Abstract | The Ising model on annealed complex networks with degree distribution decaying algebraically as p(K)∼K−λ has a second-order phase transition at finite temperature if λ>3. In the absence of space dimensionality, λ controls the transition strength; mean-field theory applies for λ>5 but critical exponents are λ-dependent if λ<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 λ<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 λ=5. |