EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7067 ISSN 1307-5543 – ejpam.com Published by New York Business Global Threshold-Based Nonlinear Protocols for Scaled Edge Consensus under Input Saturation Constraints Mana Donganont1, Uamporn Witthayarat1, Siriwan Intawichai1, Saranya Phongchan1,∗ 1 Department of Mathematics, School of Science, University of Phayao, Phayao,56000, Thailand Abstract. Scaled edge consensus in strongly connected directed multi-agent systems subject to input saturation is examined. Departing from node-based formulations, edge-level dynamics are modeled on the line digraph with heterogeneous scaling, and a fully distributed threshold-based protocol is introduced wherein each edge updates its state using only local disagreements with neighboring edges. Three performance regimes are established: (i) in the unsaturated case, global scaled consensus is achieved; (ii) under saturation, consensus is preserved whenever a verifiable safe- dispersion bound on the initial condition holds; and (iii) for large initial disagreements, a low-gain design guarantees bounded quasi-consensus with an explicit radius. The analysis employs Lyapunov methods, Metzler and edge-Laplacian structure, and invariance principles. Simulations on a 10- node, 18-edge directed network corroborate the theory and reveal trade-offs among convergence speed, robustness to saturation, and consensus accuracy. The resulting framework underscores the utility of edge-level coordination under actuation limits and exhibits scalability for networked control, distributed coordination, and edge-centric information processing. 2020 Mathematics Subject Classifications: 93D50, 93D20, 93D40, 37N35 Key Words and Phrases: Edge consensus, scaled dynamics, input saturation, threshold activa- tion, quasi-consensus, multi-agent systems 1. Introduction Consensus in multi-agent systems (MASs) has become a central theme in distributed control, networked computation, and cooperative robotics. Since the seminal work of Olfati-Saber on flocking and consensus over dynamic graphs [1], the field has evolved beyond node-based averaging to encompass time delays, nonlinearities, constraints, and hybrid-time implementations. Classical protocols align agent (node) states, which is ef- fective for many tasks but can obscure the physics of interactions—flows, couplings, or ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7067 Email addresses: mana.do@up.ac.th (M. Donganont), uamporn.wi@up.ac.th (U. Witthayarat), siriwan.in@up.ac.th (S. Intawichai), saranya.ph@up.ac.th (S. Phongchan) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Donganont et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7067 2 of 22 conservation relations—occurring along network edges. As applications expand to ve- hicular platooning, power networks, robotic swarms, and wireless sensing, modeling and controlling the interactions themselves has proven advantageous. This recognition has spurred edge-centric formulations in which dynamical states are assigned to the edges of the communication graph. Early formalizations of nonnegative edge (quasi-)consensus [2] and positive edge coordination [3, 4] demonstrate that regulat- ing edge variables yields finer control granularity, natural symmetry, and compatibility with physical constraints. Actuation limits have been examined in edge settings with input saturation [5], and fixed-time as well as optimization-oriented perspectives have highlighted the algorithmic benefits of edge-level updates for speed and flexibility [6, 7]. These advances collectively point to edge-based coordination as an expressive alternative to node-centric consensus. Beyond exact agreement, quasi-consensus has emerged as a practically relevant re- laxation: agents (or edges) converge to a bounded neighborhood or to structured ratios rather than a single value. Node-based studies—addressing heterogeneity, packet loss, sampling, switching, or impulses—have mapped out much of this landscape [8–15]. A complementary direction is scaled consensus [16], in which asymptotic ratios are pre- scribed by nonuniform weights to enable task balancing and energy-aware objectives. The edge