  In the problem of influence maximization in information networks, the
objective is to choose a set of initially active nodes subject to some budget
constraints such that the expected number of active nodes over time is
maximized. The linear threshold model has been introduced to study the opinion
cascading behavior, for instance, the spread of products and innovations. In
this paper, we we extends the classic linear threshold model [18] to capture
the non-progressive be- havior. The information maximization problem under our
model is proved to be NP-Hard, even for the case when the underlying network
has no directed cycles. The first result of this paper is negative. In general,
the objective function of the extended linear threshold model is no longer
submodular, and hence the hill climbing approach that is commonly used in the
existing studies is not applicable. Next, as the main result of this paper, we
prove that if the underlying information network is directed acyclic, the
objective function is submodular (and monotone). Therefore, in directed acyclic
networks with a specified budget we can achieve 1/2 -approximation on
maximizing the number of active nodes over a certain period of time by a
deterministic algorithm, and achieve the (1 - 1/e )-approximation by a
randomized algorithm.
