Distribution of shortest path lengths on trees of a given size in subcritical Erdos-Renyi networks

In the subcritical regime Erdos-Renyi (ER) networks consist of finite tree components, which are nonextensive in the network size. The distribution of shortest path lengths (DSPL) of subcritical ER networks was recently calculated using a topological expansion [E. Katzav, O. Biham, and A. K. Hartmann, Phys. Rev. E 98, 012301 (2018)]. The DSPL, which accounts for the distance l between any pair of nodes that reside on the same finite tree component, was found to follow a geometric distribution of the form P( L = l|L < infinity) = (1 - c)c(l-1), where 0 < c < 1 is the mean degree of the network. This result includes the contributions of trees of all possible sizes and topologies. Here we calculate the distribution of shortest path lengths P(L = l|S = s) between random pairs of nodes that reside on the same tree component of a given size s. It is found that P(L = l| S = s) = l+1/s(l) (s-2)!/(s-l-1)!. Surprisingly, this distribution does not depend on the mean degree c of the network from which the tree components were extracted. This is due to the fact that the ensemble of tree components of a given size s in subcritical ER networks is sampled uniformly from the set of labeled trees of size s and thus does not depend on c. The moments of the DSPL are also calculated. It is found that the mean distance between random pairs of nodes on tree components of size s satisfies E[L|S = s] similar to root s, unlike small-world networks in which the mean distance scales logarithmically with s.