Frustration of signed networks: How does it affect the thermodynamic properties of a system?

Signed networks with positive and negative interaction are widely observed in the real systems. The negative links would induce frustration, then affect global properties of the system. Based on previous studies, frustration of signed networks is investigated and quantified. Frustrations of $\pm J$ (Edwards-Anderson) Ising model with a concentration $p$ of negative bonds, constructed on different networks, such as triangular lattice, square lattice and random regular networks (RRN) with connectivity $k=6$ are estimated by theoretical and numerical approaches. Based on the quantitative measurement of frustration, its effects on phase transitions characterized by order parameter $q_{EA}$ are studied. The relationship of critical temperature $T_c$ with the quantified frustration $\mu$ is given by mean-field theory. It shows that $T_c$ decreases linearly with frustration $\mu$ . The theory is checked by numerical estimations, such as the Metropolis algorithm and Replica Symmetric Population Dynamics Algorithm. The numerical estimates are consistent well with the mean-field prediction.