%0 Journal Article %1 Rozenfeld2005Statistics %A Rozenfeld, Hernán D. %A Kirk, Joseph E. %A Bollt, Erik M. %A ben Avraham, Daniel %D 2005 %I Institute of Physics Publishing %J Journal of Physics A: Mathematical and General %K cycles, er-networks scale-free-networks %N 21 %P 4589--4595 %R 10.1088/0305-4470/38/21/005 %T Statistics of cycles: how loopy is your network? %U http://dx.doi.org/10.1088/0305-4470/38/21/005 %V 38 %X We study the distribution of cycles of length h in large networks (of size N 1) and find it to be an excellent ergodic estimator, even in the extreme inhomogeneous case of scale-free networks. The distribution is sharply peaked around a characteristic cycle length, h \~ N α . Our results suggest that h and the exponent α might usefully characterize broad families of networks. In addition to an exact counting of cycles in hierarchical nets, we present a Monte Carlo sampling algorithm for approximately locating h and reliably determining α. Our empirical results indicate that for small random scale-free nets of degree exponent λ, α = 1/(λ − 1), and α grows as the nets become larger.

@article{Rozenfeld2005Statistics, abstract = {{We study the distribution of cycles of length h in large networks (of size N 1) and find it to be an excellent ergodic estimator, even in the extreme inhomogeneous case of scale-free networks. The distribution is sharply peaked around a characteristic cycle length, h \~{} N α . Our results suggest that h and the exponent α might usefully characterize broad families of networks. In addition to an exact counting of cycles in hierarchical nets, we present a Monte Carlo sampling algorithm for approximately locating h and reliably determining α. Our empirical results indicate that for small random scale-free nets of degree exponent λ, α = 1/(λ − 1), and α grows as the nets become larger.}}, added-at = {2019-06-10T14:53:09.000+0200}, author = {Rozenfeld, Hern\'{a}n D. and Kirk, Joseph E. and Bollt, Erik M. and ben Avraham, Daniel}, biburl = {https://www.bibsonomy.org/bibtex/2e9d5d00deaff311e0930c08c49458dca/nonancourt}, citeulike-article-id = {201399}, citeulike-linkout-0 = {http://dx.doi.org/10.1088/0305-4470/38/21/005}, citeulike-linkout-1 = {http://www.ingentaconnect.com/content/iop/jphysa/2005/00000038/00000021/art00005}, citeulike-linkout-2 = {http://iopscience.iop.org/0305-4470/38/21/005}, day = 27, doi = {10.1088/0305-4470/38/21/005}, interhash = {c3b3a6bfbe1a2de67e862c28142b2bbb}, intrahash = {e9d5d00deaff311e0930c08c49458dca}, issn = {0305-4470}, journal = {Journal of Physics A: Mathematical and General}, keywords = {cycles, er-networks scale-free-networks}, month = may, number = 21, pages = {4589--4595}, posted-at = {2014-10-21 11:32:59}, priority = {2}, publisher = {Institute of Physics Publishing}, timestamp = {2019-08-01T16:13:01.000+0200}, title = {{Statistics of cycles: how loopy is your network?}}, url = {http://dx.doi.org/10.1088/0305-4470/38/21/005}, volume = 38, year = 2005 }