  In this paper, we study the stability problem of a star-shaped network of
elastic strings with a local Kelvin-Voigt damping. Under the assumption that
the damping coefficients have some singularities near the transmission point,
we prove that the semigroup corresponding to the system is polynomially stable
and the decay rates depends on the speed of the degeneracy. This result
improves the decay rate of the semigroup associated to the system on an earlier
result of Z.~Liu and Q.~Zhang in \cite{LZ} involving the wave equation with
local Kelvin-Voigt damping and non-smooth coefficient at interface.
