%0 Book Section %1 statphys23_0595 %A Hinczewski, M. %A Berker, A.N. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K behavior berezinskii-kosterlitz-thouless fractal hierarchical lattices networks scale-free small-world statphys23 topic-11 transition %T Unusual Phase Transitions in Scale-Free Networks: Algebraic Order and Griffiths Singularites in Small-world and Fractal Hierarchical Lattices %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=595 %X We introduce a family of hierarchical lattices which exhibit a variety of structural features seen in real-world complex networks: from scale-free degree distributions and small-world behavior in the shortest-path length, to a modular organization characterized by fractal scaling laws---recently discovered in the WWW, protein interaction, and metabolic networks, among others. By varying the parameters which govern the construction of these deterministic networks, we can control their topological properties, tuning the degree exponent, fractal dimension, clustering coefficient, and the scaling of the diameter with size. The self-similar nature of the hierarchical lattices allow us to obtain analytic expressions for these features. When we examine cooperative behavior on the networks, their structural variety translates into highly unusual phase transitions and critical phenomena, even in a simple system like the ferromagnetic Ising model. We look at two specific networks, representative of the broader family, and obtain the exact thermodynamic behavior of the Ising model on these networks through a renormalization-group (RG) approach. In the first case we consider a scale-free network with a varying probability $p$ of long-range bonds, and calculate the RG flows of the quenched bond probability distribution. For $p < 0.494$, where the network is non-small-world, we find power-law critical behavior, with critical exponents continuously varying with $p$. For $p 0.494$, coinciding with the onset of small-world scaling, there is a very interesting phase transition: a Berezinskii-Kosterlitz-Thouless (BKT) singularity between a long-range-ordered phase at low temperatures and algebraic order at high temperatures, with zero magnetization but power-law decay of correlations. This is one of the few examples of such a singularity where there is no apparent mapping of the system onto an XY magnet. In the second case the network is composed of tightly-knit communities nested hierarchically with fractal scaling, and we vary the ratio $K/J$ of inter- to intra-community couplings. At high temperatures or small $K/J$ we have a disordered phase with a Griffiths singularity in the free energy, due to the presence of rare large clusters. As the temperature is lowered, true long-range order is not seen, but there is a transition to algebraic order. The existence of slowly decaying pair correlations is unexpected in a fat-tailed scale-free network, where correlations longer than nearest-neighbor are typically suppressed. The observed thermodynamic phenomena---such as Griffiths singularities, BKT transitions, and algebraic order---are not unique to these two examples. Using RG analysis and duality arguments, we can show that these unusual behaviors characterize a much larger class of hierarchical lattice complex networks.\\ References:\\ M. Hinczewski and A.N. Berker, Phys. Rev. E 73, 066126 (2006)\\ M. Hinczewski, cond-mat/0701349.

@incollection{statphys23_0595, abstract = {We introduce a family of hierarchical lattices which exhibit a variety of structural features seen in real-world complex networks: from scale-free degree distributions and small-world behavior in the shortest-path length, to a modular organization characterized by fractal scaling laws---recently discovered in the WWW, protein interaction, and metabolic networks, among others. By varying the parameters which govern the construction of these deterministic networks, we can control their topological properties, tuning the degree exponent, fractal dimension, clustering coefficient, and the scaling of the diameter with size. The self-similar nature of the hierarchical lattices allow us to obtain analytic expressions for these features. When we examine cooperative behavior on the networks, their structural variety translates into highly unusual phase transitions and critical phenomena, even in a simple system like the ferromagnetic Ising model. We look at two specific networks, representative of the broader family, and obtain the exact thermodynamic behavior of the Ising model on these networks through a renormalization-group (RG) approach. In the first case we consider a scale-free network with a varying probability $p$ of long-range bonds, and calculate the RG flows of the quenched bond probability distribution. For $p < 0.494$, where the network is non-small-world, we find power-law critical behavior, with critical exponents continuously varying with $p$. For $p \ge 0.494$, coinciding with the onset of small-world scaling, there is a very interesting phase transition: a Berezinskii-Kosterlitz-Thouless (BKT) singularity between a long-range-ordered phase at low temperatures and algebraic order at high temperatures, with zero magnetization but power-law decay of correlations. This is one of the few examples of such a singularity where there is no apparent mapping of the system onto an XY magnet. In the second case the network is composed of tightly-knit communities nested hierarchically with fractal scaling, and we vary the ratio $K/J$ of inter- to intra-community couplings. At high temperatures or small $K/J$ we have a disordered phase with a Griffiths singularity in the free energy, due to the presence of rare large clusters. As the temperature is lowered, true long-range order is not seen, but there is a transition to algebraic order. The existence of slowly decaying pair correlations is unexpected in a fat-tailed scale-free network, where correlations longer than nearest-neighbor are typically suppressed. The observed thermodynamic phenomena---such as Griffiths singularities, BKT transitions, and algebraic order---are not unique to these two examples. Using RG analysis and duality arguments, we can show that these unusual behaviors characterize a much larger class of hierarchical lattice complex networks.\\ {\it References:}\\ M. Hinczewski and A.N. Berker, Phys. Rev. E 73, 066126 (2006)\\ M. Hinczewski, cond-mat/0701349.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Hinczewski, M. and Berker, A.N.}, biburl = {https://www.bibsonomy.org/bibtex/2c8dddcb28824777e93d5f30adb861265/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {0010efdbe5a3a1bc954b799062509947}, intrahash = {c8dddcb28824777e93d5f30adb861265}, keywords = {behavior berezinskii-kosterlitz-thouless fractal hierarchical lattices networks scale-free small-world statphys23 topic-11 transition}, month = {9-13 July}, timestamp = {2007-06-20T10:16:24.000+0200}, title = {Unusual Phase Transitions in Scale-Free Networks: Algebraic Order and Griffiths Singularites in Small-world and Fractal Hierarchical Lattices}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=595}, year = 2007 }