Power-Law Networks in the Stock Market: Stability and Dynamics

Power-Law Networks in the Stock Market: Stability and Dynamics VLADIMIR BOGINSKI, SERGIY BUTENKO, PANOS M. PARDALOS Department of Industrial and Systems Engineering University of Florida 303 Weil Hall, Gainesville, FL 32611 USA Abstract: We consider a representation of the stock market as a network, which is constructed by calculating cross-correlations between pairs of stocks based on the opening prices data over a certain period of time. This network is referred to as the market graph. We study the evolution of the structural properties of the market graph over time and draw conclusions regarding the dynamics of the stock market development based on the interpretation of the obtained results. Key-Words: Stock prices, Market graph, Cross-correlation, Degree distribution, Power law, Edge density, Evolution. 1 Introduction One of the challenging problems arising in the modern finance is finding efficient ways of summarizing and visualizing the stock market data that would allow one to obtain useful information about the behavior of the market. A large number of financial instruments are traded in the U.S. stock markets, and this number changes on a regular basis. The amount of data generated daily by the stock market is huge. This data is usually visualized by thousands of plots reflecting the price of each stock over a certain period of time. The analysis of these plots becomes more and more complicated as the number of stocks grows. An alternative way of summarizing the stock prices data, which was recently developed, is based on representing the stock market as a graph (or, a network). One can easily imagine a graph as a set of dots (vertices) and links (edges) connecting them. This graph is referred to as the market graph. A natural graph representation of the stock market is based on the cross-correlations of price fluctuations. The market graph is constructed as follows: each stock is represented by a vertex, and two vertices are connected by an edge if the correlation coefficient of the corresponding pair of stocks (calculated over a certain period of time) exceeds a specified threshold θ [ 1,1]. It should be noted that the approach of representing a dataset as a network becomes more and more extensively used in various practical applications, finance being one of them [1, 3 13, 15, 16]. The formal procedure of constructing the market graph is as follows. Let Pi () t denote the price of the stock i on day t. Then Ri() t = ln( Pi() t Pi( t 1)) defines the logarithm of return of the stock i over the one-day period from ( t 1) to t. If we denote by R i = R() t, 1 N N t= 1 i.e., the average return of the stock i over the considered period ( N denotes the number of days in the period), then the correlation coefficient between instruments i and j is calculated as [14] C ij = RR R R i j i j 2 2 2 i i j j R R R R i 2 (1) An edge connecting stocks i and j is added to the graph if > θ, which means that the prices of C ij these two stocks behaved similarly (to an extent depending of the value of θ) over the given period of time. Therefore, studying the pattern of connections in the market graph would provide valuable information about the internal structure of the stock market. In our previous research, we have investigated various properties of the market graph constructed using the data for 500 recent consecutive trading days in 2000-2002 [8]. Among the results of this work, we should mention the fact that the distribution of the correlation coefficients calculated for all possible pairs of stocks using formula (1) is

nearly symmetric with the mean approximately equal to 0.05, and its shape is somewhat similar to the shape of a normal distribution, however the distribution of cross-correlations is defined only over the interval [-1,1]. Another fundamental observation concerning the structure of the considered market graph is that for the values of the correlation threshold θ 0.2, its degree distribution is well approximated by the power-law model [2]. According to this model, the probability that a vertex has a degree k (i.e., there are k edges emanating from it) is P( k) k γ, (2) or, equivalently, log P( k) γ log k, (3) which shows that it can be plotted as a straight line in the logarithmic scale. An interesting fact is that besides the market graph, many other graphs arising in diverse areas also have a well-defined power-law structure [2, 3, 5 7, 9, 11 13, 15 16]. This fact served as a motivation to introduce a concept of selforganized networks [3, 5, 6], and it turns out that this phenomenon also takes place in finance. Another significant contribution of [8] is in a suggestion to relate some correlation-based properties of the market to certain combinatorial properties of the corresponding market graph. For example, the authors attacked the problem of finding large groups of highly-correlated instruments by applying simple algorithms for finding large cliques in the market graph constructed using a relatively large value of correlation threshold. A clique is defined as a set of completely interconnected vertices. Similarly, an independent set is a set of vertices which are not connected with any other vertex in this set. Therefore, an independent set in a market graph with a negative value of θ corresponds to a set of negatively correlated stocks, or fully diversified portfolios. The main purpose of the present paper is to reveal the dynamics of the changes in structural properties of the market graph over time. We consider the graphs constructed from the stock opening prices data observed in U.S. stock markets during different time periods within 1998-2002 and study the evolution of certain characteristics of these graphs. 2 Evolution of the Market Graph In order to investigate the dynamics of the market graph structure, we chose the period of 1000 trading days in 1998 2002 and considered eleven 500-day shifts within this period. The starting points of every two consecutive shifts are separated by the interval of 50 days. Therefore, every pair of consecutive shifts had 450 days in common and 50 days different (see Figure 1). Dates corresponding to each shift are summarized in Table 1. Period # Starting date Ending date 1 09/24/1998 09/15/2000 2 12/04/1998 11/27/2000 3 02/18/1999 02/08/2001 4 04/30/1999 04/23/2001 5 07/13/1999 07/03/2001 6 09/22/1999 09/19/2001 7 12/02/1999 11/29/2001 8 02/14/2000 02/12/2002 9 04/26/2000 04/25/2002 10 07/07/2000 07/08/2002 11 09/18/2000 09/17/2002 Table 1: Dates corresponding to each 500-day shift. Fig. 1: Time shifts used for studying the evolution of the market graph structure. The advantage of this procedure is that it allows us to accurately reflect the structural changes of the market graph using relatively small intervals between shifts, but at the same time we can maintain large sample sizes of the stock prices data for calculating cross-correlations for each shift. We should note that in our analysis we considered only stocks which were among those traded as of the last of the 1000 trading days, i.e. for practical reasons we did not take into account stocks which had been withdrawn from the market. However, these could be included in a more detailed analysis to obtain a better global picture of market evolution.

