From simple to complex networks: inherent structures, barriers and valleys in the context of spin glasses

Abstract

Given discrete degrees of freedom (spins) on a graph interacting via an energy function, what can be said about the energy local minima and associated inherent structures? Using the lid algorithm in the context of a spin glass energy function, we investigate the properties of the energy landscape for a variety of graph topologies. First, we find that the multiplicity Ns of the inherent structures generically has a lognormal distribution. In addition, the large volume limit of ln<Ns>/<lnNs> differs from unity, except for the Sherrington-Kirkpatrick model. Second, we find simple scaling laws for the growth of the height of the energy barrier between the two degenerate ground states and the size of the associated valleys. For finite connectivity models, changing the topology of the underlying graph does not modify qualitatively the energy landscape, but at the quantitative level the models can differ substantially.

10 pages, 9 figs, slightly improved presentation, more references, accepted for publication in Phys Rev E

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Cited by in corpus (14)

• Does the brain behave like a (complex) network? I. Dynamics
• Random Walk over Basins of Attraction to Construct Ising Energy Landscapes
• The energy landscape networks of spin-glasses
• Network of inherent structures in spin glasses: scaling and scale-free distributions
• Glass transition and random walks on complex energy landscapes
• Efficient exploration of discrete energy landscapes
• The Statistics of the Number of Minima in a Random Energy Landscape
• Complex networks and glassy dynamics: walks in the energy landscape
• Counting metastable states of Ising spin glasses on arbitrary graphs
• Distribution of metastable states of Ising spin glasses
• Scattering theory of transport through disordered magnets
• Phase Transition in a Random Minima Model: Mean Field Theory and Exact Solution on the Bethe Lattice
• Barriers and local minima in energy landscapes of stochastic local search
• Stochastic Neural Networks for Automatic Cell Tracking in Microscopy Image Sequences of Bacterial Colonies