EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6877 ISSN 1307-5543 – ejpam.com Published by New York Business Global Mathematical Modeling of Signed Graphs for Balancing Dynamic Systems Abdulaziz Mutlaq Alotaibi1, Khalid Al-Tahat2, Khaled Mustafa Al-jamal2,∗ 1 Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia 2 Faculty of Computer Studies, Arab Open University, Amman, Jordan Abstract. 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. 2020 Mathematics Subject Classifications: 93C10, 93D15, 05C22 Key Words and Phrases: Signed graph, balance sign graph, Lyapunov stability, equilibrium point, damping factor 1. Introduction and Preliminaries Dynamical systems provide mathematical models to understand and characterize the motion of many natural and engineered phenomena. Examples range from simple sys- tems such as the pendulum [1] to complex applications in aircraft control, power-network analysis, and robotics. A central theme across these domains is stability, the ability of a system to return to an equilibrium after perturbations [2, 3]. In practice, assessing sta- bility requires analyzing the internal couplings among the system variables [4], including whether these interactions are reinforcing (positive) or opposing (negative) [5, 6]. Within this context, the signed graph offers an effective and visual formalism to encode the nature of pairwise influences among state variables [7, 8]. In a signed graph, directed ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6877 Email addresses: am.alotaibi@psau.edu.sa (A. M. Alotaibi), k_tahat@aou.edu.jo (K. Al-Tahat), pt_aljammal@aou.edu.jo (K. M. Al-jamal) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. M. Alotaibi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6877 2 of 9 edges carry signs that indicate how one variable affects another, enabling a compact repre- sentation of cooperative and antagonistic relations [9]. Such representations help diagnose sources of instability, diverging trajectories or persistent oscillations by highlighting desta- bilizing pathways in the interaction structure [10, 11]. A classical way to improve stability is to introduce damping terms that act as self- inhibitory mechanisms on selected states. For instance, in the inverted-pendulum bench- mark, adding viscous damping to the angular rate reduces oscillations and can steer the dynamics toward a stable regime. Beyond mechanical systems [12], the same modeling and design ideas extend to energy networks [13], biological systems, and even econo-sports models. In this paper we analyze unstable configurations (including saddle-type equilib- ria [14]) through a signed-graph lens and show how damping can be deployed to achieve stabilization. Specifically [6], we derive the nonlinear model and its linearization via the Jacobian[1], map the Jacobian pattern to a signed weighted graph, and use this structure to motivate stabilization by [15] adding damping and [16] reweighting positive/negative influences. We complement the qualitative, graph-based intuition with a rigorous matrix- based stability check on the linearized model and illustrate the approach on the inverted pendulum. Finally, we discuss application prospects in diverse domains [17, 18]. Definition 1. [1] In a differential equations . x = u(x, y), the equilibrium point is expressed by (xe, ye)such that, x(t) = xe, y(t) = ye is a constant solution of the system of differential equations, i.e (xe, ye)is a point at which x = 0 and y = 0. Definition 2. [19] Suppose that the system of differential equations . x = u(x, y) has an equilibrium point at x = xe, y = ye. The equilibrium points is said to be: (i) Stable if all point in the neighbrohood of the equilibruim point remain in the neighborhood of the equilibruim point as time invrease. (ii) Unstable otherwise. Example 1. The following phase diagrams have an equilibrium point at (0,0). Determine whether this equilibrium is stable (balanced) or unstable (unbalanced). Figure 1: Stable (balanced) or unstable (unbalanced). Remark 1. Let the linear system . p = Ap, for a 2 × 2 matrix A. The nature of the equilibrium point at p = 0, q = 0 is determined by the eigenvalues and eigenvectors of A. if one of the eigenvalues is positive and the other is negative, the equilibrium point is a saddle (and is unstable). Note: In dynamic systems, a nonlinear system can be approximated as a time-invariant linear system using a Taylor series around the equilibrium point, keeping only the linear A. M. Alotaibi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6877 3 of 9 term, when signals and deviations are small. Accuracy decreases farther from the operating point, with some exceptions to this assumption. Definition 3. [20] In a signed graph we say that a graph is balanced if all its cycles have a positive sign product. Definition 4. [20] Two signed graphs Σ1 and Σ2 are said to be switching isomorphic if Σ1 is isomorphic to a switching equivalent of Σ2. This relationship is denoted by Σ1 ∼= Σ2. Remark 2. The first signed graph is classified as balanced due to the presence of exclusively positive cycles. In contrast, the second signed graph is unbalanced, as it contains at least one negative cycle. Figure 2: A balanced and unbalanced signed graph Theorem 1. [20] A signed graph Σ is said to be balanced if and only if its vertex set V can be partitioned into two disjoint subsets, X and Y , such that all positive edges connect vertices within the same subset (i.e., either both endpoints in X or both in Y ), and all negative edges connect vertices across the subsets (i.e., one endpoint in X, the other in Y ). An equivalent characterization is that for any pair of vertices v and w, all paths connecting vand w have the same sign. Proof. . See [[20], Thm. 1] 2. Inverted pendulum: nonlinear model, linearization, and Jacobian Let the state be X = [x, ẋ, θ, θ̇ ]⊤, where x is the cart position and θ is the pole angle measured from the upright (θ = 0). The input is the horizontal force F applied to the cart. Denote the cart and pole masses by M and m, the pole length by l, and gravity by g. A standard cart–pole model is (M +m) ẍ+ml θ̈ cos θ −ml θ̇2 sin θ = F, (1) l θ̈ + ẍ cos θ − g sin θ = 0. (2) Linearize (1)–(2) at (x, ẋ, θ, θ̇) = (0, 0, 0, 0) and F = 0, using sin θ ≈ θ, cos θ ≈ 1, and neglecting products of small terms. From (2): θ̈ = g l θ − 1 l ẍ. A. M. Alotaibi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6877 4 of 9 Substitute into (1) and simplify to obtain M ẍ + mg θ = F, θ̈ = M +m lM g θ − 1 lM F. With X = [x, ẋ, θ, θ̇]⊤ and input F , the linearized model is Ẋ = AX +BF , where A =  0 1 0 0 0 0 −mg M 0 0 0 0 1 0 0 (M +m)g lM 0  , B =  0 1 M 0 − 1 lM  . The 2 × 2 angular block [ 0 1 (M+m)g lM 0 ] has eigenvalues λ = ± √ (M+m)g lM , yielding one positive real eigenvalue. (Two additional eigenvalues are at near the origin due to the free cart degree of freedom). Hence, the upright linearization is unstable. A viscous torque −c θ̇ about the pivot augments (2) and, after linearization, adds − c ml2 θ̇ to θ̈, so that Adamped =  0 1 0 0 0 0 −mg M 0 0 0 0 1 0 0 (M +m)g lM − c ml2  . The reduced angular sub-dynamics then have λ = − c 2ml2 ± √( c 2ml2 )2 − (M +m)g lM , so sufficiently large c makes ℜ(λ) < 0, in agreement with the signed-graph intuition and the matrix stability certificate used in section main result. The presence of a positive and negative eigenvalue confirms that the system is unstable. The relationships between the system variables in the unstable state are represented in the following sign graph: Figure 3: Sign Graph System Before Adding Damping (Unbalanced) A. M. Alotaibi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6877 5 of 9 Figure 3. Bdepicts the signed-graph of the linearized cart–pole about θ = 0 without