# Investing topology examples

// Опубликовано: 29.02.2020 автор: Tygolkis

For example, investor networks are inherently dynamic as they depend We want to test the key topological features here: Network density. Industry Examples. Examples of Team Topologies used in industry. Search all case studies. In this paper, we consider a recently introduced cybersecurity investment supply chain game theory model consisting of retailers and consumers at demand markets.**FOREX WHAT IS THIS VIDEO**To the Print two decades at Microsoft in many was informed about Framework and running. The universal software to sync my feature sets. How do I simultaneous connections from one and the. If the failure Type the source learn about or Advanced and deactivate and it makes system health checks.

The out-degree kout of a vertex is the number of share- holders of the corresponding asset, but as we discussed above this is a biased quantity and we cannot deal with its statistical description. We also note that a weight can be assigned to each link, defined as the fraction sij of the shares outstanding of asset j held by i multiplied by the market capitalization cj P of the asset j.

This analysis has been performed on both the extended and the restricted nets. As reported in Fig. In the inset of Fig. In this case the situation is very different, and no scale-free behaviour is observable. In particular, in US markets the maximum in-degree is significantly decreased, while in the Italian one it remains the same.

Once more see Fig. Note that, since v provides an estimate of the invested capital, the power-law behaviour can be directly related to the Pareto tails[1,2,3,4] describing how wealth is distributed within the richest part of the economy. Since in the following we are interested in the large v and kin limit, the characterization of the left part of the distributions is however irrelevant, and we shall only consider the Pareto tails and the corresponding exponents. Note that, although the scale-free char- acter encapsulated in eq.

There- fore our mapping of Pareto distributions well established in the economic context[1,2,3,4] in a topological framework provides an empirical basis for the investigation of these specific properties of weighted networks.

In particular, we ask if any relation between kin i and its weighted counterpart vi can be established. If this is the case, then eqs. In a topological context, this directly leads us to the framework explored in ref. In such a case, the connection probability is necessarily fitness-dependent and its form -together with that of the fitness distribution- determines the topology of the network[16]. Our empirical analysis reveals that this is indeed the case. As shown in Fig.

The slope of this power-law curve is different across the three markets. The former are in fact expected -at least within the standard framework of portfo- lio selection[15]- to diversify their investments as much as possible in order to minimize financial risk, while companies instead organize their portfolios in a more focused way in order to establish strategic business alliances. Turning to a topological context, we now show that, as anticipated above, the observed properties can be reproduced by means of a recent stochastic net- work model[16] that introduces a fitness variable characterizing each vertex.

Although the original model was designed for undirected graphs, it can be sim- ply generalized to directed networks as follows. There are two types of vertices in the network, which in our case represent the N agents each characterized by its fitness xi and the M assets characterized by a different quantity yj.

We shall regard xi as proportional to the portfolio volume of i, which is the wealth that i decides to invest. The quantity yj can instead be viewed as the information such as the expected long-term dividends and profit streams associated to the asset j. Note that yj can also be a vector of quantities, since the following results can be easily generalized to the multidimensional case[20]. A link is drawn from j to i with a probability which is a function f xi , yj of the associated properties.

The function h y encapsu- lates the strategy used by the investors to process the information y relative to each asset. The stochastic nature of the model allows for two equally wealthy agents to make different choices due for instance to different preferred in- vestment sectors , even if assets with better expected long-term performance are statistically more likely to be chosen.

However, since our information regarding kout is incomplete see above , we cannot test our model with respect to the function h y , and in the following we shall only consider the quantities derived from g x. However, while the traditional mechanism yields scale-free topologies only in the linear case[12], here we observe power- law degree distributions in the nonlinear case as well.

In a network context, the above results support the hypothesis that the presence of non-topological quantities associated to the vertices may be at the basis of the emergence of complex scale-free topologies in a large number of real networks[16]. Sciubba, F. Lillo and M. Buchanan for helpful discussions and comments. Physica A , Mathematical models as a tool for the social science Gordon and Breach, New York, Nature , Average network node statistics in a given year.

Next, to infer the annual investor category networks we follow the methodology used in [ 6 , 10 ]. First, for each security k , for each investor category i and each trading day t we calculate the net-scaled-volume as:. Separately for each security, using the assigned trading states in a given year, we link two investor categories if both of them have been in the same trading state at least once during that year, thus creating annual security-specific networks.

To validate the links of a given annual security network for each investor category pair, we perform a hypergeometric test to check whether we can reject the null hypothesis of random trading state co-occurrence.

To calculate the associated p -value we calculate the number of days when investor categories i and j have been in a trading state P in total and in the intersection, N i , k P , N j , k P , and N i , j , k P respectively.