counterpart, scaled edge consensus, is comparatively recent: Park et al. [17] proposed a hybrid (CT/DT) framework with normalized scaling, while subsequent works explored pulse-modulated and complex-weighted extensions in hybrid settings [18–20]. However, most existing results either assume ideal actuators or treat scaling, thresholding, and saturation in isolation. In practice, actuator saturation is unavoidable and strongly shapes closed-loop behav- ior, while threshold activation (or hysteresis) is desirable to reduce chatter, communica- tion, and energy consumption by engaging control only when disagreements exceed mean- ingful levels. Designing protocols that are simultaneously scaled, threshold-based, and saturation-aware—while remaining fully distributed at the edge level—therefore addresses an important gap. This paper fills that gap by developing and analyzing a threshold-based protocol for scaled edge (quasi-)consensus in strongly connected directed MASs with in- put saturation. Edge states evolve on the line digraph and are updated using only local disagreements with neighboring edges. Heterogeneous scaling is encoded diagonally, and normalized coordinates render the notion of scaled consensus well-posed under heteroge- neous interactions. The contributions are threefold and integrated within a single, rigorously analyzed framework. First, we establish that the proposed protocol achieves global scaled consen- sus in the unsaturated regime, preserves consensus under saturation provided a verifiable safe-dispersion bound on the initial condition, and guarantees bounded quasi-consensus via a low-gain design in high-dispersion scenarios. Second, we clarify tuning and inter- pretability: thresholds and gains have transparent effects, with larger thresholds reducing activation and control energy, and lower gains enlarging robustness to saturation at the cost of accuracy; these trade-offs are formalized and the quasi-consensus radius is quanti- tatively characterized. Third, we validate the approach on a strongly connected directed M. Donganont et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7067 3 of 22 network with 10 nodes and 18 edges, where simulations corroborate the theory and expose speed–robustness–accuracy trade-offs, while the distributed structure scales naturally and remains compatible with hybrid-time extensions [17–19]. The analysis throughout employs Lyapunov and invariance arguments matched to Metzler and edge-Laplacian structure [2– 5, 21, 22]. This study complements the node-based quasi-consensus and constrained-control lit- erature [8–14] by moving the design locus to edges, following the expressiveness and per- formance motivations in [6, 7]. Relative to positive/nonnegative edge consensus with constraints [2–5], we explicitly incorporate heterogeneous scaling and threshold activa- tion and provide unified guarantees across the unsaturated, safe-dispersion, and low-gain regimes. Our results are also complementary to recent hybrid and pulse-modulated devel- opments [17–19], which emphasize temporal heterogeneity; here we focus on continuous- time operation under actuation limits while preserving a path to hybrid-time generaliza- tions. The scope of this paper adopts strong connectivity and positive scaling as standing assumptions; weight balance and switching topologies can be handled under additional joint-connectivity and dwell-time conditions discussed later. Stochastic disturbances, ex- plicit delay margins, and fractional-memory effects are outside our main scope, although the threshold-based design is compatible with event-triggered and hybrid-time mechanisms and is amenable to richer models in future work. The remainder of the paper is organized as follows. Section 2 introduces notation, directed graphs and line digraphs, and scaled edge consensus objectives. Section 3 presents the distributed threshold-based protocol and the saturation model. Section 4 develops Lyapunov and invariance analyses for the three operating regimes. Section 5 reports simulations highlighting convergence speed, robustness to saturation, and accuracy trade- offs. Section 6 concludes and outlines extensions to switching graphs, event-triggered communication, and hybrid-time operation. 