  In this work, we give a characterization of the reservoir computer (RC) by
the network structure, especially the probability distribution of random
coupling constants. First, based on the path integral method, we clarify the
universal behavior of the random network dynamics in the thermodynamic limit,
which depends only on the asymptotic behavior of the second cumulant generating
functions of the network coupling constants. This result enables us to classify
the random networks into several universality classes, according to the
distribution function of coupling constants chosen for the networks.
Interestingly, it is revealed that such a classification has a close
relationship with the distribution of eigenvalues of the random coupling
matrix. We also comment on the relation between our theory and some practical
choices of random connectivity in the RC. Subsequently, we investigate the
relationship between the RC's computational power and the network parameters
for several universality classes. We perform several numerical simulations to
evaluate the phase diagrams of the steady reservoir states,
common-signal-induced synchronization, and the computational power in the
chaotic time series inference tasks. As a result, we clarify the close
relationship between these quantities, especially a remarkable computational
performance near the phase transitions, which is realized even near a
non-chaotic transition boundary. These results may provide us with a new
perspective on the designing principle for the RC.
