  The metric structure of bosonic scale-free networks and fermionic Cayley-tree
networks is analyzed focousing on the directed distance of nodes from the
origin. The topology of the netwoks strongly depends on the dynamical parameter
$T$, called temperature. At $T=\infty$ we show analytically that the two
networks have a similar behavior: the distance of a generic node from the
origin of the network scales as the logarithm of the number of nodes in the
network. At T=0 the two networks have an opposite behavior: the bosonic network
remains very clusterized (the distance from the origin remains constant as the
network increases the number of nodes) while the fermionic network grows
following a single branch of the tree and the distance from the origin grows as
a power-law of the number of nodes in the network.
