Unknottedness of graphs of pairwise stable networks & network dynamics

We extend Bich–Fixary’s topological structure theorem about graphs of pairwise stable networks. Specifically, we show that certain graphs of pairwise stable networks are not only homeomorphic to their underlying space of societies but are, in fact, ambient isotopic to a trivial copy of this space. This result aligns with Demichelis–Germano’s unknottedness theorem and Predtetchinski’s unknottedness theorem. Furthermore, we introduce the notion of network dynamics which refers to families of vector fields on the set of weighted networks whose zeros correspond to pairwise stable networks. We leverage our version of the unknottedness theorem to demonstrate that most network dynamics can be continuously connected to one another without introducing additional zeros. Finally, we show that this result has a significant consequence on the indices of these network dynamics at any pairwise stable network — a concept that we connect to genericity using Bich–Fixary’s oddness theorem.