Complex networks in a stock market

We consider cross-correlations among stock prices in the Korean stock-market[1,2]. We use the daily Korean stock-market prices of KOSPI 200 for 4 years from January 3, 2000 to December 29, 2004. Let us define logarithmic return as ri(t) = log pi(t)− log pi(t−∆t) where pi(t) is the stock prices of a company i at a time t and ∆t is the return time. We use one day return time ∆t = 1day. The total number of stocks is N = 200. From the logarithmic return we calculate the cross correlations Cij between stock prices i and j. The metric distance between a pair of stocks is defined by dij(∆t) = √ 2(1− Cij(∆t)). We construct a mimimal spanning tree(MST) from the distance matrix. In MST graph, we obtain the average shortest path length < l >= 5.05. We observe that the degree distribution of the MST follows a power-law p(k) ∼ k−γ with γ = 2.7(1). In MST graph there is a unique hub node. Consider complex networks in stock market by using cross-correlation coefficient. We introduce a specified threshold C× in the cross-correlation coefficient. In the correlation network two stocks are connected by a link when the cross-correlation coefficient is greater than the threshold value C×. We observe a scale-free network at a restricted range of the threshold. In Fig. 1 we present a scale-free network at the threshold C× = 0.45 If the threshold is small, almost all companies are connected each other. If the threshold is large, only a small number of companies are connected. We get very sparse scale-free network at the threshold 0.4 ≤ q ≤ 0.6. The degree distribution of the correlation network follows a power law and exponents depend on the threshold value.