to the correlation when it was included: the median of multiple correlations went from approximately 0.25 if only one lag was included, to nearly 0.5 with five lags of Qj,t-l(s). The results thus suggest the possibility of gaining extra insights from micro-features for financial risk management, if we construct networks out of the financial stock prices and observe the dates when the networks are highly assortative locally, we potentially will be able to foresee future losses and take preparedness action for risk management.

We saw that the standardized local assortativity coefficient complements other network statistics: when it is included, its correlation becomes stronger with future losses in dynamic financial networks. Whereas other network statistics have their own significance, we advocate the use of a local assortativity coefficient because it provides a summary of both the homophilic and resilient characteristics of a network.

We also may be able to draw implications for financial risks from the local assortativity coefficient. When nodes are more locally assortative, it implies that stocks will have a higher coherence of degrees with their neighbor nodes in the network, and we could use that coherence to provide early warning signals of severe market downturns or to predict a loss situation in the foreseeable future, as is shown in Figs 4 and 7. The visualization in Fig 3a shows that this high coherence of degrees or local assortative mixing can occur locally in many clusters that global assortativity cannot capture. The big mismatch of global assortativity and standardized local assortativity could be a potential indicator to inform us of unusual homophilic characteristics forming in a network in the future, and providing an early signal of increased systemic risk in financial markets. Further research needs to be conducted next on the modeling of local assortative patterns on financial risks.

Conclusions

This study applied the standardized local assortativity coefficient on generated benchmarks in social networks, and in dynamic financial networks, but we believe it is possible that this concept can be extended to other networks in other fields, such as biology, epidemiology, and technology, to take into account the local features that otherwise can be easily omitted in complex networks. The results from the financial networks have suggested the importance of micro-features in risk management, and we believe that the same characteristics can be also found in other fields where groups of local nodes might help us answer important questions. Fig 8 shows the number of confirmed COVID-19 cases, deaths due to COVID, the global assortativity and the local assortative coefficient Qj,t(s) obtained from the dynamic pandemic networks in [30]. When the COVID-19 pandemic was evolving quickly from March to July 2020, most of the values of Qj,t(s) were larger than the corresponding global assortativity was. After July 2020, the trend of the median Qj,t(s) and the trend of the global assortativity were similar. By following the trends of the standardized local assortativity of the dynamic pandemic networks especially when the local and global assortativities are very different, we may find hints as to when the COVID-19 pandemic is likely to be over.

Data Availability

The COVID-19 dataset, which includes daily case and death counts for each of 164 countries, is publicly accessible on Zenodo at https://doi.org/10.5281/zenodo.8255535. Financial data, encompassing daily prices and constituents, can be accessed from Reuters Pricing and Market Data through the company’s subscription service. There are two ways to gain access: download the daily prices and list of constituents from the Reuters Terminal (now known as the Eikon desktop) or subscribe to Refinitiv’s Reuters Pricing and Market Data.

Funding Statement

This work was partially supported by the Hong Kong RGC Theme-based Research Scheme, grant number T31-604/18-N and The Hong Kong University of Science and Technology research grant “Risk Analytics and Applications” (grant number SBMDF21BM07). The funding recipient was MKPS. The funder had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. There was no additional external funding received for this study.

Data Availability Statement

The COVID-19 dataset, which includes daily case and death counts for each of 164 countries, is publicly accessible on Zenodo at https://doi.org/10.5281/zenodo.8255535. Financial data, encompassing daily prices and constituents, can be accessed from Reuters Pricing and Market Data through the company’s subscription service. There are two ways to gain access: download the daily prices and list of constituents from the Reuters Terminal (now known as the Eikon desktop) or subscribe to Refinitiv’s Reuters Pricing and Market Data.