  We use an exact renormalization-group transformation to study the Ising model
on a complex network composed of tightly-knit communities nested hierarchically
with the fractal scaling recently discovered in a variety of real-world
networks. Varying the ratio K/J of of inter- to intra-community coupling, we
obtain an unusual phase diagram: at high temperatures or small K/J we have a
disordered phase with a Griffiths singularity in the free energy, due to the
presence of rare large clusters, which we analyze through the Yang-Lee zeros in
the complex magnetic field plane. As the temperature is lowered, true
long-range order is not seen, but there is a transition to algebraic order,
where pair correlations have power-law decay with distance, reminiscent of the
XY model. The transition is infinite-order at small K/J, and becomes
second-order above a threshold value (K/J)_m. The existence of such slowly
decaying correlations is unexpected in a fat-tailed scale-free network, where
correlations longer than nearest-neighbor are typically suppressed.
