Size of quantum networks

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.