  Kempe, Kleinberg and Tardos (KKT) proposed the following conjecture about the
general threshold model in social networks: local monotonicity and
submodularity imply global monotonicity and submodularity. That is, if the
threshold function of every node is monotone and submodular, then the spread
function $\sigma(S)$ is monotone and submodular, where $S$ is a seed set and
the spread function $\sigma(S)$ denotes the expected number of active nodes at
termination of a diffusion process starting from $S$. The correctness of this
conjecture has been proved by Mossel and Roch. In this paper, we first provide
the concept AD-k (Alternating Difference-$k$) as a generalization of
monotonicity and submodularity. Specifically, a set function $f$ is called \adk
if all the $\ell$-th order differences of $f$ on all inputs have sign
$(-1)^{\ell+1}$ for every $\ell\leq k$. Note that AD-1 corresponds to
monotonicity and AD-2 corresponds to monotonicity and submodularity. We propose
a refined version of KKT's conjecture: in the general threshold model, local
AD-k implies global AD-k. The original KKT conjecture corresponds to the case
for AD-2, and the case for AD-1 is the trivial one of local monotonicity
implying global monotonicity. By utilizing continuous extensions of set
functions as well as social graph constructions, we prove the correctness of
our conjecture when the social graph is a directed acyclic graph (DAG).
Furthermore, we affirm our conjecture on general social graphs when $k=\infty$.
