Stochastic Stability of Nonsymmetric Threshold Networks

For a nonsymm etri c threshold network equippe d with an asynchronous dynamics, we show that if the product of weights in any cycle of units is nonnegative, then each tr ajectory converges to a stable state with probability one. We also show that such networks have a natural feed-forward layer struc ture and stabilit y is achieved in a hierarchical order due to this struct ure. It follows then that this new class of networks can perform similar tasks as symmetric networks as well as new tas ks due to its relation to directed graphs.