  We use scale-free networks to study properties of the infected mass $M$ of
the network during a spreading process as a function of the infection
probability $q$ and the structural scaling exponent $\gamma$. We use the
standard SIR model and investigate in detail the distribution of $M$, We find
that for dense networks this function is bimodal, while for sparse networks it
is a smoothly decreasing function, with the distinction between the two being a
function of $q$. We thus recover the full crossover transition from one case to
the other. This has a result that on the same network a disease may die out
immediately or persist for a considerable time, depending on the initial point
where it was originated. Thus, we show that the disease evolution is
significantly influenced by the structure of the underlying population.
