%0 Book Section %1 statphys23_0935 %A Agrawal, H. %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 complex homogeneity network networks order phase statphys23 systems topic-11 transitions %T Topological phase transitions in networks give truncated power-law tails in degree distributions %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=935 %X In any given suitably ordered collection of networks $\G\$ major topological changes can be detected systematically by analyzing the variation of homogeneity $Łambda$ with the parameter affecting the ordering, say $\xi$, where \ Łambda(G) = 1+\barz+121-P(z_\max)1+z_\max, \ where the network $G$ has $P(z_\max)$ fraction of hubs of degree $z_\max$ and average degree in the network is $z$. These changes, detectable as phase transitions, are signaled by singular behavior of the second and higher derivatives of homogeneity with respect to the ordering parameter or another parameter that is smoothly isomorphic to it, i.e., \ d^nŁambdad\xi^n = \pmınfty \ where $n 2$. The case $n = 2$ corresponds to first-order and $n > 2$ corresponds to continuous phase transitions. We show that irrespective of whether the phase transitions are first-order or continuous, the networks corresponding to the point of transitions, or the flat regions surrounded by such points, have truncated power-law tails with the cumulative distributions behaving like $a x^-\alpha \exp(-b x^\beta)$, where $x = (z+1)/(z_\max+1)$. Furthermore, for all finite sized networks the truncation function is an stretched exponential, i.e., $1$. We also determine the order of the phase transitions from the observed behavior of the tails.

@incollection{statphys23_0935, abstract = {In any given suitably ordered collection of networks $\{G\}$ major topological changes can be detected systematically by analyzing the variation of homogeneity $\Lambda$ with the parameter affecting the ordering, say $\xi$, where \[ \Lambda(G) = \frac{1+\bar{z}+\frac{1}{2}[1-P(z_{\max})]}{1+z_{\max}}, \] where the network $G$ has $P(z_{\max})$ fraction of hubs of degree $z_{\max}$ and average degree in the network is $\bar{z}$. These changes, detectable as phase transitions, are signaled by singular behavior of the second and higher derivatives of homogeneity with respect to the ordering parameter or another parameter that is smoothly isomorphic to it, i.e., \[ \frac{d^{n}\Lambda}{d\xi^{n}} = \pm\infty \] where $n \ge 2$. The case $n = 2$ corresponds to first-order and $n > 2$ corresponds to continuous phase transitions. We show that irrespective of whether the phase transitions are first-order or continuous, the networks corresponding to the point of transitions, or the flat regions surrounded by such points, have truncated power-law tails with the cumulative distributions behaving like $a x^{-\alpha} \exp(-b x^{\beta})$, where $x = (z+1)/(z_{\max}+1)$. Furthermore, for all finite sized networks the truncation function is an stretched exponential, i.e., $\beta \ne 1$. We also determine the order of the phase transitions from the observed behavior of the tails.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Agrawal, H.}, biburl = {https://www.bibsonomy.org/bibtex/23982ee523e43fa2f2d378309f2f5f823/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {385f33a33cf529f2adffbf69f6fb89d1}, intrahash = {3982ee523e43fa2f2d378309f2f5f823}, keywords = {complex homogeneity network networks order phase statphys23 systems topic-11 transitions}, month = {9-13 July}, timestamp = {2007-06-20T10:16:34.000+0200}, title = {Topological phase transitions in networks give truncated power-law tails in degree distributions}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=935}, year = 2007 }