  We use the linear threshold model to study the diffusion of information on a
network generated by the stochastic block model. We focus our analysis on a two
community structure where the initial set of informed nodes lies only in one of
the two communities and we look for optimal network structures, i.e. those
maximizing the asymptotic extent of the diffusion. We find that, constraining
the mean degree and the fraction of initially informed nodes, the optimal
structure can be assortative (modular), core-periphery, or even disassortative.
We then look for minimal cost structures, i.e. those such that a minimal
fraction of initially informed nodes is needed to trigger a global cascade. We
find that the optimal networks are assortative but with a structure very close
to a core-periphery graph, i.e. a very dense community linked to a much more
sparsely connected periphery.
