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arXiv · 1202.1153

Boolean decision problems with competing interactions on scale-free networks: Critical thermodynamics

Helmut G. Katzgraber·Katharina Janzen·Creighton K. Thomas
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Abstract

We study the critical behavior of Boolean variables on scale-free networks with competing interactions (Ising spin glasses). Our analytical results for the disorder-network-decay-exponent phase diagram are verified using Monte Carlo simulations. When the probability of positive (ferromagnetic) and negative (antiferromagnetic) interactions is the same, the system undergoes a finite-temperature spin-glass transition if the exponent that describes the decay of the interaction degree in the scale-free graph is strictly larger than 3. However, when the exponent is equal to or less than 3, a spin-glass phase is stable for all temperatures. The robustness of both the ferromagnetic and spin-glass phases suggests that Boolean decision problems on scale-free networks are quite stable to local perturbations. Finally, we show that for a given decay exponent spin glasses on scale-free networks seem to obey universality. Furthermore, when the decay exponent of the interaction degree is larger than 4 in the spin-glass sector, the universality class is the same as for the mean-field Sherrington-Kirkpatrick Ising spin glass.

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Helmut G. Katzgraber, Katharina Janzen, Creighton K. Thomas. 2012-09-13. Boolean decision problems with competing interactions on scale-free networks: Critical thermodynamics. https://doi.org/10.1103/physreve.86.031116

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