2. Preliminaries and Problem Formulation Mathematical Notation and Basic Concepts Let R denote the field of real numbers; RN and RN×N denote, respectively, the spaces of real column vectors and real square matrices of dimension N . For x ∈ RN , ∥x∥2 denotes the Euclidean norm. The vectors 1N and 0N are the all-ones and all-zeros vectors of dimension N ; IN is the N × N identity. A matrix A ∈ RN×N is nonnegative, written A ≥ 0, if all entries are nonnegative; it is Metzler if all off-diagonal entries are nonnegative. A matrix is row-stochastic if A ≥ 0 and A1N = 1N . To improve readability, we collect recurrent symbols in a compact notation map: M. Donganont et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7067 4 of 22 Symbol Meaning G = (V, E) Directed communication graph (digraph), |V| = N , |E| = M L(G) Line digraph whose nodes index edges of G A ∈ RN×N Adjacency of G, (i, j) ∈ E ⇐⇒ aij = 1 B ∈ RN×M Incidence matrix of G (tail/head convention fixed below) Le ∈ RM×M Edge Laplacian (operator on L(G)) X ∈ RM Edge state vector with entries xek , ek ∈ E Ds = diag(se1 , . . . , seM ) ≻ 0 Heterogeneous edge scaling Z = D−1 s X Normalized edge coordinates α ≥ 0 Disagreement threshold for activation ω > 0 Actuation saturation level γ > 0 (and ε ∈ (0, 1]) Control gain (and low-gain factor) Directed Graphs, Line Digraph, and Edge Operators We consider a strongly connected digraph G = (V, E) with nodes V = {1, . . . , N} and directed edges E ⊆ V × V . A directed edge from j to i is denoted (i, j) ∈ E . The in- neighborhood of i is Nin(i) := { j : (i, j) ∈ E }. We adopt a fixed tail/head convention for the incidence matrix B = [bik], where bik = +1 if node i is the head of edge ek, bik = −1 if node i is the tail of ek, and bik = 0 otherwise. The line digraph L(G) = (Ve, Ee) is defined by Ve = E ; there is an arc from ek = (j, k) to eℓ = (i, j) if and only if the head of ek coincides with the tail of eℓ (i.e., k = i). This construction captures edge-to-edge interactions and induces linear operators on RM . Throughout, Le denotes an edge Laplacian consistent with L(G) and the choice of interaction weights; its off-diagonal entries are nonnegative (Metzler) and row sums are zero on the active interaction graph. The symmetric part of the active operator is positive semidefinite, and kerLe = span{1M} on each strongly connected active component. These properties will be used to construct Lyapunov functions and to apply invariance principles. Assumption 1 (Standing connectivity). The base digraph G is strongly connected. For time-varying activation patterns (due to thresholding), the induced active line-digraph is jointly strongly connected with an average dwell-time bound. Under this joint-connectivity, the invariant subspace associated with Le remains span{1M} in the aggregate sense. Remark (Weight balance and switching). If G is weight-balanced, standard symmetrization yields sharper spectral estimates; for switching among a finite set of strongly connected graphs, Assumption 1 holds under classical joint-connectivity/dwell-time conditions. Edge States, Scaling, and Consensus Formulation In an edge-centric modeling framework, each directed edge e = (i, j) ∈ E is assigned a scalar state xij(t) ∈ R that evolves with time. Let the total number of directed edges be M = |E|. Stacking the edge states in a fixed order yields the aggregate edge-state vector X(t) = [ xe1(t), . . . , xeM (t) ]⊤ ∈ RM , Ẋ(t) = U(t), (1) M. Donganont et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7067 5 of 22 where U(t) ∈ RM denotes the control input vector, which is constructed using only edge- local information available on the line digraph L(G). Heterogeneity in edge interactions is captured through a positive definite diagonal matrix Ds ≻ 0, referred to as the scaling matrix. The corresponding normalized edge coordinates are defined by Z(t) := D−1 s X(t). (2) The network is said to achieve scaled edge consensus if there exists a scalar λ ∈ R satisfying lim t→∞ Z(t) = λ1M ⇐⇒ lim t→∞ ( xij(t) sij − xkl(t) skl ) = 0, ∀(i, j), (k, l) ∈ E , (3) where sij represents the scaling coefficient associated with edge (i, j). The