  In this work, we investigate the temporal evolution of the degree of a given
vertex in a network by mapping the dynamics into a random walk problem in
degree space. We analyze when the degree approximates a pre-established value
through a parallel with the first-passage problem of random walks. The method
is illustrated on the time-dependent versions of the Erd\H{o}s-R\'enyi and
Watts-Strogatz models, which originally were formulated as static networks. We
have succeeded in obtaining an analytic form for the first and the second
moments of the first-passage time and showing how they depend on the size of
the network. The dominant contribution for large networks with $N$ vertices
indicates that these quantities scale on the ratio $N/p$, where $p$ is the
linking probability.