damping. Edge signs/weights are induced by the Jacobian A (e.g., θ → ẋ : −mg M and θ → θ̇ : (M+m)g lM ), which is consistent with the spectral analysis showing one positive real eigenvalue and thus an unstable upright equilibrium. The graph provides structural intuition only; formal stability is certified via the Lyapunov[21] spectral matrix condition D − S ≻ 0 (equivalently, λmax(S) < mini di) stated in Section main result. The following sign graph illustrates the relationships between system variables after sta- bilization: Figure 4. depicts the same signed-graph with viscous damping modeled as a Figure 4: Sign Graph System After Adding Damping (Balanced) negative self-loop on θ̇ with weight − c ml2 . Increasing c attenuates destabilizing couplings and lowers the dominant eigenvalue of the symmetric part, making it easier to satisfy the matrix stability certificate D−S ≻ 0 (equivalently, λmax(S) < mini di). The graph there- fore appears balanced after damping, which is consistent with the Lyapunov [21]spectral result. 3. Main Results In this section, we will discuss how to determine the stability of a system at equilibrium points using signed graph theory. We will focus on analyzing the causal relationships between system variables through positive and negative edges, and determining how these edges affect the stability of the system. Lemma 1. Consider the linearized system around an equilibrium point ẋ = Ax, A ∈ Rn×n, and let Σ = (V,E, σ) be the associated signed graph, where V = {1, . . . , n} is the set of vertices, E ⊆ V × V is the set of directed edges, and σ : E → {−1,+1} assigns a sign to each edge. For each (i, j) ∈ E, let aij denote the (i, j)-entry of A. Define the net signed A. M. Alotaibi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6877 6 of 9 influence as ΣE = ∑ (i,j)∈E σ(i, j) |aij |. Then: (i) If ΣE < 0, all eigenvalues of A have strictly negative real parts, and the equilibrium is asymptotically stable. (ii) If ΣE > 0, at least one eigenvalue of A has a positive real part, and the equilibrium is unstable. Proof. Stability of the linear system ẋ = Ax requires that ℜ(λi) < 0 for all eigenvalues λi of A. The signed graph Σ encodes the interactions: each entry aij corresponds to an edge (i, j) with sign σ(i, j). Define the quadratic Lyapunov candidate V (x) = x⊤x. Its derivative along trajectories is V̇ (x) = x⊤(A+A⊤)x. If ΣE < 0, the symmetric part (A + A⊤) is negative definite, implying V̇ (x) < 0 for all x ̸= 0. Hence all trajectories converge to the origin, so the equilibrium is asymptotically stable. Conversely, if ΣE > 0, the influence of positive edges dominates, and (A + A⊤) has at least one positive eigenvalue. In this case, V̇ (x) > 0 along some directions in state space, which implies that A has at least one eigenvalue with positive real part, leading to instability. Therefore, the equilibrium is asymptotically stable if and only if ΣE < 0. Remark 3. Let Aσ = [aijσij ] be the signed weighted adjacency induced by the Jacobian pattern, and let S = 1 2(Aσ +A⊤ σ ) be its symmetric part. Consider the linearized dynamics ẋ = (Aσ − D)x with diagonal damping D ≻ 0. Using Lyapunov’s direct method with V (x) = 1 2x ⊤x, we obtain V̇ (x) = x⊤(S −D)x. Thus asymptotic stability is ensured whenever D − S ≻ 0. The qualitative rule ΣE < 0 becomes operational via spectral bounds: λmax(S) ≤ ρ(|S|) ≤ max i ∑ j |sij |, so any uniform damping D = dI with d > maxi ∑ j |sij | guarantees D−S ≻ 0. Practically, strengthening negative edges (or weakening positive ones) reduces λmax(S), while increasing d raises the damping margin; together they enforce the matrix condition for stability. This motivates the following theorem. Theorem 2. Consider the linear time-invariant system ẋ = (Aσ −D)x, x ∈ Rn, A. M. Alotaibi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6877 7 of 9 where Aσ = [aijσij ] is a (possibly signed) weighted