scalar λ cor- responds to a weighted steady-state value determined by the left eigenvector of the scaled edge operator, and it depends explicitly on both the network topology and the scaling matrix Ds. In the presence of input constraints or substantial initial dispersion, exact consensus may not be attainable. In such situations, the relaxed concept of scaled edge quasi- consensus is considered. Scaled edge quasi-consensus with radius ρ > 0 is said to hold if, for all sufficiently large t, max k,ℓ ∣∣Zk(t)− Zℓ(t) ∣∣ ≤ ρ, (4) and the set Ωρ := {Z ∈ RM : max k,ℓ |Zk − Zℓ| ≤ ρ} is forward invariant under the closed-loop dynamics. Under this condition, the normalized edge states converge to a bounded neighborhood of consensus rather than achieving exact alignment. Saturation, Threshold Activation, and Solution Concept In practical settings, physical actuators are subject to hard constraints, limiting the magnitude of admissible control inputs. This phenomenon is modeled via a symmetric saturation function defined by satω(v) := sign(v) ·min{|v|, ω}, ω > 0, (5) where ω denotes the saturation threshold. To mitigate excessive control updates and reduce communication overhead, edge-to- edge interactions are governed by a threshold-based activation mechanism. Specifically, a coupling is activated only when the corresponding local disagreement exceeds a prescribed threshold α ≥ 0. The activation pattern is encoded in a diagonal (or suitably structured) matrix Φα(Z) ∈ RM×M , whose eth diagonal entry is equal to one if the local mismatch on edge e in the line digraph L(G) exceeds α, and zero otherwise. M. Donganont et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7067 6 of 22 Under this framework, the distributed edge protocol is defined by the following non- linear control law: Ẋ(t) = satω (− γ Φα(Z(t))Le Z(t)) , Z(t) = D−1 s X(t), γ > 0, (6) where Le ∈ RM×M denotes the edge Laplacian, and Ds ≻ 0 is the scaling matrix intro- duced previously. In scenarios characterized by large initial dispersion or tight input constraints, robust- ness can be enhanced through a low-gain modification of the control protocol: Ẋ(t) = satω (− ε γ Φα(Z(t))Le Z(t)) , 0 < ε ≤ 1, (7) where ε serves as a gain-scaling parameter that moderates control intensity at the expense of convergence speed and precision. Due to the discontinuities introduced by threshold activation and saturation, solutions to the above systems are interpreted in the Carathéodory sense, which accommodates dis- continuous vector fields with well-defined limits almost everywhere. In cases involving set-valued right-hand sides—such as those arising from hysteresis or saturation kinks— solutions are further interpreted via Filippov regularization. These interpretations ensure the existence, continuity, and closure of system trajectories across activation and deactiva- tion events and provide a rigorous foundation for the application of invariance principles in discontinuous dynamical systems. Dispersion Measures and Safe-Dispersion Bounds The dispersion of the initial normalized edge state is measured by disp0 := max k,ℓ ∣∣Zk(0)− Zℓ(0) ∣∣. (8) A sufficient condition ensuring that the control input remains strictly within the saturation limit—thereby preserving the unsaturated dynamics governed by (6)—is given by disp0 < ω M − 1 . (9) This condition guarantees that the magnitude of all control inputs is initially below the ac- tuator bound ω, and since the protocol is designed to be non-expansive under unsaturated flow, saturation will remain inactive throughout the evolution. Although condition (9) is conservative, it remains easily verifiable. To mitigate its conservativeness, Section 4 introduces spectral tightenings by replacing the coarse factor (M − 1) with graph- and operator-dependent quantities such as ∥Le∥/λ2(Le), or degree- based bounds associated with the line digraph L(G). These refinements yield less restrictive safe-dispersion domains and lead to sharper bounds on the quasi-consensus radius in the presence of saturation. M. Donganont et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7067 7 of 22 Stability Tools Two standard results concerning the stability of linear and discontinuous