  We study the problem of a particle/message that travels as a biased random
walk towards a target node in a network in the presence of traps. The bias is
represented as the probability $p$ of the particle to travel along the shortest
path to the target node. The efficiency of the transmission process is
expressed through the fraction $f_g$ of particles that succeed to reach the
target without being trapped. By relating $f_g$ with the number $S$ of nodes
visited before reaching the target, we firstly show that, for the unbiased
random walk, $f_g$ is inversely proportional to both the concentration $c$ of
traps and the size $N$ of the network. For the case of biased walks, a simple
approximation of $S$ provides an analytical solution that describes well the
behavior of $f_g$, especially for $p>0.5$. Also, it is shown that for a given
value of the bias $p$, when the concentration of traps is less than a threshold
value equal to the inverse of the Mean First Passage Time (MFPT) between two
randomly chosen nodes of the network, the efficiency of transmission is
unaffected by the presence of traps and almost all the particles arrive at the
target. As a consequence, for a given concentration of traps, we can estimate
the minimum bias that is needed to have unaffected transmission, especially in
the case of Random Regular (RR), Erd\H{o}s-R\'{e}nyi (ER) and Scale-Free (SF)
networks, where an exact expression (RR and ER) or an upper bound (SF) of the
MFPT is known analytically. We also study analytically and numerically, the
fraction $f_g$ of particles that reach the target on SF networks, where a
single trap is placed on the highest degree node. For the unbiased random walk,
we find that $f_g \sim N^{-1/(\gamma-1)}$, where $\gamma$ is the power law
exponent of the SF network.
