Unknottedness of the Graph of Pairwise Stable Networks & Network Dynamics

We extend Bich-Fixary's theorem about the topological structure of the graph of pairwise stable networks. Namely, we show that the graph of pairwise stable networks is not only homeomorphic to the space of societies, but that it is ambient isotopic to a trivial copy of this space (a result in the line of Demichelis-Germano's unknottedness theorem. Furthermore, we introduce the notion of (extended) network dynamics which refers to families of vector fields on the set of weighted networks whose zeros correspond to pairwise stable networks. We use our version of the unknottedness theorem to show that most of network dynamics can be continuously connected to each other, without adding additional zeros. Finally, we prove that this result has an important consequence on the indices of these network dynamics at any pairwise stable network, a concept that we link to genericity using Bich-Fixary's oddness theorem