Renormalization group for critical phenomena in complex networks

Frontiers in Physiology

Volumn 2 DEC, Issue , 2011, Pages

  Boettcher, S ^a   Brunson, C T ^a  

^a EMORY UNIVERSITY   (United States)

Author keywords

Complex networks; Critical phenomena; Hanoi networks; Ising model; Renormalization group

Indexed keywords

EID: 84866145991     PISSN: None     EISSN: 1664042X     Source Type: Journal    
DOI: 10.3389/fphys.2011.00102     Document Type: Article

References (28)

1. Andrade, J. S., Herrmann, H. J., Andrade, R. R S., and Da Silva, L. R. (2005). Apollonian networks: simultaneously scale-free, small world, Euclidean, space filling, and with matching graphs. Phys. Rev. Lett. 94, 018702.
2. Barabasi, A.-L. (2003). Linked: How Everything is Connected to Everything Else and What it Means for Business, Science, and Every day Life. New York: Plume.
3. Barabasi, A.-L., and Albert, R. (1999). Emergence of scaling in random networks. Science 286, 509.
4. Barthelemy, M. (2011). Spatial networks. Phys. Rep. 499, 1.
5. Bender, C. M., and Orszag, S. A. (1978). Advanced Mathematical Methods for Scientists and Engineers. New York: McGraw-Hill.
6. Berker, A. N., Hinczewski, M., and Netz, R. R. (2009). Critical percolation phase and thermal Bere zinskii-Kosterlitz-Thouless transition in a scale-free network with short-range and long-range random bonds. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80, 041118.
7. Berker, A. N., and Ostlund, S. (1979). Renormalisation-group calculations of finite systems: order parameter and specific heat for epitaxial ordering. J. Phys. Condens. Matter 12, 4961.
8. Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., and Hwang, D.-U. (2006). Complex networks: structure and dynamics. Phys. Rep. 424, 175.
9. Boettcher, S.,and Brunson, C. T. (2011). Fixed-point properties of the Ising ferromagnet on the Hanoi networks. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83, 021103.
10. Boettcher, S., Cook, J. L., and Ziff, R. M. (2009). Patch percolation on a hierarchical network with small-world bonds. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80, 041115.
11. Boettcher, S., and Goncalves, B. (2008). Anomalous diffusion on the Hanoi networks. Europhys. Lett. 84, 30002.
12. Boettcher, S., Gonçalves, B., and Guclu, H. (2008). Hierarchical regular small-world networks. J. Phys. A Math. Theor. 41, 252001.
13. Dorogovtsev, S. N., Goltsev, A. V., and Mendes, J. F. F. (2008). Critical phenomena in complex networks. Rev. Mod. Phys. 80, 1275.
14. Goldenfeld, N. (1992). Lectures on Phase Transitions and the Renormalization Group. Reading: Addison-Wesley.
15. Hasegawa, T., Sato, M., and Nemoto, K. (2010). Generating-function approach for bond percolation in hierarchical networks. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 82, 046101.
16. Hinczewski, M. (2007). Griffiths singularities and algebraic order in the exact solution of an Ising model on a fractal modular network. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75, 061104.
17. Hinczewski, M., and Berker, A. N. (2006). Inverted Berezinskii Kosterlitz-Thouless singularity and high-temperature algebraic order in an Ising model on a scale-free hierarchical-lattice small-world network. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73, 066126.
18. Kadanoff, L. P. (1966). Scaling laws for Ising models near Tc. Physics 2, 263.
19. Katzgraber, H. G. and Young, A. P. (2003). Monte Carlo studies of the one-dimensional Ising spin glass with power-law interactions. Phys. Rev. B Condens. Matter Mater. Phys. 67, 134410.
20. Kleinert, H. (2004). Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets. Singapore: World Scientific.
21. Kotliar, G., Anderson, P. W., and Stein, D. L. (1983). One-dimensional spin-glass model with long-range random interactions. Phys. Rev. B Condens. Matter Mater. Phys. 27,R602.
22. Mantegna, R. N., and Stanley, H. E. (1999). An Introduction to Econo-physics: Correlations and Complexity in Finance. Cambridge: Cambridge University Press.
23. Nogawa, T., and Hasegawa, T. (2009). Monte Carlo simulation study of the two-stage percolation transition in enhanced binary trees. J. Phys. A Math. Theor. 42, 145001.
24. Plischke, M., and Bergersen, B. (1994). Equilibrium Statistical Physics, 2nd Edn. Singapore: World Scientific.
25. Trusina, A., Maslov, S., Minnhagen, P., and Sneppen, K. (2004). Hierarchy measures in complex networks. Phys. Rev. Lett. 92, 178702.
26. Watts, D. J., and Strogatz, S. H. (1998). Collective dynamics of "small-world" networks. Nature 393, 440.
27. Wilson, K. G. (1971). Renormalization group and critical phenomena I. Renormalization group and the Kadanoff scaling picture. Phys. Rev. B Condens. Matter Mater. Phys. 4, 3174.
28. Wilson, K. G., and Fisher, M. E. (1972). Critical exponents in 3.99 dimensions. Phys. Rev. Lett. 28, 240.