  Let $G$ be a graph, and let $\rho\in (0,1)$. For a set $D$ of vertices of
$G$, let the set $H_{\rho}(D)$ arise by starting with the set $D$, and
iteratively adding further vertices $u$ to the current set if they have at
least $\lceil \rho d_G(u)\rceil$ neighbors in it. If $H_{\rho}(D)$ contains all
vertices of $G$, then $D$ is known as an irreversible dynamic monopoly or a
perfect target set associated with the threshold function $u\mapsto \lceil \rho
d_G(u)\rceil$. Let $h_{\rho}(G)$ be the minimum cardinality of such an
irreversible dynamic monopoly.
  For a connected graph $G$ of maximum degree at least $\frac{1}{\rho}$, Chang
(Triggering cascades on undirected connected graphs, Information Processing
Letters 111 (2011) 973-978) showed $h_{\rho}(G)\leq 5.83\rho n(G)$, which was
improved by Chang and Lyuu (Triggering cascades on strongly connected directed
graphs, Theoretical Computer Science 593 (2015) 62-69) to $h_{\rho}(G)\leq
4.92\rho n(G)$. We show that for every $\epsilon>0$, there is some
$\rho(\epsilon)>0$ such that $h_{\rho}(G) \leq(2+\epsilon)\rho n(G)$ for every
$\rho$ in $(0,\rho(\epsilon))$, and every connected graph $G$ that has maximum
degree at least $\frac{1}{\rho}$ and girth at least $5$. Furthermore, we show
that $h_{\rho}(T) \leq \rho n(T)$ for every $\rho$ in $(0,1]$, and every tree
$T$ that has order at least $\frac{1}{\rho}$.
