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Networks that are fragmented into small disconnected components are prevalentin a large variety of systems. These include the secure communication networksof commercial enterprises, government agencies and illicit organizations, aswell as networks that suffered multiple failures, attacks or epidemics. Theproperties 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 istypical in supercritical random networks, where the mean distance between pairsof nodes scales logarithmically with the network size. Unlike supercriticalnetworks whose structure has been studied extensively, subcritical networkshave attracted little attention. A special feature of these networks is thatthe statistical and geometric properties vary between different components anddepend on their sizes and topologies. The overall statistics of the network canbe obtained by a summation over all the components with suitable weights. Weuse a topological expansion to perform a systematic analysis of the degreedistribution and the distribution of shortest path lengths (DSPL) on componentsof given sizes and topologies in subcritical Erdos-Renyi (ER) networks. Fromthis expansion we obtain an exact analytical expression for the DSPL of theentire subcritical network, in the asymptotic limit. The DSPL, which accountsfor all the pairs of nodes that reside on the same finite component (FC), isfound to follow a geometric distribution of the form $P_{\rmFC}(L=\ell|L<\infty)=(1-c)c^{\ell-1}$, where $c<1$ is the mean degree. Weconfirm 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 percolationtransition.

The distribution of shortest path lengths in subcritical Erd\H{o}s-R\'enyi networks