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

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.

[1]  Sergey N. Dorogovtsev,et al.  Critical phenomena in complex networks , 2007, ArXiv.

[2]  A. Bray,et al.  Phase diagrams for dilute spin glasses , 1985 .

[3]  R. Pastor-Satorras,et al.  Generation of uncorrelated random scale-free networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Moore,et al.  Critical behavior of the three-dimensional Ising spin glass. , 1985, Physical review. B, Condensed matter.

[5]  UNIVERSAL FINITE-SIZE SCALING FUNCTIONS IN THE 3D ISING SPIN GLASS , 1999, cond-mat/9904246.

[6]  C. Herrero.  Antiferromagnetic Ising model in scale-free networks , 2009, 0908.3560.

[7]  Kurt Binder,et al.  Critical Properties from Monte Carlo Coarse Graining and Renormalization , 1981 .

[8]  P. L. Krapivsky,et al.  Social balance on networks : The dynamics of friendship and enmity , 2006 .

[9]  Yoseph Imry,et al.  Random-Field Instability of the Ordered State of Continuous Symmetry , 1975 .

[10]  R. Monasson.  Optimization problems and replica symmetry breaking in finite connectivity spin glasses , 1997, cond-mat/9707089.

[11]  H. Kappen,et al.  Spin-glass phase transitions on real-world graphs , 2004, cond-mat/0408378.

[12]  D. Thouless,et al.  Stability of the Sherrington-Kirkpatrick solution of a spin glass model , 1978 .

[13]  Albert-László Barabási,et al.  Internet: Diameter of the World-Wide Web , 1999, Nature.

[14]  A P Young,et al.  Study of the de Almeida-Thouless line using power-law diluted one-dimensional Ising spin glasses. , 2008, Physical review letters.

[15]  K. Adkins.  Theory of spin glasses , 1974 .

[16]  R. Zecchina,et al.  Ferromagnetic ordering in graphs with arbitrary degree distribution , 2002, cond-mat/0203416.

[17]  A. L. Ferreira,et al.  Critical behavior and correlations on scale-free small-world networks. Application to network design , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  B Wemmenhove,et al.  Replica symmetry breaking in the 'small world' spin glass , 2005 .

[19]  F. L. Metz,et al.  The phase diagram of Lévy spin glasses , 2009, 0910.1231.

[20]  Shore,et al.  Hysteresis and hierarchies: Dynamics of disorder-driven first-order phase transformations. , 1992, Physical review letters.

[21]  H. Katzgraber,et al.  Monte Carlo simulations of spin glasses at low temperatures , 2000, cond-mat/0007113.

[22]  M. Bartolozzi,et al.  Spin-glass behavior of the antiferromagnetic Ising model on a scale-free network , 2006 .

[23]  M. Mézard,et al.  The Bethe lattice spin glass revisited , 2000, cond-mat/0009418.

[24]  M. Mézard,et al.  Thermodynamics of the Lévy spin glass. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  W. Marsden.  I and J , 2012 .

[26]  Andrea Pelissetto,et al.  Critical structure factor in using systems. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  Edward A. Bender,et al.  The Asymptotic Number of Labeled Graphs with Given Degree Sequences , 1978, J. Comb. Theory A.

[28]  S. Redner,et al.  Connectivity of growing random networks. , 2000, Physical review letters.

[29]  J. Bouchaud,et al.  Mean field theory of dilute spin-glasses with power-law interactions , 1993 .

[30]  K. Hukushima,et al.  Exchange Monte Carlo Method and Application to Spin Glass Simulations , 1995, cond-mat/9512035.

[31]  Sang Hoon Lee,et al.  Random field Ising model on networks with inhomogeneous connections. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  K. Binder,et al.  Spin glasses: Experimental facts, theoretical concepts, and open questions , 1986 .

[33]  A. Hartmann,et al.  Bias in generation of random graphs. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  S. Bornholdt,et al.  Handbook of Graphs and Networks , 2012 .

[35]  Uncorrelated random networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  H. Katzgraber,et al.  Numerical studies of a one-dimensional three-spin spin-glass model with long-range interactions , 2009, 0908.2224.

[37]  S. Redner,et al.  Organization of growing random networks. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  Helmut G. Katzgraber,et al.  Universality in three-dimensional Ising spin glasses: A Monte Carlo study , 2006, cond-mat/0602212.

[39]  Frustration effects in antiferromagnets on planar random graphs , 2007, cond-mat/0701550.

[40]  S. Kirkpatrick,et al.  Solvable Model of a Spin-Glass , 1975 .

[41]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.