  We study degree-penalized contact processes on Galton-Watson trees (GW) and
the configuration model. The model we consider is a modification of the usual
contact process on a graph. In particular, each vertex can be either infected
or healthy. When infected, each vertex heals at rate one. Also, when infected,
a vertex $v$ with degree $d_v$ infects its neighboring vertex $u$ with degree
$d_u$ with rate $\lambda/f(d_u, d_v)$ for some positive function $f$. In the
case $f(d_u, d_v)=\max(d_u, d_v)^\mu$ for some $\mu>0$, the infection is slowed
down to and from high degree vertices. This is in line with arguments used in
social network science: people with many contacts do not have the time to
infect their neighbors at the same rate as people with fewer contacts.
  We show that new phase transitions occur in terms of the parameter $\mu$ (at
$1/2$) and the degree distribution $D$ of the GW tree.
  - When $\mu\ge 1$, the process goes extinct for all distributions $D$ for all
sufficiently small $\lambda>0$;
  - When $\mu\in(1/2, 1)$, and the tail of $D$ weakly follows a power law with
tail-exponent less than $1-\mu$, the process survives globally but not locally
for all $\lambda$ small enough;
  - When $\mu\in(1/2, 1)$, and $\mathbb{E}[D^{1-\mu}]<\infty$, the process goes
extinct almost surely, for all $\lambda$ small enough;
  - When $\mu<1/2$, and $D$ is heavier then stretched exponential with
stretch-exponent $1-2\mu$, the process survives (locally) with positive
probability for all $\lambda>0$.
  We also study the product case $f(x,y)=(xy)^\mu$. In that case, the situation
for $\mu < 1/2$ is the same as the one described above, but $\mu\ge 1/2$ always
leads to a subcritical contact process for small enough $\lambda>0$ on all
graphs. Furthermore, for finite random graphs with prescribed degree sequences,
we establish the corresponding phase transitions in terms of the length of
survival.
