Network geometry and market instability.

The complexity of financial markets arise from the strategic interactions among agents trading stocks, which manifest in the form of vibrant correlation patterns among stock prices. Over the past few decades, complex financial markets have often been represented as networks whose interacting pairs of nodes are stocks, connected by edges that signify the correlation strengths. However, we often have interactions that occur in groups of three or more nodes, and these cannot be described simply by pairwise interactions but we also need to take the relations between these interactions into account. Only recently, researchers have started devoting attention to the higher-order architecture of complex financial systems, that can significantly enhance our ability to estimate systemic risk as well as measure the robustness of financial systems in terms of market efficiency. Geometry-inspired network measures, such as the Ollivier-Ricci curvature and Forman-Ricci curvature, can be used to capture the network fragility and continuously monitor financial dynamics. Here, we explore the utility of such discrete Ricci curvatures in characterizing the structure of financial systems, and further, evaluate them as generic indicators of the market instability. For this purpose, we examine the daily returns from a set of stocks comprising the USA S&P-500 and the Japanese Nikkei-225 over a 32-year period, and monitor the changes in the edge-centric network curvatures. We find that the different geometric measures capture well the system-level features of the market and hence we can distinguish between the normal or 'business-as-usual' periods and all the major market crashes. This can be very useful in strategic designing of financial systems and regulating the markets in order to tackle financial instabilities.

Frequency spectra of turbulent thermal convection with uniform rotation.

The frequency spectra of the entropy and kinetic energy along with the power spectrum of the thermal flux are computed from direct numerical simulations for turbulent Rayleigh-Bénard convection with uniform rotation about a vertical axis in low-Prandtl-number fluids (Pr<0.6). Simulations are done for convective Rossby numbers Ro≥0.2. The temporal fluctuations of these global quantities show two scaling regimes: (i) ω(-2) at higher frequencies for all values of Ro and (ii) ω(-γ1) at intermediate frequencies with γ1≈4 for Ro>1, while 4<γ1<6.6 for 0.2≤Ro<1.

Entropy and energy spectra in low-Prandtl-number convection with rotation.

We present results for entropy and kinetic energy spectra computed from direct numerical simulations for low-Prandtl-number (Pr < 1) turbulent flow in Rayleigh-Bénard convection with uniform rotation about a vertical axis. The simulations are performed in a three-dimensional periodic box for a range of the Taylor number (0 ≤ Ta ≤ 10(8)) and reduced Rayleigh number r = Ra/Ra(∘)(Ta,Pr) (1.0 × 10(2) ≤ r ≤ 5.0 × 10(3)). The Rossby number Ro varies in the range 1.34 ≤ Ro ≤ 73. The entropy spectrum E(θ)(k) shows bisplitting into two branches for lower values of wave number k. The entropy in the lower branch scales with k as k(-1.4 ± 0.1) for r>10(3) for the rotation rates considered here. The entropy in the upper branch also shows scaling behavior with k, but the scaling exponent decreases with increasing Ta for all r. The energy spectrum E(v)(k) is also found to scale with the wave number k as k(-1.4 ± 0.1) for r>10(3). The scaling exponent for the energy spectrum and the lower branch of the entropy spectrum vary between -1.7 and -2.4 for lower values of r (<10(3)). We also provide some simple arguments based on the variation of the Kolmogorov picture to support the results of simulations.

Turbulence in rotating Rayleigh-Bénard convection in low-Prandtl-number fluids.

The heat flux in rotating Rayleigh-Bénard convection in a fluid of Prandtl number Pr=0.1 enclosed between free-slip top and bottom boundaries is investigated using direct numerical simulation in a wide range of Rayleigh numbers (10(4)≤Ra≤10(8)) and Taylor numbers (0≤Ta≤10(8)). The Nusselt number Nu scales with the Rayleigh number Ra as Ra(ß) with ß=2/7 for values of Nu greater than a critical value Nu(c), which occurs for Ta/Ra∼1. The exponent ß is not universal for Nu1) but a function of Ta showing a minimum for some intermediate value of Ta. The critical Nusselt number Nu(c) and the corresponding critical Rossby number Ro(c) scale with Ta as Ta(0.277±0.001) and Ta(-0.015±0.003), respectively.