%0 Journal Article %1 Molloy1995Critical %A Molloy, Michael %A Reed, Bruce %C Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15213; Equipe Combinatoire, CNRS, Universite Pierre et Marie Curie, Paris, France %D 1995 %I Wiley Subscription Services, Inc., A Wiley Company %J Random Struct. Alg. %K degree\_sequence, er-networks critical-points %N 2-3 %P 161--180 %R 10.1002/rsa.3240060204 %T A critical point for random graphs with a given degree sequence %U http://dx.doi.org/10.1002/rsa.3240060204 %V 6 %X Given a sequence of nonnegative real numbers λ0, λ1… which sum to 1, we consider random graphs having approximately λin vertices of degree i. Essentially, we show that if Σ i(i - 2)λi > 0, then such graphs almost surely have a giant component, while if Σ i(i-2)λ. < 0, then almost surely all components in such graphs are small. We can apply these results to Gn,p,Gn.M, and other well-known models of random graphs. There are also applications related to the chromatic number of sparse random graphs.

@article{Molloy1995Critical, abstract = {{Given a sequence of nonnegative real numbers λ0, λ1… which sum to 1, we consider random graphs having approximately λin vertices of degree i. Essentially, we show that if Σ i(i - 2)λi > 0, then such graphs almost surely have a giant component, while if Σ i(i-2)λ. < 0, then almost surely all components in such graphs are small. We can apply these results to Gn,p,Gn.M, and other well-known models of random graphs. There are also applications related to the chromatic number of sparse random graphs.}}, added-at = {2019-06-10T14:53:09.000+0200}, address = {Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15213; Equipe Combinatoire, CNRS, Universite Pierre et Marie Curie, Paris, France}, author = {Molloy, Michael and Reed, Bruce}, biburl = {https://www.bibsonomy.org/bibtex/2a8d7d5e58b97b7f031a716f9555ba780/nonancourt}, citeulike-article-id = {6544405}, citeulike-linkout-0 = {http://dx.doi.org/10.1002/rsa.3240060204}, citeulike-linkout-1 = {http://www3.interscience.wiley.com/cgi-bin/abstract/114287738/ABSTRACT}, day = 1, doi = {10.1002/rsa.3240060204}, interhash = {0998c00ecea7c5a7ea384898aa6d137c}, intrahash = {a8d7d5e58b97b7f031a716f9555ba780}, issn = {1098-2418}, journal = {Random Struct. Alg.}, keywords = {degree\_sequence, er-networks critical-points}, month = mar, number = {2-3}, pages = {161--180}, posted-at = {2010-01-15 15:37:21}, priority = {2}, publisher = {Wiley Subscription Services, Inc., A Wiley Company}, timestamp = {2019-08-23T10:59:34.000+0200}, title = {{A critical point for random graphs with a given degree sequence}}, url = {http://dx.doi.org/10.1002/rsa.3240060204}, volume = 6, year = 1995 }