  We consider the problem of maximizing the spread of influence in a social
network by choosing a fixed number of initial seeds, formally referred to as
the influence maximization problem. It admits a $(1-1/e)$-factor approximation
algorithm if the influence function is submodular. Otherwise, in the worst
case, the problem is NP-hard to approximate to within a factor of
$N^{1-\varepsilon}$. This paper studies whether this worst-case hardness result
can be circumvented by making assumptions about either the underlying network
topology or the cascade model. All of our assumptions are motivated by many
real life social network cascades.
  First, we present strong inapproximability results for a very restricted
class of networks called the (stochastic) hierarchical blockmodel, a special
case of the well-studied (stochastic) blockmodel in which relationships between
blocks admit a tree structure. We also provide a dynamic-program based
polynomial time algorithm which optimally computes a directed variant of the
influence maximization problem on hierarchical blockmodel networks. Our
algorithm indicates that the inapproximability result is due to the
bidirectionality of influence between agent-blocks.
  Second, we present strong inapproximability results for a class of influence
functions that are "almost" submodular, called 2-quasi-submodular. Our
inapproximability results hold even for any 2-quasi-submodular $f$ fixed in
advance. This result also indicates that the "threshold" between submodularity
and nonsubmodularity is sharp, regarding the approximability of influence
maximization.
