  This paper deals with different models of random walks with a reinforced
memory of preferential attachment type. We consider extensions of the Elephant
Random Walk introduced by Sch\"utz and Trimper [2004] with a stronger
reinforcement mechanism, where, roughly speaking, a step from the past is
remembered proportional to some weight and then repeated with probability $p$.
With probability $1-p$, the random walk performs a step independent of the
past. The weight of the remembered step is increased by an additive factor
$b\geq 0$, making it likelier to repeat the step again in the future. A
combination of techniques from the theory of urns, branching processes and
$\alpha$-stable processes enables us to discuss the limit behavior of
reinforced versions of both the Elephant Random Walk and its $\alpha$-stable
counterpart, the so-called Shark Random Swim introduced by Businger [2018]. We
establish phase transitions, separating subcritical from supercritical regimes.
