  We consider an idealized model in which individuals' changing opinions and
their social network coevolve, with disagreements between neighbors in the
network resolved either through one imitating the opinion of the other or by
reassignment of the discordant edge. Specifically, an interaction between $x$
and one of its neighbors $y$ leads to $x$ imitating $y$ with probability
$(1-\alpha)$ and otherwise (i.e., with probability $\alpha$) $x$ cutting its
tie to $y$ in order to instead connect to a randomly chosen individual.
Building on previous work about the two-opinion case, we study the
multiple-opinion situation, finding that the model has infinitely many phase
transitions. Moreover, the formulas describing the end states of these
processes are remarkably simple when expressed as a function of $\beta =
\alpha/(1-\alpha)$.
