Network of inherent structures in spin glasses: scaling and scale-free distributions

Ground-state phase-space structures of two-dimensional ±J spin glasses: A network approach

We illustrate a complex-network approach to study the phase spaces of spin glasses. By mapping the whole ground-state phase spaces of two-dimensional Edwards-Anderson bimodal (±J) spin glasses exactly into networks for analysis, we discovered various phase-space properties. The Gaussian connectivity distribution of the phase-space networks demonstrates that both the number of free spins and the visiting frequency of all microstates follow the Gaussian distribution. The spectra of phase-space networks are Gaussian, which is proven to be exact when the system is infinitely large. The phase-space networks exhibit community structures. By coarse graining to the community level, we constructed a network representing the entropy landscape of the ground state and discovered its scale-free property. The phase-space networks exhibit fractal structures, as a result of the rugged entropy landscape. Moreover, we show that the connectivity distribution, community structures, and fractal structures change drastically at the ferromagnetic-to-glass phase transition. These quantitative measurements of the ground states provide new insight into the study of spin glasses. The phase-space networks of spin glasses share a number of common features with those of lattice gases and geometrically frustrated spin systems and form a new class of complex networks with unique topology.

Glass transition and random walks on complex energy landscapes

We present a simple mathematical model of glassy dynamics seen as a random walk in a directed weighted network of minima taken as a representation of the energy landscape. Our approach gives a broader perspective to previous studies focusing on particular examples of energy landscapes obtained by sampling energy minima and saddles of small systems. We point out how the relation between the energies of the minima and their number of neighbors should be studied in connection with the network's global topology and show how the tools developed in complex network theory can be put to use in this context.

Partition of networks into basins of attraction

We study partition of networks into basins of attraction based on a steepest ascent search for the node of highest degree. Each node is associated with, or "attracted" to its neighbor of maximal degree, as long as the degree is increasing. A node that has no neighbors of higher degree is a peak, attracting all the nodes in its basin. Maximally random scale-free networks exhibit different behavior based on their degree distribution exponent gamma : For small gamma (broad distribution) networks are dominated by a giant basin, whereas for large gamma (narrow distribution) there are numerous basins, with peaks attracting mainly their nearest neighbors. We derive expressions for the first two moments of the number of basins. We also obtain the complete distribution of basin sizes for a class of hierarchical deterministic scale-free networks that resemble random nets. Finally, we generalize the problem to regular networks and lattices where all degrees are equal, and thus the attractiveness of a node must be determined by an assigned weight, rather than the degree. We derive the complete distribution of basins of attraction resulting from randomly assigned weights in one-dimensional chains.