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

Violation of Lee-Yang circle theorem for Ising phase transitions on complex networks