Small-world phenomenon can occur in coupled dynamical systems which are highly clustered at a local level and yet strongly coupled at the global level. We show that cellular neural networks (CNN's) can exhibit "small-world phenomenon". We generalize the "characteristic path length" from previous works on "small-world phenomenon" into a "characteristic coupling strength" for measuring the average coupling strength of the outputs of CNN's. We also provide a simplified algorithm for calculating the "characteristic coupling strength" with a reasonable amount of computing time. We define a "clustering coefficient" and show how it can be calculated by a horizontal "hole detection" CNN, followed by a vertical "hole detection" CNN. Evolutions of the game-of-life CNN with different initial conditions are used to illustrate the emergence of a "small-world phenomenon". Our results show that the well-known game-of-life CNN is not a small-world network. However, generalized CNN life games whose individuals have strong mobility and high survival rate can exhibit small-world phenomenon in a robust way. Our simulations confirm the conjecture that a population with a strong mobility is more likely to qualify as a small world. CNN games whose individuals have weak mobility can also exhibit a small-world phenomenon under a proper choice of initial conditions. However, the resulting small worlds depend strongly on the initial conditions, and are therefore not robust.
It's Hard for Neural Networks to Learn the Game of Life