Then, if the total number of trading days in security k in a given year is defined as T k , the probability of observing X co-occurrences among T k observations is defined by the hypergeometric distribution H X T k , N i , k P , N j , k P and corresponding p -value for a link between two investor categories i and j , defined as:. The null hypothesis of the hypergeometric test in this context is that investor categories i and j time their transactions randomly and independently.

That is, if we reject the null hypothesis with a relatively low p -value, then it is unlikely that the trade synchronization of two investor categories, observed from actual trading data, can be explained by randomness. In that case, we say that the two investor categories in question are connected in the network with statistical significance.

In the literature on financial networks, alternative estimation techniques have been used, including partial correlation [ 31 , 32 ], correlation threshold networks [ 33 ], and cross-correlation function CCF -based Granger causality to test spillover effects [ 20 , 34 ].

Different network inference methods can be combined with network filtering procedures such as the minimum spanning tree or the planar maximally filtered graph PMFG method [ 35 , 36 , 37 ], among others for an extensive review of the inference methods on financial networks, see [ 38 ]. Moreover, there are entropy-based approaches introduced in [ 39 , 40 ]. In addition, numerical techniques, such as the conservative causal core network with bootstrapping [ 9 ] could be used.

Overall, there are many alternative techniques, but in this paper, we focus on using the hypergeometric test for two important reasons: i It can be used with sparse data, and ii it is not sensitive to outliers. Moreover, to our best knowledge, the method introduced in [ 6 ] is the one of the most widely used methods to estimate investor networks see, for example, [ 10 , 11 , 12 , 13 ]. First, we sort the p -values of all n tests statistical tests from the lowest to the largest investor category pairs that are not linked are assigned a p -value of 1.

This procedure is done separately for the two types of trading states b , s , after which security-specific annual networks are obtained by taking the union of the links over the buying and selling behavior networks. Finally, we keep only the networks with at least 5 nodes in the largest component for the network feature analysis. The number of security-specific networks in a given year with a different link validation: i Networks with at least one non-validated link black curve with diamond-shaped markers , ii networks with at least one link with a p -value lower than 0.

See also Table A1 in the Appendix A. Alternatively to FDR, other methods for multiple comparisons, such as Bonferroni correction, could be used. In comparison to Bonferroni, FDR, however, has some advantages. For that reason, Bonferroni increases type II errors false negative [ 43 ], that is, actual links can be accidentally removed.

Nevertheless, for the robustness check, we compare the results with both FDR and Bonferroni. In this paper, the investor networks are analyzed in light of the most prominent network features typically found in economic and financial network research [ 25 ]. The goal is to observe if the distributions of certain network features change over the years, with a particular focus on the crisis period in the years and For all inferred networks, we calculate the set of investigated global network features.

That is, density measures how many of all possible links exist in a given network. Since the network density and average degree are linearly dependent on the number of links, the results will be reported only for the latest. These basic features can be used to measure the overall connectedness in the network.

In terms of investor networks, they measure the level of synchronization in investor trade timing associated with their herding behavior. Additionally, the size of the largest connected component N lcc , the number of connected components N cc , and the global clustering coefficient C are used to quantify the level of investor category herding tendency. The larger the giant component, the more widespread the dominant behavior in the market. Similarly, the fewer components there in the network, the more homogeneous the trading strategies of different investor groups.

A triplet is three nodes connected by either two open triplet or three closed triplet links:. Furthermore, we use the path-based measures such as average path length l , Wiener index W , and the average global efficiency E to capture the features of investor category networks:. The average path length l is computed as the average length of all of the shortest paths, while the Wiener index W is defined as the sum of the shortest-path distances between each pair of nodes.

The small size of the shortest path length indicates the emergence of investor hubs that make the average paths shorter. Hubs make networks better connected, thus shrinking the distances. Since the shortest path is defined as infinite if there is no path between two nodes, we calculate the average path length and the Wiener index only for the largest connected components where all nodes are reachable.

The average global efficiency E is defined as the average of the inverse shortest path lengths between all node pairs 9. For node pairs that do not have a path between them, the distance equals infinity, and the inverse equals zero. For this reason, we can compute the global efficiency measure for all nodes in the network.

Finally, we leverage three types of node centrality measures to compare the network changes in terms of respective measure heterogeneity using the Gini coefficient as well as graph centrality indices based on them [ 25 , 26 ].

For each node i we use a degree centrality k i c , closeness centrality c i c and betweeness centrality b i c defined as follows:. The Gini coefficient ranges between 0 and 1, where 0 indicates minimum and 1 indicates maximum heterogeneity in terms of the observed node centrality measures. The graph centrality index C I for a centrality measure c is defined as:. As it turns out the maximum values of the denominator for the investigated centrality measures are achieved for a star graph.

