  A fundamental question related to innovation diffusion is how the social
network structure influences the process. Empirical evidence regarding
real-world influence networks is very limited. On the other hand, agent-based
modeling literature reports different and at times seemingly contradictory
results. In this paper we study innovation diffusion processes for a range of
Watts-Strogatz networks in an attempt to shed more light on this problem. Using
the so-called Sznajd model as the backbone of opinion dynamics, we find that
the published results are in fact consistent and allow to predict the role of
network topology in various situations. In particular, the diffusion of
innovation is easier on more regular graphs, i.e. with a higher clustering
coefficient. Moreover, in the case of uncertainty - which is particularly high
for innovations connected to public health programs or ecological campaigns - a
more clustered network will help the diffusion. On the other hand, when social
influence is less important (i.e. in the case of perfect information), a
shorter path will help the innovation to spread in the society and - as a
result - the diffusion will be easiest on a random graph.
