  Let $G$ be a graph and ${\mathcal{\tau}}: V(G)\rightarrow \Bbb{N}$ be an
assignment of thresholds to the vertices of $G$. A subset of vertices $D$ is
said to be dynamic monopoly (or simply dynamo) if the vertices of $G$ can be
partitioned into subsets $D_0, D_1,..., D_k$ such that $D_0=D$ and for any
$i=1,..., k-1$ each vertex $v$ in $D_{i+1}$ has at least $t(v)$ neighbors in
$D_0\cup ...\cup D_i$. Dynamic monopolies are in fact modeling the irreversible
spread of influence such as disease or belief in social networks. We denote the
smallest size of any dynamic monopoly of $G$, with a given threshold
assignment, by $dyn(G)$. In this paper we first define the concept of a
resistant subgraph and show its relationship with dynamic monopolies. Then we
obtain some lower and upper bounds for the smallest size of dynamic monopolies
in graphs with different types of thresholds. Next we introduce
dynamo-unbounded families of graphs and prove some related results. We also
define the concept of a homogenious society that is a graph with probabilistic
thresholds satisfying some conditions and obtain a bound for the smallest size
of its dynamos. Finally we consider dynamic monopoly of line graphs and obtain
some bounds for their sizes and determine the exact values in some special
cases.
