  We study the stabilization time of two common types of influence propagation.
In majority processes, nodes in a graph want to switch to the most frequent
state in their neighborhood, while in minority processes, nodes want to switch
to the least frequent state in their neighborhood. We consider the sequential
model of these processes, and assume that every node starts out from a uniform
random state.
  We first show that if nodes change their state for any small improvement in
the process, then stabilization can last for up to $\Theta(n^2)$ steps in both
cases. Furthermore, we also study the proportional switching case, when nodes
only decide to change their state if they are in conflict with a
$\frac{1+\lambda}{2}$ fraction of their neighbors, for some parameter $\lambda
\in (0,1)$. In this case, we show that if $\lambda < \frac{1}{3}$, then there
is a construction where stabilization can indeed last for $\Omega(n^{1+c})$
steps for some constant $c>0$. On the other hand, if $\lambda > \frac{1}{2}$,
we prove that the stabilization time of the processes is upper-bounded by $O(n
\cdot \log{n})$.
