Connection sparsity versus orbit stability in dynamic binary neural networks

This paper considers two basic problems in artificial neural networks that can generate various binary periodic orbits. The first problem is relation between sparsity of network connection and stability of a target periodic orbit. The second problem is comparison between digital circuits and artificial neural networks in the orbit stability. We consider these problems in dynamic binary neural networks characterized by the signum activation function and ternary connection matrix. Performing basic numerical experiments, we give conjectures for the two problems. First, as the connection sparsity increases, the orbit stability varies. There exists suitable sparsity in which the orbit stability is very strong. Second, as the connection matrix approaches to the most sparse case, the dynamic binary neural network approaches to an equivalent system to the shift register that has no stable periodic orbit.