We derive the exact form of the eigenvalue spectra of correlation matrices derived from a set of time-shifted, finite Brownian random walks (time-series). These matrices can be seen as random, real, asymmetric matrices with a special structure superimposed due to the time-shift. We demonstrate that the associated eigenvalue spectrum is circular symmetric in the complex plane for large matrices. This fact allows us to exactly compute the eigenvalue density via an inverse Abel-transform of the density of the symmetrized problem. We demonstrate the validity of this approach by numerically computing eigenvalue spectra of lagged correlation matrices based on uncorrelated, Gaussian distributed time-series. We then compare our theoretical findings with eigenvalue densities obtained from actual high frequency (5 min) data of the S&P500 and discuss the observed deviations. We identify various non-trivial, non-random patterns and find asymmetric dependencies associated with eigenvalues departing strongly from the Gaussian prediction in the imaginary part. For the same time-series, with the market contribution removed, we observe strong clustering of stocks, i.e. causal sectors. We finally comment on the time-stability of the observed patterns.

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Abstract. We consider social systems in which agents are not only characterized by their states but also have the freedom to choose their interaction partners to maximize their utility. We map such systems onto an Ising model in which spins are dynamically coupled by links in a dynamical network. In this model there are two dynamical quantities which arrange towards a minimum energy state in the canonical framework: the spins, si, and the adjacency matrix elements, cij. The model is exactly solvable because microcanonical partition functions reduce to products of binomial factors as a direct consequence of the cij minimizing energy. We solve the system for finite sizes and for the two possible thermodynamic limits and discuss the phase diagrams.