Diagrammatic expansion of information flows in stochastic Boolean networks

Accurate information transfer is essential for biological, social, and technological networks. Computing transfer entropy (TE), a measure of information flow, usually relies on numerical methods even in small networks, which obscures the origin of accurate information transfer. In this study, we establish a diagrammatic formula that analytically computes TE for a stochastic Boolean network, where the intermediate network from signal to output can have arbitrary topology and nonidentical Boolean functions. By expressing TE in terms of network components, we elucidate the mechanism of information flow and provide optimal design principles of network architectures applicable to real networks.