Definition Erdős–Rényi model

a graph generated binomial model of erdős , rényi (p = 0.01)



in g(n, m) model, graph chosen uniformly @ random collection of graphs have n nodes , m edges. example, in g(3, 2) model, each of 3 possible graphs on 3 vertices , 2 edges included probability 1/3.
in g(n, p) model, graph constructed connecting nodes randomly. each edge included in graph probability p independent every other edge. equivalently, graphs n nodes , m edges have equal probability of









p

m


(
1
−
p

)




(


n
2


)



−
m


.


{\displaystyle p^{m}(1-p)^{{n \choose 2}-m}.}






the parameter p in model can thought of weighting function; p increases 0 1, model becomes more , more include graphs more edges , less , less include graphs fewer edges. in particular, case p = 0.5 corresponds case




2



(


n
2


)






{\displaystyle 2^{\binom {n}{2}}}

graphs on n vertices chosen equal probability.

the behavior of random graphs studied in case n, number of vertices, tends infinity. although p , m can fixed in case, can functions depending on n. example, statement

almost every graph in g(n, 2ln(n)/n) connected.

means

as n tends infinity, probability graph on n vertices edge probability 2ln(n)/n connected, tends 1.