  The tail of a quantum spin network in the two-sphere is a $q$-series
associated to the network. We study the existence of the head and tail
functions of quantum spin networks colored by $2n$. We compute the $q$-series
for an infinite family of quantum spin networks and give the relation between
the tail of these networks and the tail of the colored Jones polynomial.
Finally, we show that the family of quantum spin networks under study satisfies
a natural product structure, making these networks satisfy a natural product
structure.
