Growth and arrest of topological cycles in small physical networks

The chordless cycle sizes of spatially embedded networks are demonstrated to follow an exponential growth law similar to random graphs if the number of nodes [Formula: see text] is below a critical value [Formula: see text]. For covalent polymer networks, increasing the network size, as measured by the number of cross-link nodes, beyond [Formula: see text] results in a crossover to a new regime in which the characteristic size of the chordless cycles [Formula: see text] no longer increases. From this result, the onset and intensity of finite-size effects can be predicted from measurement of [Formula: see text] in large networks. Although such information is largely inaccessible with experiments, the agreement of simulation results from molecular dynamics, Metropolis Monte Carlo, and kinetic Monte Carlo suggests the crossover is a fundamental physical feature which is insensitive to the details of the network generation. These results show random graphs as a promising model to capture structural differences in confined physical networks.