Mathematical Modeling of Signed Graphs for Balancing Dynamic Systems
DOI:
https://doi.org/10.29020/nybg.ejpam.v18i4.6877Keywords:
Signed graph, Balance sign graph, Equilibrium point, Damping factorAbstract
Signed graphs encode cooperative and antagonistic interactions in dynamical systems. In this paper we will study how explicit damping transforms unstable linearized dynamics into stable behavior, mapping the system Jacobian to a signed weighted matrix to diagnose destabilizing pathways and guide damping or edge reweighting. Beyond graphical intuition, stability is certified by a Lyapunov/spectral test on the symmetric part S with diagonal damping D, namely D −S ≻ 0 (equivalently, λmax(S) < mini di). Using the inverted pendulum as a benchmark, we derive a cleaned state-space model, show undamped instability, and demonstrate how viscous damping enforces the certificate. We provide a concise numerical verification (eigenvalue check
and time-response illustration). The results clarify when signed graphs aid design and placement of damping, while the Lyapunov/spectral certificate supplies the formal guarantee of stability through a computable criterion.
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Copyright (c) 2025 Abdulaziz Alotaibi, Khalid Al-Tahat, Khaled Mustafa Aljamal
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