To verify this assumption, we have calculated the degree distribution of the graphs constructed for all considered time shifts. The correlation threshold θ = 0.5 was chosen. Our experiments show that the degree distribution is similar for all intervals, and in all cases it is well described by a power law. Figures 3 6 show the degree distributions (in the logarithmic scale) for some instances of the market graph corresponding to different intervals. As one can see, all these plots can be well approximated by straight lines, which means that they represent the power-law distribution, as it follows from formula (3). Fig. 2: Distribution of the correlation coefficients between all considered pairs of stocks in the market, for odd-numbered time shifts. The first subject of our consideration is the distribution of correlation coefficients between all pairs of stocks in the market. As it was mentioned above, this distribution was nearly symmetric around 0.05 and had a shape similar to a normal distribution for the sample data considered in [8]. The interpretation of this fact is that the correlation of most pairs of stocks is close to zero, therefore, the structure of the stock market is substantially random, and one can make a reasonable assumption that the prices of most stocks change independently. As we consider the evolution of the correlation distribution over time, it turns out that this distribution remains almost unchanged for all time intervals, which is illustrated by Figure 2. Fig. 4: Degree distribution of the market graph for period 4 (logarithmic scale). Fig. 5: Degree distribution of the market graph for period 7 (logarithmic scale). Fig. 3: Degree distribution of the market graph for period 1 (logarithmic scale). The stability of the correlation coefficients distribution of the market graph intuitively motivates the hypothesis that the degree distribution should also remain stable. The cross-correlation distribution and the degree distribution of the market graph represent the global characteristics of the market, and the aforementioned results lead us to the conclusion that the general structure of the market is stable over time. However, as we will see now, some global changes in the stock market structure do take place. In order to demonstrate it, we look at another

characteristic of the market graph its edge density. The edge density of a graph is the ratio of the number of edges in this graph to the maximum possible number of edges, which is equal to nn ( 1)/2, where n is the number of vertices in the graph. Fig. 6: Degree distribution of the market graph for period 11 (logarithmic scale). For studying the edge density of the market graph, we chose a relatively high correlation threshold θ = 0.5 that would ensure that we consider only the edges corresponding to the pairs of stocks, which are significantly correlated with each other. In this case, the edge density of the market graph would represent the proportion of those pairs of stocks in the market, whose price fluctuations are somewhat similar and correspondingly influence each other. The subject of our interest is to study how this proportion changes during the considered period of time. Table 2 summarizes the obtained results. As it can be seen from this table, both the number of vertices and the number of edges in the market graph increase as time goes. Obviously, the number of vertices grows since new stocks appear in the market, and we don t consider those stocks which ceased to exist by the last of 1000 trading days used in our analysis, so the maximum possible number of edges in the graph increases as well. However, it turns out that the number of edges grows faster; therefore, the edge density of the market graph increases from period to period. As one can see from Figure 7, the greatest increase of the edge density corresponds to the last two periods. In fact, the edge density for the latest interval is approximately 8.5 times higher than for the first interval! This dramatic jump suggests that there is a trend to the globalization of the modern stock market, which means that nowadays more and more stocks significantly affect the behavior of the others, and the structure of the market becomes not purely random. Period Number of Vertices Number of Edges Edge density 1 5430 2258 0.015% 2 5507 2614 0.017% 3 5593 3772 0.024% 4 5666 5276 0.033% 5 5768 6841 0.041% 6 5866 7770 0.045% 7 6013 10428 0.058% 8 6104 12457 0.067% 9 6262 12911 0.066% 10 6399 19707 0.096% 11 6556 27885 0.130% Table 2: Number of vertices and number of edges in the market graph ( θ = 0.5 ) for different periods. Fig. 7: Growth dynamics of the edge density of the market graph over time. It should be noted that the increase of the edge density could be predicted from the analysis of the distribution of the cross-correlations between all pairs of stocks, which was mentioned above. From Figure 2, one can observe that even though the distributions corresponding to different periods have a similar shape and the same mean, the tail of the distribution corresponding to the latest period (period 11) is significantly heavier than for the earlier periods, which means that there are more pairs of stocks with higher values of the correlation coefficient. Another natural question that can be asked is, which stock in the market graph has the highest degree? Recall that the degree of the vertex is the number of edges emanating from it; therefore, if one is looking for a single stock which would most precisely reflect the behavior of the market, then a

