On the evolution of random graphs

(n) k edges have equal probabilities to be chosen as the next one . We shall 2 study the "evolution" of such a random graph if N is increased . In this investigation we endeavour to find what is the "typical" structure at a given stage of evolution (i . e . if N is equal, or asymptotically equal, to a given function N(n) of n) . By a "typical" structure we mean such a structure the probability of which tends to 1 if n -* + when N = N(n) . If A is such a property that lim Pn,N,(n ) ( A) = 1, we shall say that „almost all" graphs Gn,N(n) n--possess this property .