Mesoscopic Community Structure of Financial Markets Revealed by Price and Sign Fluctuations

The mesoscopic organization of complex systems, from financial markets to the brain, is an intermediate between the microscopic dynamics of individual units (stocks or neurons, in the mentioned cases), and the macroscopic dynamics of the system as a whole. The organization is determined by "communities" of units whose dynamics, represented by time series of activity, is more strongly correlated internally than with the rest of the system. Recent studies have shown that the binary projections of various financial and neural time series exhibit nontrivial dynamical features that resemble those of the original data. This implies that a significant piece of information is encoded into the binary projection (i.e. the sign) of such increments. Here, we explore whether the binary signatures of multiple time series can replicate the same complex community organization of the financial market, as the original weighted time series. We adopt a method that has been specifically designed to detect communities from cross-correlation matrices of time series data. Our analysis shows that the simpler binary representation leads to a community structure that is almost identical with that obtained using the full weighted representation. These results confirm that binary projections of financial time series contain significant structural information.Comment: 15 pages, 7 figure

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<p>Each community is labelled with the number of stocks, and the pie chart represents the relative composition of each community based on the industry sectors of the constituent stocks (color legend in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0133679#pone.0133679.t001" target="_blank">Table 1</a>). The inter-community link weights are negative, indicating that the communities are all residually anti-correlated.</p

Communities of the Nikkei 225 (daily closing prices from 2001 to 2011) generated using the modified Louvain algorithm [7].

<p>Each community is labelled with the number of stocks, and the pie chart represents the relative composition of each community based on the industry sectors of the constituent stocks (colour legend in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0133679#pone.0133679.t001" target="_blank">Table 1</a>). The link weights are negative, indicating that the communities are all residually anti-correlated.</p

The 10 industry sectors in the Global Industry Classification Standard (GICS), with the color representation used to highlight the sectors in the following Figures.

<p>The 10 industry sectors in the Global Industry Classification Standard (GICS), with the color representation used to highlight the sectors in the following Figures.</p

The eigenvalue density distribution of the Pearson correlation matrices where the upper panels are for the weighted series and the lower panels are for the binary series.

<p>The red curve is the empirical eigenvalue distribution and the blue curve the Marchenko-Pastur distribution.</p

‘Weighted’ (left) versus ‘Binary’ (right) time series of log-returns of the Apple stock over a period of 40 days starting from 7/5/2011.

<p>‘Weighted’ (left) versus ‘Binary’ (right) time series of log-returns of the Apple stock over a period of 40 days starting from 7/5/2011.</p

The variation of information between the binary and weighted partitions for a sliding window of 600 trading days (approximately 28 moths) starting at Q3 2001.

<p>The VI is measured between the frequent partitions for the different algorithms: Potts (blue), Louvain(red) and Spectral (green).</p

The eigenvalue density distribution (of the cross-correlation matrix) for the different indexes, where the upper panels are for the weighted series and the lower panels are for the binary series.

<p>The red curve is the empirical eigenvalue distribution and the blue curve the Marchenko-Pastur distribution. The largest empirical eigenvalue <i>λ</i>_<i>m</i> is not shown in the plots, but the its value is reported in each panel.</p