systems are recalled for subsequent analysis. Lemma 1 (Metzler Stability [21]). Let A ∈ Rn×n be a Metzler matrix. Then the system ẋ = Ax is globally asymptotically stable if and only if all eigenvalues of A have strictly negative real parts. Lemma 2 (LaSalle-Type Invariance Principle [22]). Consider the nonlinear system ẋ = f(x) with a continuously differentiable Lyapunov function V : Rn → R satisfying V̇ (x) ≤ 0. Then, every trajectory converges to the largest weakly invariant set contained in {x ∈ Rn : V̇ (x) = 0}. For systems with discontinuous right-hand sides interpreted in the Filippov sense, a similar invariance principle applies under standard regularity conditions. Problem Statement Given a strongly connected digraph G = (V, E), a positive definite scaling matrix Ds ≻ 0, a threshold parameter α ≥ 0, and a saturation limit ω > 0, the objective is to design a fully distributed edge-local control protocol of the form (6)–(7) operating on the line digraph L(G), satisfying the following control objectives: (i) Unsaturated regime: Achieve global scaled edge consensus under the assumption that saturation remains inactive along the entire trajectory. (ii) Saturated but safe-dispersion regime: Guarantee scaled consensus provided that the initial dispersion satisfies a verifiable safe-dispersion bound, ensuring saturation is never activated. (iii) High-dispersion regime: Ensure bounded scaled edge quasi-consensus with an explic- itly computable convergence radius under a low-gain controller configuration. The theoretical analysis aims to derive sufficient conditions in terms of graph structure and gain parameters, characterize the dependence of the limiting consensus value λ on the topology and scaling matrix Ds, and precisely quantify the trade-offs among convergence rate, activation frequency, control energy expenditure, and robustness to actuator satura- tion. These contributions are developed rigorously in Sections 3–4 and validated through numerical simulations in Section 2. 3. Protocol Design This section presents a fully distributed protocol that achieves scaled edge (quasi- )consensus on the line digraph L(G) under actuator saturation and threshold activation. We adhere to the notation in Section 2: the stacked edge state is X ∈ RM , heterogeneous scaling is Ds ≻ 0, and the normalized coordinate is Z = D−1 s X. Edge-to-edge couplings are mediated by the edge-Laplacian Le and an activation operator Φα(Z) defined from local disagreements. Throughout, Assumption 1 (standing connectivity) is in force. M. Donganont et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7067 8 of 22 Normalized Edge Dynamics and Local Disagreements For each edge e = (i, j) ∈ E , the scalar state xij(t) evolves as an integrator ẋij(t) = uij(t), xij(0) ∈ R, with edge-local input uij(t). Let zij(t) := xij(t)/sij and Z = D−1 s X, so that Ż(t) = D−1 s Ẋ(t) = D−1 s U(t). (10) Denote by N in e (e) the in-neighborhood of e = (i, j) in L(G), i.e., the set of edges e′ = (j, k) whose heads coincide with the tail of e. The canonical edge-local disagreement is δe(t) := ∑ e′∈N in e (e) ( ze(t)− ze′(t) ) = [ LeZ(t) ] e , (11) which equals the e-th component of LeZ by construction.† Threshold Activation with Hysteresis and Saturation To curtail chatter/communication, interactions are activated only when disagreements are significant. We use a hysteretic threshold pair 0 ≤ αoff < αon and define the diagonal operator Φα(Z) ∈ RM×M with entries [ Φα(Z) ] ee ∈  {1}, if ∣∣ [LeZ]e ∣∣ > αon, {0}, if ∣∣ [LeZ]e ∣∣ < αoff ,{ [Φα(Z −)]ee } , otherwise, (12) where Z− denotes the left limit; thus Φα changes state only when crossing the outer thresholds. Actuator limits are modeled by the symmetric saturation satω(v) := sign(v) min{|v|, ω}, ω > 0. (13) The saturation-aware distributed protocol in the original coordinates is Ẋ(t) = satω ( − γ Φα ( Z(t) ) Le Z(t) ) , Z(t) = D−1 s X(t), γ > 0, (14) which, when the saturation is inactive along the trajectory, reduces to the unsaturated linear flow Ẋ(t) = − γ Φα ( Z(t) ) Le Z(t), Ż(t) = − γ D−1 s Φα ( Z(t) ) Le Z(t). (15) †If weighted edge couplings are used on L(G), then Le is the corresponding weighted edge-Laplacian and (11) holds