stock with the highest degree seems to be a perfect candidate since it is correlated with many other stocks. For the sample market graph considered in [8] we have found that the vertex with the highest degree corresponds to NASDAQ-100 index tracking stock, which is certainly not surprising from the financial point of view, and serves as a good verification point. Table 3 presents the stocks with the highest degrees in the market graph with the correlation threshold θ = 0.5, for different time intervals. As one would expect, these stocks are exchange traded funds: NASDAQ-100 index tracking stock (QQQ), MidCap SPDR Trust Series I (MDY), and SPDR Trust Series I (SPY). Note, that the values of the highest degree in the market graph increase from period to period, which is another confirmation of the fact that the globalization of the stock market actually takes place. Period 1 SPY 2 QQQ 3 QQQ 4 QQQ 5 QQQ 6 QQQ 7 QQQ 8 QQQ 9 QQQ 10 MDY 11 MDY Stock with the highest degree Type of stock Degree 106 130 181 224 252 267 318 338 337 402 514 Table 3: Vertices with the highest degrees in the market graph for different periods ( θ = 0.5 ). 3 Conclusion Graph representation of the stock market and interpretation of the properties of this graph gives a new insight into the internal structure of the stock market. In this paper, we have studied different characteristics of the market graph and their evolution over time and came to several interesting conclusions based on our analysis. It turns out that the power-law structure of the market graph is quite stable over the considered time intervals; therefore one can say that the concept of self-organized networks, which was mentioned above, is applicable in finance, and in this sense the stock market can be considered as a self-organized system. Another important result is the fact that the edge density of the market graph and the highest degrees of the vertices steadily increase during the last several years, which supports the well-known idea about the globalization of economy which has been widely discussed recently. References: [1] J. Abello, P.M. Pardalos, M.G.C. Resende, On maximum clique problems in very large graphs, DIMACS Series, 50, American Mathematical Society, 1999, pp. 119-130. [2] W. Aiello, F. Chung, L. Lu, A random graph model for power law graphs, Experimental Math, 10, 2001, pp. 53-66. [3] R. Albert, A.-L. Barabasi, Statistical mechanics of complex networks, Reviews of Modern Physics 74, 2002, pp. 47-97. [4] G. Avondo-Bodeno, Economic applications of the theory of graphs, Gordon and Breach Science Publishers, 1962. [5] A.-L. Barabasi, R. Albert, Emergence of scaling in random networks, Science 286, 1999, pp. 509 511. [6] A.-L. Barabasi, Linked, Perseus Publishing, 2002. [7] V. Boginski, S. Butenko, P.M. Pardalos, Modeling and Optimization in Massive Graphs, In: Novel Approaches to Hard Discrete Optimization (P. M. Pardalos and H. Wolkowicz, eds.), American Mathematical Society, 2003, 17-39. [8] V. Boginski, S. Butenko, P.M. Pardalos, On Structural Properties of the Market Graph, In: Innovations in Financial networks (A. Nagurney, ed.), Edward Elgar Publishers, 2003, to appear. [9] A. Broder, R. Kumar, F. Maghoul, P. Raghavan, S. Rajagopalan, R. Stata, A. Tomkins, J. Wiener, Graph structure in the Web, Computer Networks, 33, 2000, pp. 309 320. [10] N. Deo, Graph theory with applications to engineering and computer science, Prentice- Hall, 1974. [11] M. Faloutsos, P. Faloutsos, C. Faloutsos, On power-law relationships of the Internet topology, ACM SICOMM, 1999. [12] B. Hayes, Graph Theory in Practice, American Scientist, 88: 9-13 (Part I), 104-109 (Part II), 2000. [13] H. Jeong, B. Tomber, R. Albert, Z.N. Oltvai, A.-L. Barabasi, The large-scale organization of

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