  A regular lattice in which the sites can have long range connections at a
distance l with a probabilty $P(l) \sim l^{-\delta}$, in addition to the short
range nearest neighbour connections, shows small-world behaviour for $0 \le
\delta < \delta_c$. In the most appropriate physical example of such a system,
namely the linear polymer network, the exponent $\delta$ is related to the
exponents of the corresponding n-vector model in the $n \to 0$ limit, and its
value is less than $\delta_c$. Still, the polymer networks do not show
small-world behaviour. Here, we show that this is due a (small value)
constraint on the number q of long range connections per monomer in the
network. In the general $\delta - q$ space, we obtain a phase boundary
separating regions with and without small-world behaviour, and show that the
polymer network falls marginally in the regular lattice region.
