Bounds on the $k$-conversion number

We consider a graphical model of the spread of influence through social networks, where the goal is to find a set of vertices in the network, such that if this initial set is ``influenced'', then after the application of a certain propagation process eventually every vertex in the graph will also be influenced. In particular, we seek a minimum set of vertices to be initially influenced and follow an iterative process, where for a fixed integer threshold $k \ge0$, a vertex outside the influenced set becomes influenced if at least $k$ of its neighbors are influenced. We determine bounds on the minimum number of vertices required in such a set for every integer $k\ge0$ and focus our study on the case for $k= 2$.