  Randomly coupled Ising spins constitute the classical model of collective
phenomena in disordered systems, with applications covering ferromagnetism,
combinatorial optimization, protein folding, stock market dynamics, and social
dynamics. The phase diagram of these systems is obtained in the thermodynamic
limit by averaging over the quenched randomness of the couplings. However, many
applications require the statistics of activity for a single realization of the
possibly asymmetric couplings in finite-sized networks. Examples include
reconstruction of couplings from the observed dynamics, learning in the central
nervous system by correlation-sensitive synaptic plasticity, and representation
of probability distributions for sampling-based inference. The systematic
cumulant expansion for kinetic binary (Ising) threshold units with strong,
random and asymmetric couplings presented here goes beyond mean-field theory
and is applicable outside thermodynamic equilibrium; a system of approximate
non-linear equations predicts average activities and pairwise covariances in
quantitative agreement with full simulations down to hundreds of units. The
linearized theory yields an expansion of the correlation- and response
functions in collective eigenmodes, leads to an efficient algorithm solving the
inverse problem, and shows that correlations are invariant under scaling of the
interaction strengths.
