Unknottedness of the Graph of Pairwise Stable Networks & Network Dynamics

Abstract

We extend Bich-Fixary's topological structure theorem about 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 network dynamic 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 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 consequenceon the indices of these network dynamics at any pairwise stable network, a concept that we link to genericity using Bich-Fixary's oddness.

Keywords: Pairwise Stability, Unknottedness Theorem, Network Dynamic, Genericity