Synchronization of Dirac-Bianconi driven oscillators

In dynamical systems on networks, one assigns the dynamics to nodes, which are then coupled via links. This approach does not account for group interactions and dynamics on links and other higher dimensional structures. Higher-order network theory addresses this by considering variables defined on nodes, links, triangles, and higher-order simplices, called topological signals (or cochains). Moreover, topological signals of different dimensions can interact through the Dirac-Bianconi operator, which allows coupling between topological signals defined, for example, on nodes and links. Such interactions can induce various dynamical behaviors, for example, periodic oscillations. The oscillating system consists of topological signals on nodes and links whose dynamics are driven by the Dirac-Bianconi coupling, hence, which we call it Dirac-Bianconi driven oscillator. Using the phase reduction method, we obtain a phase description of this system and apply it to the study of synchronization between two such oscillators. This approach offers a way to analyze oscillatory behaviors in higher-order networks beyond the node-based paradigm, while providing a ductile modeling tool for node- and edge-signals.

💡 Research Summary

The paper introduces a novel class of oscillatory systems called “Dirac‑Bianconi driven oscillators,” which arise when dynamical variables are placed on both nodes (0‑cochains) and edges (1‑cochains) of a simplicial complex and are coupled exclusively through the Dirac‑Bianconi operator. Traditional network dynamics assign state variables only to nodes, treating links merely as conduits for pairwise interactions. This node‑centric view cannot capture phenomena where higher‑order structures (e.g., edges, triangles) carry their own dynamics. By employing algebraic‑topological tools—boundary operators B₁ and their transposes—the authors construct the Dirac‑Bianconi operator

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