Existence of Solutions for Hybrid Dynamical Systems on Graph State-Space
This paper proposes a framework to ensure the existence of dynamical system trajectories in the state space of labeled, weighted, and attributed graphs. The evolution of such a system exhibits hybrid behavior: discrete jumps affecting the topology-the emergence and disappearance of vertices and edges over time-as well as vertex attributes and edge weights, combined with a continuous evolution of these same attributes and weights. To address the discrete behavior, an appropriate algebraic structure for the graph space is proposed; the analysis of its properties shows the existence of a new mathematical structure: a semi-vector space over the field of real numbers, whereas the literature only describes semi-vector spaces over semi-fields. The continuous behavior is modeled by differential equations. To facilitate formal treatment, the graph space is embedded, via a semi-linear mapping, into a new space, called variable-basis space, introduced in this work. The system's evolution model is then formulated within this space-the image of the graph space in the variable-basis space-under the framework of hybrid dynamical systems theory. A general result on the existence of solutions is established. Finally, this framework is applied to model and simulate the dynamics of the gut microbiota under antibiotic treatment followed by bacteriotherapy, where a generalized Lotka-Volterra (gLV) model describes the evolution of species abundances.