Simple models of small-world networks with directed links.

We investigate the effect of directed short- and long-range connections in a simple model of a small-world network. Our model is one in which we can determine many quantities of interest by an exact analytical method. We calculate the function V(T), defined as the number of sites affected up to time T when a naive spreading process starts in the network. As opposed to shortcuts, the presence of unfavorable bonds has a negative effect on this quantity. Hence, the spreading process may not be able to affect all of the network. We define and calculate a quantity identified as the average size of the accessible world in our model. The interplay of shortcuts and unfavorable bonds on the small world properties is studied.