  We consider the question of existence of a unique invariant probability
distribution which satisfies some evolutionary property. The problem arises
from the random graph theory but to answer it we treat it as a dynamical system
in the functional space, where we look for a global attractor. We consider the
following bifurcation problem: Given a probability measure $\mu$, which
corresponds to the weight distribution of a link of a random graph we form a
positive linear operator $\Phi$
  (convolution) on distribution functions and then we analyze a family of its
exponents with a parameter $\lambda$ which corresponds to connectivity of a
sparse random graph. We prove that for every measure $\mu$ (\emph{i.e.},
convolution $\Phi$) and every $\lambda< e$ there exists a unique globally
attracting fixed point of the operator, which yields the existence and
uniqueness of the limit probability distribution on the random graph. This
estimate was established earlier \cite{KarpSipser} for deterministic weight
distributions (Dirac measures $\mu$) and is known as $e$-cutoff phenomena, as
for such distributions and
  $\lambda>e$ there is no fixed point attractor. We thus establish this
phenomenon in a much more general sense.
