  The Ising model on annealed complex networks with degree distribution
decaying algebraically as $p(K)\sim K^{-\lambda}$ has a second-order phase
transition at finite temperature if $\lambda> 3$. In the absence of space
dimensionality, $\lambda$ controls the transition strength; mean-field theory
applies for $\lambda >5$ but critical exponents are $\lambda$-dependent if
$\lambda < 5$. Here we show that, as for regular lattices, the celebrated
Lee-Yang circle theorem is obeyed for the former case. However, unlike on
regular lattices where it is independent of dimensionality, the circle theorem
fails on complex networks when $\lambda < 5$. We discuss the importance of this
result for both theory and experiments on phase transitions and critical
phenomena. We also investigate the finite-size scaling of Lee-Yang zeros in
both regimes as well as the multiplicative logarithmic corrections which occur
at $\lambda=5$.
