Cellular neural networks can mimic small-world networks

Small-world phenomenon can occur in coupled dynamical systems which are hig hly 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 ph enomenon". We generalize the "characteristic path length" from previous wor ks on "small-world phenomenon" into a "characteristic coupling strength" fo r 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 "cluster ing coefficient" and show how it can be calculated by a horizontal "hole de tection" CNN, followed by a vertical "hole detection" CNN. Evolutions of th e game-of-life CNN with different initial conditions are used to illustrate the emergence of a "small-world phenomenon". Our results show that the wel l-known game-of-life CNN is not a small-world network. However, generalized CNN life games whose individuals have strong mobility and high survival ra te can exhibit small-world phenomenon in a robust way. Our simulations conf irm 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 ini tial conditions, and are therefore not robust.