  If we consider the contact process with infection rate $\lambda$ on a random
graph on $n$ vertices with power law degree distributions, mean field
calculations suggest that the critical value $\lambda_c$ of the infection rate
is positive if the power $\alpha>3$. Physicists seem to regard this as an
established fact, since the result has recently been generalized to bipartite
graphs by G\'{o}mez-Garde\~{n}es et al. [Proc. Natl. Acad. Sci. USA 105 (2008)
1399--1404]. Here, we show that the critical value $\lambda_c$ is zero for any
value of $\alpha>3$, and the contact process starting from all vertices
infected, with a probability tending to 1 as $n\to\infty$, maintains a positive
density of infected sites for time at least $\exp(n^{1-\delta})$ for any
$\delta>0$. Using the last result, together with the contact process duality,
we can establish the existence of a quasi-stationary distribution in which a
randomly chosen vertex is occupied with probability $\rho(\lambda)$. It is
expected that $\rho(\lambda)\sim C\lambda^{\beta}$ as $\lambda \to0$. Here we
show that $\alpha-1\le\beta\le2\alpha-3$, and so $\beta>2$ for $\alpha>3$. Thus
even though the graph is locally tree-like, $\beta$ does not take the mean
field critical value $\beta=1$.
