Let
$G$
be a graph and let
$\tau $
be an assignment of nonnegative integer thresholds to the vertices of
$G$
. A subset of vertices,
$D$
, is said to be a
$\tau $
-dynamicmonopoly if
$V\left( G \right)$
can be partitioned into subsets
${{D}_{0}},\,{{D}_{1}},\,\ldots \,,\,{{D}_{k}}$
such that
${{D}_{0}}\,=\,D$
and for any
$i\,\in \,\left\{ 0,\,\ldots \,,\,k-1 \right\}$
, each vertex
$v$
in
${{D}_{i+1}}$
has at least
$\tau \left( v \right)$
neighbors in
${{D}_{0}}\cup \cdots \cup {{D}_{i}}$
. Denote the size of smallest
$\tau $
-dynamic monopoly by
$\text{dy}{{\text{n}}_{\tau }}\left( G \right)$
and the average of thresholds in
$\tau $
by
$\bar{\tau }$
. We show that the values of
$\text{dy}{{\text{n}}_{\tau }}\left( G \right)$
over all assignments
$\tau $
with the same average threshold is a continuous set of integers. For any positive number
$t$
, denote the maximum
$\text{dy}{{\text{n}}_{\tau }}\left( G \right)$
taken over all threshold assignments
$\tau $
with
$\bar{\tau }\,\le \,t$
, by
$\text{Ldy}{{\text{n}}_{t}}\left( G \right)$
. In fact,
$\text{Ldy}{{\text{n}}_{t}}\left( G \right)$
shows the worst-case value of a dynamicmonopoly when the average threshold is a given number
$t$
. We investigate under what conditions on
$t$
, there exists an upper bound for
$\text{Ldy}{{\text{n}}_{t}}\left( G \right)$
of the form
$c\left| G \right|$
, where
$c\,<\,1$
. Next, we show that
$\text{Ldy}{{\text{n}}_{t}}\left( G \right)$
is
$\text{coNP}$
-hard for planar graphs but has polynomial-time solution for forests.