  We examine a global disorder transition when identifying community structure
in an arbitrary complex network. Earlier, we illustrated [Phil. Mag. 92, 406
(2012)] that "community detection" (CD) generally exhibits disordered (or
unsolvable) and ordered (solvable) phases of both high and low computational
complexity along with corresponding transitions from regular to chaotic
dynamics in derived systems. Using an exact generalized dimensional reduction
inequality, multivariate Tutte polynomials, and other considerations, we
illustrate how increasing the number of communities q emulates increasing the
heat bath temperature T for a general weighted Potts model, leading to global
disorder in the community structure of arbitrary large graphs. Dimensional
reduction bounds lead to results similar to those suggested by mean-field type
approaches. Large systems tend toward global insolvability in the limit of
large q above a crossover temperature $T_\times\approx L|J_e|/[N\ln{} q]$ where
|J_e| is a typical interaction strength, L is the number of edges, and N is the
number of nodes. For practical system sizes, a solvable phase is generally
accessible at low T. The global nature of the disorder transition does not
preclude solutions by local CD algorithms (even those that employ global cost
function parameters) as long as community evaluations are locally determined.