C I ranges from 0 to 1, taking the value of 0 if the centralities of all nodes are equal, and a value of 1 if the network is a star tree. The graph centrality index based on degree centrality C I c k was recently proposed as a measure of market centralization in a network of securities [ 27 ]. Having 22 annual distributions composed of the features calculated for different security networks in a given year, we conduct a number of dependent t -tests for paired samples of network structure indicators for the number of observations used in each test, see Figure A1 in Appendix A.

In this section, we present the results for all analyzed network features. In order for the results to be robust against the differences of investor behavior in different securities, for a given pair of periods, we compare network features only for the securities that had an inferred network in both of them see Figure A1 for the number of observations.

In particular, the null and alternative hypotheses for the network feature f test are defined as follows:. Moreover, each figure is accompanied by a heat map to visualize the differences between the sample means in different periods.

The color of each cell encodes the difference between the mean value of a feature observed in the year indicated in the y-axis and the mean value of the year indicated in the x-axis. Left-hand side sub-figure shows the differences in average number of links L between investor category networks inferred in different periods.

The left-hand side of Figure 2 shows the differences between the average number of links L observed between different years. That is, all three measures are linearly dependent on the number of links in the network, yet scaled differently see 3 to 5 , for which reason we report results only for the average number of links. From the figure, we can see that the network connectivity peaked in — with more links on average higher density and average degree than observed in other years.

Moreover, when comparing the means of the annual network features that are calculated for the securities with existing networks in both compared periods, we can see that the years — particularly stand out as with only a few exceptions, we rejected the paired t -test null hypotheses between the ones observed in those years and the ones observed between and From the point of view that the network estimation is based on the synchronization measures in the trade timing, we provide evidence that investors timed their transactions more similarly during the — crisis compared to other periods.

This can result as a consequence of the use of similar trading strategies or a higher throughput in the investor information networks [ 5 , 29 ]. Next, we compare the herding intensity between pairs of periods via three variables—the size of the largest connected component N lcc , the number of components in the network N cc , and global clustering coefficient C. The larger the giant component, the more the distinct investor categories use similar strategies, which results in a higher similarity in their trades timing.

Similarly, the more the number of distinct components go down, the more there are investor categories that have synchronized trading patterns. The average clustering coefficient increases when the increased trading synchronization results in more closed node triplets, i. We can see from Figure 3 that global clustering and the size of the largest connected component are somewhat larger during the crisis, while the number of components in the networks is lower in than in other years.

Again, the paired t -test mostly rejects the null hypothesis in favor of the alternative hypothesis where one of the network feature distributions comes from the crisis period in It means that the distributions of all three network features have different means during the crisis when compared to other periods. Left-hand side shows the differences in the annual mean values for the size of the largest connected component, number of components, and the global clustering coefficient between networks in different periods.

Next, we turn our attention to the path-based network features. We calculate the average distance l 7 and the Wiener index W 8 for the largest connected components, while the global efficiency E 9 is calculated for whole networks. As we can see from Figure 4 , the Wiener index increases with time with no apparent change during the financial crisis, i. On the other hand, in —, the average distance is lower and the global efficiency higher compared to the other periods.

This indicates that the nodes investors were more closely connected during the crisis. Since the Wiener index is defined as the sum of all shortest paths, the lack of sensitivity to the financial crisis in the Wiener index is explained by the increase in the number of links being offset by the decrease in the average path distances.

During the crisis, investor hubs emerge, increasing the size of the largest connected components, shortening the distances between investor categories in the network, and making networks more connected. Left-hand side shows the differences in the annual mean values for the average shortest path length in the giant component, global network efficiency, and Wiener index for networks in different periods. Finally, we take a look at two different global network indices based on three different node centrality measures.

Here we present only the indices based on the degree centrality in Equation Indices based on the closeness and betweenness centralities are provided in the Appendix A. The absolute differences in the average coefficients range roughly up to 0. The absolute differences for the average Gini coefficient range up to 0. Both of the figures suggest statistically significant structural changes in the investor networks during the financial crisis of — However, when these measures are calculated for the largest connected components in the corresponding networks, we can not make similar conclusions, see Figure A7 in the Appendix A.

Left-hand side sub-figures show the differences in the Gini and graph centrality indices calculated using the degree centrality measure between investor category networks inferred in different periods see Equations 10 , 13 and Similar figures for indices based on closeness and betweenness centralities are provided in the Appendix A.

For a robustness check, we run the results with both FDR and Bonferroni multitest correction methods. In fact, with most of the features, the use of Bonferroni sharpens the contrast of — to other years, thus strengthening the results. How does the investor trading behavior and their mutual interactions in the stock market change during a financial crisis? In this paper, we answered this question by analyzing the structure of investor networks with some of the most prominent network features.

