  Polymer networks invariably possess topological inhomogeneities in the form
of loops and dangling ends. The macroscopic properties of such materials are
directly dependent on the local cyclic topology around nodes and chains. Here,
a new formalism to model polymer network topology is presented based on the
concept of 3D nets that enables computation of these local cyclic topologies. A
cycle counting algorithm is developed which characterizes the relevant local
topological cycles around every node, enabling comparison of networks to ideal
nets based on local cycle distributions. Comparison of networks formed by
different simulation algorithms, using a topology-based distance metric,
reveals that polymer networks possess a fundamental cycle size which depends on
the topological proximity of crosslinkers during bond formation. This parameter
identifies distinct topological classes of networks and provides an effective
method to simulate a wide array of networks starting from suitable 3D nets.
This approach can be readily generalized beyond polymer networks, enabling
quantification of the cyclic topology and facilitating study of
topology-property correlations.
