Peter Ronhovde, Dandan Hu, Zohar Nussinov
We examine a global disorder transition when identifying structure in an arbitrary complex network ("community detection"). Earlier, we illustrated that this problem 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 \emph{general} weighted Potts model, leading to global disorder in the community structure of \emph{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 $T_\times\approx L|J_e|/[N\log q]$ 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 localcommunity detection algorithms (even those that employ global cost function parameters) so long as community evaluations are locally determined.
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http://arxiv.org/abs/1204.3649
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