  Most studies on influence maximization focus on one-shot propagation, i.e.
the influence is propagated from seed users only once following a probabilistic
diffusion model and users' activation are determined via single cascade. In
reality it is often the case that a user needs to be cumulatively impacted by
receiving enough pieces of information propagated to her before she makes the
final purchase decision. In this paper we model such cumulative activation as
the following process: first multiple pieces of information are propagated
independently in the social network following the classical independent cascade
model, then the user will be activated (and adopt the product) if the
cumulative pieces of information she received reaches her cumulative activation
threshold. Two optimization problems are investigated under this framework:
seed minimization with cumulative activation (SM-CA), which asks how to select
a seed set with minimum size such that the number of cumulatively active nodes
reaches a given requirement $\eta$; influence maximization with cumulative
activation (IM-CA), which asks how to choose a seed set with fixed budget to
maximize the number of cumulatively active nodes. For SM-CA problem, we design
a greedy algorithm that yields a bicriteria $O(\ln n)$-approximation when
$\eta=n$, where $n$ is the number of nodes in the network. For both SM-CA
problem with $\eta<n$ and IM-CA problem, we prove strong inapproximability
results. Despite the hardness results, we propose two efficient heuristic
algorithms for SM-CA and IM-CA respectively based on the reverse reachable set
approach. Experimental results on different real-world social networks show
that our algorithms significantly outperform baseline algorithms.
