  Consider a graph $G$ and an initial configuration where each node is black or
white. Assume that in each round all nodes simultaneously update their color
based on a predefined rule. One can think of graph $G$ as a social network,
where each black/white node represents an individual who holds a
positive/negative opinion regarding a particular topic. In the $r$-threshold
(resp. $\alpha$-threshold) model, a node becomes black if at least $r$ of its
neighbors (resp. $\alpha$ fraction of its neighbors) are black, and white
otherwise. The $r$-monotone (resp. $\alpha$-monotone) model is the same as the
$r$-threshold (resp. $\alpha$-threshold) model, except that a black node
remains black forever.
  What is the number of rounds that the process needs to stabilize? How many
nodes must be black initially so that black color takes over or survives? Our
main goal in the present paper is to address these two questions