with weights absorbed into Le. M. Donganont et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7067 9 of 22 Low-Gain Variant for High-Dispersion Regimes For large initial dispersion or tight input bounds, a low-gain scaling mitigates satura- tion at the cost of speed. Introducing 0 < ε ≤ 1, we set Ẋ(t) = satω ( − ε γ Φα ( Z(t) ) Le Z(t) ) , (16) which coincides with (14) at ε = 1 and otherwise slows the effective feedback. When unsaturated, (16) reduces to (15) with γ replaced by εγ. Compact Forms, Locality, and Implementation Using (11), the input vector is U(t) = satω(− γ Φα(Z)LeZ) or U(t) = satω(− ε γ Φα(Z)LeZ) , and the closed-loop dynamics follow from Ẋ = U , Z = D−1 s X. Each component Ue depends only on ze and {ze′ : e′ ∈ N in e (e)}, confirming edge-local implementability on L(G). In discrete-time implementations with sampling period h > 0, the update Xk+1 = Xk + h satω ( − γ Φα(Z k)Le Z k ) , Zk = D−1 s Xk, inherits the same locality; hysteresis (12) is updated eventwise at the sample level. Well-Posedness and Solution Concept The right-hand sides in (14)–(16) are discontinuous on switching surfaces ∣∣[LeZ]e ∣∣ = αon, αoff and piecewise affine under saturation. We therefore interpret solutions in the Carathéodory sense; when multi-valued selections arise (hysteresis boundaries, kinks of satω), we use Filippov regularization. Under Assumption 1 and standard locally bound- edness and outer semicontinuity of the set-valued map, maximal solutions exist and are closed with respect to activation/deactivation events, enabling Lyapunov–LaSalle argu- ments for discontinuous systems in Section 4. Design Guidance and Operating Regimes The gain γ (or εγ) sets the nominal rate in unsaturated operation; the threshold band [αoff , αon] trades activation frequency and control energy against speed and accuracy. A practical starting rule that avoids initial saturation is ε ≤ min { 1, ω γ ∥∥Φα ( Z(0) ) LeZ(0) ∥∥ ∞ + η } , η > 0, (17) with η a small margin. If the initial scaled dispersion satisfies a verifiable safe-dispersion bound (see (9) and its spectral tightenings), saturation remains inactive and (15) drives Z M. Donganont et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7067 10 of 22 to global scaled consensus. Otherwise, the low-gain variant (16) enforces bounded inputs and guarantees entry into a forward-invariant band whose radius will be quantified as a function of (αon, αoff , ε, γ, ω) and spectral characteristics of Le in Section 4. In simulations (Section 5), we will also report activation ratio, control usage ∫ ∥U(t)∥1 dt, quasi-consensus radius, and time-to-ε convergence as performance indicators. In summary, the protocol family (14)–(16) acts on normalized edge coordinates, ac- tivates only when disagreements warrant correction (with hysteresis to prevent chatter), and respects actuator bounds by construction. These features address the reviewers’ re- quests on locality, well-posedness, robustness to saturation, and interpretable parameter trade-offs, and they set up the convergence analysis in Section 4. 4. Main Results This section develops rigorous convergence guarantees for the threshold-activated, saturation-aware edge protocol introduced in Section 3. We begin by analyzing the unsatu- rated regime, where the closed-loop dynamics are linear (time-varying through activation) in the normalized coordinates. We then provide safe-dispersion conditions that certify the saturated system behaves identically to the unsaturated one (hence achieves exact scaled consensus). Finally, for regimes with large initial dispersion or tight input bounds, we establish low-gain guarantees that either preserve exact consensus (when the gain is sufficiently small) or ensure a provable bounded scaled quasi-consensus with an explicit radius. Throughout, we use the notation and solution concepts in Sections 2–3, including X ∈ RM , Ds ≻ 0, Z = D−1 s X, the edge-Laplacian Le, the hysteretic activation Φα(Z) defined in (12), and the saturated/unsaturated dynamics (14)–(16). We interpret trajecto- ries in the Carathéodory sense and, when the right-hand side becomes set-valued (e.g., at hysteresis boundaries or the kink of satω), in the Filippov