Topology counts: Force distributions in random spring networks

Under large stresses, filamentous polymer networks often exhibit highly inhomogeneous force distributions. Forces propagate along nontrivial loops that make up a network's topology, where each nontrivial loop is a path that connects one boundary of the system to another. By introducing a toy model comprising ensembles of one-dimensional, periodic, linear spring networks, we demonstrate that network topology is a crucial determinant of force distributions in elastic spring networks. In contrast to a mean-field approach, our graph-theoretic approach explicitly accounts for the full network topology and is in excellent agreement with numerical simulations. Our results are a first step towards understanding the concentration of forces along a few nontrivial loops, known as force chains, that has been observed in stressed 2D and 3D nonlinear networks.