We investigate the effect of directed short and long range connections in a
simple model of small world network. Our model is such that 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 un-favorable bonds has a negative effect on this quantity. Hence
the spreading process may not be able to affect all the network. We define and
calculate a quantity named the average size of accessible world in our model.
The interplay of shortcuts, and un-favorable bonds on the small world
properties is studied.Comment: 15 pages, 9 figures, published versio