Emergence of frustration signals systemic risk

We show that the emergence of systemic risk in complex systems can be understood from the evolution of functional networks representing interactions inferred from fluctuation correlations between macroscopic observables. Specifically, we analyze the long-term collective dynamics in the New York Stock Exchange, the largest financial market in the world, for almost a century and show that periods marked by systemic crisis are associated with emergence of frustration. This is indicated by the loss of structural balance in the networks of interaction between stocks. Moreover, the mesoscopic organization of the networks during these periods exhibits prominent core-periphery organization. This suggests an increased degree of coherence in the collective dynamics of the system, which is reinforced by our observation of the transition to delocalization in the dominant eigenmodes when the systemic risk builds up. While frustration has been associated with phase transitions in physical systems such as spin glasses, its role as a signal for systemic risk buildup leading to severe crisis as shown here provides a novel perspective into the dynamical processes leading to catastrophic failures in complex systems.

Figure 1

(a–b) Interaction networks of stocks traded in the New York Stock Exchange (NYSE) showing the top 300 in terms of average price during (a) Period 80: Dec. 2002–Dec. 2006 and (b) Period 85: Feb. 2008–Feb. 2012, corresponding to time-intervals prior to and during the Great Recession. Edges represent significant interactions obtained by spectral filtering of the cross-correlation matrix removing the common “global” mode and noise which mask these interactions. The crisis period is characterized by the emergence of a large number of negative edges (indicated by red color) representing anti-correlation between pairs of stocks. This period also exhibits dense clustering evident from the presence of many connected triads. (c–d) The time-evolution of the number of total edges ($N_E$, in blue) and negative edges ($N_E^{(-)}$, in red) is shown in (c) and that of the number of connected triads $\left[N\right(\mathrm{\Delta })$, blue: all triads, red: frustrated triads] is shown in (d) for the interaction networks of NYSE stocks between 1926–2016. The intervals corresponding to systemic crisis (viz., the Great Depression of 1929–1933 and the Great Recession of 2007–9) in the economy are indicated by the shaded regions. The red inverted triangles indicate the periods where structural balance in the interaction network is lost. Node colors in (a) and (b) indicate the business sectors to which the stocks belong (key to the color code is as in Fig. 4).

Figure 2

(a) The evolution of the probability distribution of significant correlations between stocks (after spectral filtering) during 1926–2016. The adjoining contour plot shows the temporal variation in the dispersion of the distribution across the period of about 90 y, the different contours corresponding to integral multiples of the standard deviation. (b) The inverse participation ratio (IPR) of the global mode corresponding to the largest eigenvalue ($u_1$) and that for the next few leading “group” modes corresponding to the second, third, and fourth highest eigenvalues ($u_2,u_3,u_4$). The lower bound of IPR, viz., $1/N$ corresponds to a completely delocalized mode in which all components contribute equally while the broken line represents the expected IPR for random noise ($=3/N$). The group modes exhibit stronger localization during periods of relative economic calm, while systemic crisis is associated with increased coherent movement among stocks implied by delocalization.

Figure 3

(a–d) The evolution of the fraction of connected triads (red triangles) in the network of significant interactions resolved into the four possible distinct signed triads, (a) and (b) corresponding to balanced (viz., $+++$ and $+--$, respectively), and (c) and (d) to unbalanced triads (viz., $++-$ and $---$). For each, the fraction of triads expected from degree-preserved randomized networks are also shown (gray inverted triangles). Note that the unbalanced triads are much less likely to occur in the empirical network. (e) The number of frustrated triads $[N\left({\mathrm{\Delta }}_{\mathrm{frust}}\right)$, indicated by the color bar] in each of the eigenmodes corresponding to the deviating eigenvalues (i.e., ${\lambda }_i>{\lambda }_{\max }^W$) of the cross-correlation matrix for the periods 1–89 spanning the entire duration being investigated. Eigenmodes falling within the bounds set by the corresponding Wishart matrix spectrum, and which are therefore indistinguishable from random modes, are not considered (shown in white). While frustrated triads appear in individual eigenmodes at many periods, the network of significant interactions exhibit such triads only prior to and during systemic crisis in the economy [indicated by negative growth rate, i.e., $\mathrm{\Delta }g<0$, in (f)]. (f) Relative change $\mathrm{\Delta }g$ in the inflation-adjusted gross domestic product (GDP) per capita of USA measured in terms of logarithmic returns over successive intervals for the same period as in (a–d).

Figure 4

(a) The evolution of the degree assortativity $r_{\mathrm{LCC}}$ of the largest connected component of the interaction networks over the duration under study. Periods associated with financial crisis are seen to have positive values of $r_{\mathrm{LCC}}$ suggesting that the corresponding networks can exhibit core-periphery organization. This is supported by $k$-core analysis of the networks with (b) showing the evolution of the order of the innermost core $k_{\max }$ of the networks. It is apparent that the depth of the core increases during systemic crisis, indicating higher degree of coherence in the movement of stocks in the market. (c–d) Multi-level pie chart for (c) Period 80 and (d) Period 85 showing the sector composition of each of the $k$-shell quartiles. The radii of the quartiles are proportional to the fraction of stocks belonging to them. The color code in the adjoining key indicating the business sectors to which stocks belong correspond to (1) Mining and Construction, (2) Manufacture (basic), (3) Manufacture (advanced), (4) Transportation, (5) Trade, (6) Finance, Insurance and Real Estate, (7) Services (Business), (8) Services (Public), and (9) Public Administration.