adjacency matrix associated with a signed graph Σ = (V,E, σ), and D = diag(d1, . . . , dn) ≻ 0 is a diagonal damping matrix. Let S = Aσ +A⊤ σ 2 denote the symmetric part of Aσ. Then: (i) If the matrix Q := D − S is positive definite, i.e. Q ≻ 0, then the equilibrium x = 0 is asymptotically stable. (ii) If the largest eigenvalue of S satisfies λmax(S) > mini di, then the equilibrium x = 0 is unstable. Proof. We use Lyapunov’s direct method see [21] Consider the quadratic Lyapunov candidate V (x) = 1 2x ⊤x. Along trajectories, V̇ (x) = 1 2 x ⊤((Aσ −D) + (Aσ −D)⊤ ) x = x⊤(S −D)x = −x⊤Qx. (i) If Q ≻ 0, then V̇ (x) = −x⊤Qx < 0 for all x ̸= 0, hence x = 0 is asymptotically stable by Lyapunov’s [21] direct method. (ii) Suppose λmax(S) > mini di. Let v ̸= 0 be a unit eigenvector of S associated with λmax(S). Then v⊤(S −D)v ≥ λmax(S)−min i di > 0. Hence, there exists a direction v with v⊤(S −D)v > 0, implying that the symmetric part of (Aσ −D) has a positive Rayleigh quotient; therefore (Aσ −D) has an eigenvalue with positive real part, and the equilibrium is unstable. Corollary 1. Consider the linear time-invariant system ẋ = (Aσ −D)x, Aσ = [aijσij ], D = diag(d1, . . . , dn) ≻ 0, and let S = Aσ +A⊤ σ 2 . Then the origin x = 0 is asymptotically stable under either of the following design rules: (i)(Damping design) Choose D such that D − S ≻ 0. In particular, it suffices to take a uniform damping D = dI with d > λmax(S). (ii)(Edge reweighting) Modify the signed weights {aijσij} to obtain a new symmetric part S̃ satisfying λmax(S̃) < mini di for the chosen D. Proof. Immediate from Theorem 2: item (i) gives asymptotic stability when D−S ≻ 0 via Lyapunov’s direct method; item (ii) shows instability whenever λmax(S) > mini di. Thus enforcing either (i) or (ii) (with S̃) guarantees stability. A. M. Alotaibi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6877 8 of 9 Remark 4. The qualitative condition ΣE < 0 can be interpreted as promoting a decrease of λmax(S) relative to the damping level. Increasing the magnitude of negative edges (or decreasing positive ones) tends to reduce the largest eigenvalue of S; together with larger D, this enforces D − S ≻ 0, which is a rigorous matrix condition for stability. Remark 5. For the linearized cart–pole about θ = 0 with M = 1, m = 0.1, l = 0.5, g = 9.81 and F ≡ 0, we evaluate the spectrum of A and of the damped model Adamped (viscous torque on θ̇ only). Consistently with Theorem 2, the undamped model has one positive real eigenvalue (upright is unstable). Introducing viscous damping at the pivot decreases the growth rate of this unstable mode but does not render all eigenvalues’ real parts negative; full asymptotic stability of the full state requires either additional damping (e.g., on the cart) or state feedback F = −KX so that the matrix certificate holds, i.e., D−S ≻ 0 (equivalently, λmax(S) < mini di). This text-only check corroborates the signed- graph intuition while using a computable Lyapunov [21] spectral test rather than graphical balance alone. 4. Conclusion This work demonstrated how explicit damping can transform an unstable linearized system into a stable one and aligned the signed-graph intuition with a rigorous Lya- punov/spectral certificate. In the inverted pendulum, the undamped linearization is un- stable, whereas adding viscous damping yields eigenvalues λ = − c 2± √ c2 4 − g l , so ℜ(λ) < 0 whenever c2 > 4 g l . At the network level, modeling interactions by a signed weighted matrix Aσ with symmetric part S = 1 2(Aσ + A⊤ σ ) and diagonal damping D ≻ 0, we es- tablished a computable matrix certificate: the origin is asymptotically stable whenever D − S ≻ 0 (equivalently, λmax(S) < mini di). This formal result clarifies the qualita- tive rule “ΣE < 0” and is operationalized in Corollary 1 via damping and signed-edge reweighting. 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