Symmetries of weighted networks: weight approximation method and its application to food webs

Knowing which parts of a complex system have identical roles simplifies computations and reveals patterns in its network structure. Group theory has been applied to study symmetries in unweighted networks. However, in real-world weighted networks, edge weights are rarely equal, making exact symmetry uncommon. To study symmetries in weighted networks, we aggregate edge weights into a small number of discrete categories. The symmetries of these aggregated networks identify vertices with similar roles in the original weighted network. In food webs, this approach helps to quantify ecological co-existence and competition by assessing the functional substitutability of species. We apply our method to 250 empirical food webs, finding that symmetric vertices emerge even under weak approximations, typically forming small orbits of size two or three. These symmetric vertices can appear at any trophic level or network position. We also apply three symmetry measures to compare structural patterns at the network level.