We illustrate a complex-network approach to study the phase spaces, or the microscopic state spaces, of spin glasses. By mapping the whole ground-state phase spaces of two-dimensional Edwards-Anderson bimodal (±J) spin glasses into networks, we discovered various structural properties of phase spaces via complex-network analysis. The Gaussian connectivity distributions of the phase-space networks show that the free spins exhibit a Gaussian distribution, whose variance provide a measure of frustration., The spectra of networks are Gaussian, which is proved to be exact when the system is large. The phase-space networks exhibit community structures whose strength is characterized by the modularity. In addition, we find the community structure of phase-space networks dramatically c...[ Read more ]

We illustrate a complex-network approach to study the phase spaces, or the microscopic state spaces, of spin glasses. By mapping the whole ground-state phase spaces of two-dimensional Edwards-Anderson bimodal (±J) spin glasses into networks, we discovered various structural properties of phase spaces via complex-network analysis. The Gaussian connectivity distributions of the phase-space networks show that the free spins exhibit a Gaussian distribution, whose variance provide a measure of frustration., The spectra of networks are Gaussian, which is proved to be exact when the system is large. The phase-space networks exhibit community structures whose strength is characterized by the modularity. In addition, we find the community structure of phase-space networks dramatically changes at the glass transition point where the concentration of anti-ferromagnetic bond is 0.103. Moreover, the phase-space networks exhibit fractal structures, which provide a real example of the conjecture that systems with long-range correlations have fractal phase spaces. These quantitative measurements of the ground states cast new light to the study of spin glasses. On the other hand, the phase-space networks of spin glasses share some common features with those of lattice gases and geometrically frustrated spin systems and establish a new class of complex networks with unique topology.