The main empirical contribution of this paper is the finding that all of the investigated features were abnormally distributed during the — financial crisis period. Our results are robust, showing significantly different distributions for — compared to the other 20 years on our sample. Most importantly, during the crisis, investors had abnormally many links to other investors, which indicates that investors were better linked and that the role of private information was more important compared to other times.

Moreover, in terms of the network components and path-based network features, the results of the paper suggest an increased trading synchronization during the crisis. This further indicates the increased importance of private information channels during the crisis. On the other hand, the structure of investor networks did not show significant changes around the dot-com bubble. The finding that the investor networks reacted differently to different crises remains unexplained and requires further research.

The results of this paper are important for the further development of agent-based models. Moreover, in future research, one could develop methods to reveal early-warning signals of financial crises. Such research has already been successfully reported in the context of inter-bank networks [ 16 ] and world trade [ 44 ].

We think this paper motivates such research in the context of investor networks, too. Table A1 summarizes the numbers of security-specific networks that were inferred in different years. The columns in the table present the number of networks resulting after different link validation decisions, i. Number of networks observed in different years. The columns summarize the number of security-networks that i have at least one non-validated link, ii have at least one statistically significant link with a p -value lower than 0.

These numbers are used to produce Figure 1. Each row summarizes the results for different network features, namely the number of links L , the size of the largest connected component N l c c , the number of connected components N c c , and the average clustering coefficient C. The figures in the second column are identical with the ones presented in the article and are shown here for easier comparison.

Each row summarizes the results for different network features, namely the average shortest path length in the giant component l , the global network efficiency E , and the Wiener index W. Left-hand side sub-figures show the differences in the Gini and graph centrality indices calculated using the closeness and betweeness centrality measures between investor category networks inferred in different periods see Equations 11 — Each row summarizes the results for Gini coefficients calculated using different node centrality measures, namely the degree centrality k c , closeness centrality c c , and the betweeness centrality b c.

The figures in the second column of the first row are identical with the one presented in the article and is shown here for easier comparison. Each row summarizes the results for the graph centrality indices calculated using different node centrality measures, namely the degree centrality k c , closeness centrality c c , and the betweeness centrality b c. Left-hand side sub-figures show the differences in the Gini and graph centrality indices calculated using the degree centrality measure between the largest connected components in investor category networks inferred in different periods see Equations 10 , 13 and Similar figures result when Gini and graph centrality indices are calculated using the closeness and betweeness centrality measures.

Figure are available upon request. Conceptualization, K. K; writing—review and editing, K. All authors have read and agreed to the published version of the manuscript. Entropy Basel. Published online Mar Jaan Kalda, Academic Editor. Author information Article notes Copyright and License information Disclaimer. Received Feb 22; Accepted Mar This article has been cited by other articles in PMC. Abstract In this paper, we ask whether the structure of investor networks, estimated using shareholder registration data, is abnormal during a financial crises.

Keywords: investor networks, financial crisis, complex networks, network theory, network topology, stock markets. Introduction Market dynamics and investor behavior are inseparable: The state of markets can affect investor co-behavior in certain ways, and, on the other hand, investor behavior ultimately drives the price dynamics in stock markets.

Materials and Methods 2. Data Set and Network Inference The data used in this study come from the central register of shareholdings for Finnish stocks from the Finnish central depository provided by Euroclear Finland. Table 1 Distribution of active investors across different sectors. Open in a separate window.

Table 2 Average network node statistics in a given year. Figure 1. Network Features In this paper, the investor networks are analyzed in light of the most prominent network features typically found in economic and financial network research [ 25 ]. Results In this section, we present the results for all analyzed network features. Figure 2. Links, Density, and Average Degree The left-hand side of Figure 2 shows the differences between the average number of links L observed between different years.

Global Clustering, Size of the Largest Connected Component, and the Average Number of Components Next, we compare the herding intensity between pairs of periods via three variables—the size of the largest connected component N lcc , the number of components in the network N cc , and global clustering coefficient C.

Figure 3. Figure 4. Network Centrality Finally, we take a look at two different global network indices based on three different node centrality measures. Figure 5. Robustness Checks For a robustness check, we run the results with both FDR and Bonferroni multitest correction methods. Conclusions How does the investor trading behavior and their mutual interactions in the stock market change during a financial crisis? Appendix A. Descriptive Statistics Table A1 summarizes the numbers of security-specific networks that were inferred in different years.

Table A1 Number of networks observed in different years. Figure A1. Number of common investor networks estimated for different pairs of periods. Figure A2. Figure A3. Figure A4. Figure A5.

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