  Networks that are fragmented into small disconnected components are prevalent
in a large variety of systems. These include the secure communication networks
of commercial enterprises, government agencies and illicit organizations, as
well as networks that suffered multiple failures, attacks or epidemics. The
properties of such networks resemble those of subcritical random networks,
which consist of finite components, whose sizes are non-extensive.
Surprisingly, such networks do not exhibit the small-world property that is
typical in supercritical random networks, where the mean distance between pairs
of nodes scales logarithmically with the network size. Unlike supercritical
networks whose structure has been studied extensively, subcritical networks
have attracted little attention. A special feature of these networks is that
the statistical and geometric properties vary between different components and
depend on their sizes and topologies. The overall statistics of the network can
be obtained by a summation over all the components with suitable weights. We
use a topological expansion to perform a systematic analysis of the degree
distribution and the distribution of shortest path lengths (DSPL) on components
of given sizes and topologies in subcritical Erdos-Renyi (ER) networks. From
this expansion we obtain an exact analytical expression for the DSPL of the
entire subcritical network, in the asymptotic limit. The DSPL, which accounts
for all the pairs of nodes that reside on the same finite component (FC), is
found to follow a geometric distribution of the form $P_{\rm
FC}(L=\ell|L<\infty)=(1-c)c^{\ell-1}$, where $c<1$ is the mean degree. We
confirm the convergence to this asymptotic result using computer simulations.
Using the duality relations between subcritical and supercritical ER networks,
we obtain the DSPL on the non-giant components above the percolation
transition.
