  Networks of elastic fibers are ubiquitous in biological systems and often
provide mechanical stability to cells and tissues. Fiber reinforced materials
are also common in technology. An important characteristic of such materials is
their resistance to failure under load. Rupture occurs when fibers break under
excessive force and when that failure propagates. Therefore it is crucial to
understand force distributions. Force distributions within such networks are
typically highly inhomogeneous and are not well understood. Here we construct a
simple one-dimensional model system with periodic boundary conditions by
randomly placing linear springs on a circle. We consider ensembles of such
networks that consist of $N$ nodes and have an average degree of connectivity
$z$, but vary in topology. Using a graph-theoretical approach that accounts for
the full topology of each network in the ensemble, we show that, surprisingly,
the force distributions can be fully characterized in terms of the parameters
$(N,z)$. Despite the universal properties of such $(N,z)$-ensembles, our
analysis further reveals that a classical mean-field approach fails to capture
force distributions correctly. We demonstrate that network topology is a
crucial determinant of force distributions in elastic spring networks.
