  Phase transitions in spin glass type systems and, more recently, in related
computational problems have gained broad interest in disparate arenas. In the
current work, we focus on the "community detection" problem when cast in terms
of a general Potts spin glass type problem. As such, our results apply to
rather broad Potts spin glass type systems. Community detection describes the
general problem of partitioning a complex system involving many elements into
optimally decoupled "communities" of such elements. We report on phase
transitions between solvable and unsolvable regimes. Solvable region may
further split into "easy" and "hard" phases. Spin glass type phase transitions
appear at both low and high temperatures (or noise). Low temperature
transitions correspond to an "order by disorder" type effect wherein
fluctuations render the system ordered or solvable. Separate transitions appear
at higher temperatures into a disordered (or an unsolvable) phase. Different
sorts of randomness lead to disparate behaviors. We illustrate the spin glass
character of both transitions and report on memory effects. We further relate
Potts type spin systems to mechanical analogs and suggest how chaotic-type
behavior in general thermodynamic systems can indeed naturally arise in
hard-computational problems and spin-glasses. The correspondence between the
two types of transitions (spin glass and dynamic) is likely to extend across a
larger spectrum of spin glass type systems and hard computational problems. We
briefly discuss potential implications of these transitions in complex many
body physical systems.
