  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.
