Existence of Solutions for Hybrid Dynamical Systems on Graph State-Space

TL;DR

This paper introduces a mathematical framework for hybrid dynamical systems on graph state-spaces, ensuring the existence of solutions for systems with both discrete and continuous changes, and applies it to microbiota dynamics.

Contribution

It develops a novel semi-vector space structure for graph state-spaces and embeds it into a variable-basis space, enabling formal analysis of hybrid graph dynamics.

Findings

Established a general existence theorem for solutions of hybrid graph systems.

Modeled microbiota dynamics under antibiotics using the framework.

Introduced a semi-vector space structure over real numbers for graph analysis.

Abstract

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.