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In the subcritical regime Erd\H{o}s-R\'enyi (ER) networks consist of finitetree components, which are non-extensive in the network size. The distributionof shortest path lengths (DSPL) of subcritical ER networks was recentlycalculated using a topological expansion [E. Katzav, O. Biham and A.K.Hartmann, Phys. Rev. E 98, 012301 (2018)]. The DSPL, which accounts for thedistance $\ell$ between any pair of nodes that reside on the same finite treecomponent, was found to follow a geometric distribution of the form $P(L=\ell |L < \infty) = (1-c) c^{\ell - 1}$, where $0 < c < 1$ is the mean degree of thenetwork. This result includes the contributions of trees of all possible sizesand topologies. Here we calculate the distribution of shortest path lengths$P(L=\ell | S=s)$ between random pairs of nodes that reside on the same treecomponent of a given size $s$. It is found that $P(L=\ell | S=s) =\frac{\ell+1}{s^{\ell}} \frac{(s-2)!}{(s-\ell-1)!}$. Surprisingly, thisdistribution does not depend on the mean degree $c$ of the network from whichthe tree components were extracted. This is due to the fact that the ensembleof tree components of a given size $s$ in subcritical ER networks is sampleduniformly 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 meandistance between random pairs of nodes on tree components of size $s$ satisfies${\mathbb E}[L|S=s] \sim \sqrt{s}$, unlike small-world networks in which themean distance scales logarithmically with $s$.

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