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Abstract

Filamentous polymer networks govern the mechanical properties of many biological materials. Force
distributions within these networks are typically highly inhomogeneous, and, although the importance of
force distributions for structural properties is well recognized, they are far from being understood
quantitatively. Using a combination of probabilistic and graph-theoretical techniques, we derive force
distributions in a model system consisting of ensembles of random linear spring networks on a circle. We
show that characteristic quantities, such as the mean and variance of the force supported by individual springs,
can be derived explicitly in terms of only two parameters: (i) average connectivity and (ii) number of nodes.
Our analysis shows that a classical mean-field approach fails to capture these characteristic quantities
correctly. In contrast, we demonstrate that network topology is a crucial determinant of force distributions in
an elastic spring network. Our results for 1D linear spring networks readily generalize to arbitrary dimensions.