  Natural and man-made networks often possess locally tree-like sub-structures.
Taking such tree networks as our starting point, we show how the addition of
links changes the synchronization properties of the network. We focus on two
different methods of link addition. The first method adds single links that
create cycles of a well-defined length. Following a topological approach we
introduce cycles of varying length and analyze how this feature, as well as the
position in the network, alters the synchronous behaviour. We show that in
particular short cycles can lead to a maximum change of the Laplacian's
eigenvalue spectrum, dictating the synchronization properties of such networks.
The second method connects a certain proportion of the initially unconnected
nodes. We simulate dynamical systems on these network topologies, with the
nodes' local dynamics being either a discrete or continuous. Here our main
result is that a certain amount of additional links, with the relative position
in the network being crucial, can be beneficial to ensure stable
synchronization.
