Optimal Connectivity of Random Boolean Networks for Reservoir Computing

Reservoir computing shifts the focus from training a system's internal dynamics to designing dynamics that expose useful computational capabilities via linear projection. We study random Boolean networks as reservoirs: cellular automata with random recurrent wiring and a random rule at each node. Their main parameter is the connectivity, the mean number of edges into a node. Without input, these networks pass from ordered to chaotic dynamics at a connectivity of 2, where their dynamics are expected to facilitate computation. How the input is connected is itself a design factor: if the input shares the network's edges, the connectivity sets both the dynamics and how many nodes receive the input, the input fan-out. We set the input fan-out separately and compare 7,560 reservoir designs, all combinations of six design factors, each built as many random reservoirs, on recalling the majority or the parity of a few input bits seen several steps earlier. The optimal connectivity falls from 2.5 to 1.75-2 as more nodes receive the input, but hardly depends on the task or on the other design factors, except self-loops, which connect a node to itself and raise the optimal connectivity. A measure computed from the reservoir states without a trained readout peaks at a connectivity that also falls with the fan-out and rises with self-loops, but does not replace training a readout. Reservoirs of one design vary widely, so a design should be judged by its typical reservoir.