**Growth and arrest of topological cycles in small physical networks**

TW Sirk, PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA, 117, 15394-15396 (2020).

DOI: 10.1073/pnas.2006305117

The chordless cycle sizes of spatially embedded networks are demonstrated to follow an exponential growth law similar to ran- dom graphs if the number of nodes N x is below a critical value N-*. For covalent polymer networks, increasing the network size, as measured by the number of cross-link nodes, beyond N-* results in a crossover to a new regime in which the characteristic size of the chordless cycles h(*) no longer increases. From this result, the onset and intensity of finite -size effects can be predicted from measurement of h(*) in large networks. Although such informa- tion 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 fun- damental 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.

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