  Analytical results for the distribution of first hitting times of random
walks on Erd\H{o}s-R\'enyi networks are presented. Starting from a random
initial node, a random walker hops between adjacent nodes until it hits a node
which it has already visited before. At this point, the path terminates. The
path length, namely the number of steps, $d$, pursued by the random walker from
the initial node up to its termination is called the first hitting time or the
first intersection length. Using recursion equations, we obtain analytical
results for the tail distribution of the path lengths, $P(d > \ell)$. The
results are found to be in excellent agreement with numerical simulations. It
is found %turns out that the distribution $P(d > \ell)$ follows a product of an
exponential distribution and a Rayleigh distribution. The mean, median and
standard deviation of this distribution are also calculated, in terms of the
network size and its mean degree. The termination of an RW path may take place
either by backtracking to the previous node or by retracing of its path, namely
stepping into a node which has been visited two or more time steps earlier. We
obtain analytical results for the probabilities, $p_b$ and $p_r$, that the
cause of termination will be backtracking or retracing, respectively. It is
shown that in dilute networks the dominant termination scenario is backtracking
while in dense networks most paths terminate by retracing. We also obtain
expressions for the conditional distributions $P(d=\ell | b)$ and $P(d=\ell |
r)$, for those paths which are terminated by backtracking or by retracing,
respectively. These results provide useful insight into the general problem of
survival analysis and the statistics of mortality rates when two or more
termination scenarios coexist.
