  When three species compete cyclically in a well-mixed, stochastic system of
$N$ individuals, extinction is known to typically occur at times scaling as the
system size $N$. This happens, for example, in rock-paper-scissors games or
conserved Lotka-Volterra models in which every pair of individuals can interact
on a complete graph. Here we show that if the competing individuals also have a
"social temperament" to be either introverted or extroverted, leading them to
cut or add links respectively, then long-living state in which all species
coexist can occur when both introverts and extroverts are present. These states
are non-equilibrium quasi-steady states, maintained by a subtle balance between
species competition and network dynamcis. Remarkably, much of the phenomena is
embodied in a mean-field description. However, an intuitive understanding of
why diversity stabilizes the co-evolving node and link dynamics remains an open
issue.