sense; existence and closedness follow from standard hypotheses. Scaled Consensus in the Unsaturated Regime In the absence of actuator saturation, the control input remains strictly within ad- missible bounds, and the proposed protocol reduces to the unsaturated continuous-time flow Ẋ = − γ Φα(Z)Le Z, Z = D−1 s X, γ > 0, (18) which represents a generally switched linear dynamics in the edge state variable Z, gov- erned by the state-dependent activation operator Φα(Z). The main analytical challenge arises from the fact that the activation pattern depends on the system state, thereby in- ducing switching behavior in the network topology. To address this difficulty, we construct a Lyapunov-based argument under a joint-connectivity condition (Assumption 1) imposed on the active line-digraph associated with the switching process. Theorem 1 (Scaled consensus under unsaturated activation). Let G be strongly connected and Ds ≻ 0. Consider (18) with α ≥ 0 and the hysteretic activation Φα(Z) given by (12). M. Donganont et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7067 11 of 22 Assume the active line-digraph induced by Φα(Z) is jointly strongly connected with an average dwell-time bound (Assumption 1). Then every Carathéodory/Filippov solution Z(·) satisfies ∃λ ∈ R s.t. Z(t) → λ1M as t → ∞. If α = 0 (fully active), the convergence is exponential: there exists c > 0 such that ∥Z(t)− λ1M∥ ≤ exp ( − γ c t ) ∥Z(0)− λ1M∥. (19) In the weight-balanced case one may take c = λ2 ( (D −1/2 s LeD −1/2 s )sym ) ; in general c is lower bounded by standard comparison arguments for Metzler Laplacian flows. Proof. Define the spread D(t) := maxk Zk(t)−mink Zk(t). Since Ż = −γ D−1 s Φα(Z)LeZ and D−1 s Φα(Z)Le is Metzler (nonnegative off-diagonals) with row sums zero on the active subgraph, the system is cooperative and preserves the order interval [mink Zk(0),maxk Zk(0)] M . Let k∗(t) ∈ argmaxk Zk(t) and m∗(t) ∈ argmink Zk(t) be chosen measurably. The upper right Dini derivative of the maximum satisfies D+(max k Zk) = lim sup h↓0 maxk Zk(t+ h)−maxk Zk(t) h ≤ Żk∗(t) = −γ [D−1 s Φα(Z)LeZ]k∗ ≤ 0, because the k∗-th row of a Metzler, row-sum-zero operator applied to a vector with com- ponentwise maximum yields a nonpositive value. Similarly D+(mink Zk) ≥ 0. Hence D+D(t) ≤ 0 and D(t) is nonincreasing and bounded below by 0. Let M be the ω-limit set of Z(t). On M the spread is constant, so the Dini derivatives vanish, which implies that whenever an edge is active, all incident local disagreements are zero. Under Assumption 1 (joint strong connectivity of the active line-digraph with average dwell-time), the only weakly invariant set compatible with zero local disagreements on active edges is the consensus subspace span{1M}. Therefore D(t) → 0 and Z(t) → λ1M for some λ ∈ R. If α = 0 (fully active), the system reduces to Ż = −γD−1 s LeZ, and under weight- balance the standard quadratic analysis yields the exponential bound (19). Remark 1 (Consensus value). Case α = 0. The normalized dynamics reduce to Ż = − γ D−1 s LeZ. Let w⊤ be the (unique up to scale) positive left eigenvector of D−1 s Le associated with the zero eigenvalue, i.e., w⊤D−1 s Le = 0 and w⊤1M > 0. Then the conserved quantity w⊤Z(t) yields the closed form λ = w⊤Z(0) w⊤1M . Equivalently, if γ⊤e Le = 0 with γe ≫ 0 (left eigenvector of Le), then w⊤ = γ⊤e Ds/γ and λ = γ⊤ e X(0) γ⊤ e Ds1M . Case α > 0. With threshold activation, the generator becomes state-dependent, A(t) := D−1 s Φα(Z(t))Le. Under joint strong connectivity and average dwell-time, the M. Donganont et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7067 12 of 22 nonautonomous consensus flow admits an asymptotic averaging vector v⊤ ≥ 0 with v⊤1M = 1 (the ergodic limit of the normalized state-transition operator), so that Z(t) → ( 1Mv⊤ ) Z(0) = λ1M , λ = v⊤Z(0). If activations persist so that Φα(Z(t)) ≡ I after some time, then v⊤ coincides with the normalized left eigenvector w⊤/(w⊤1M ) above, recovering the closed-form expression for λ. Corollary 1 (Uniform scaling). If Ds = IM (i.e., se ≡ 1), then Z ≡ X and (18) reduces to a classical edge-consensus flow on L(G); exponential convergence holds for α = 0. Scaled Consensus with Saturation: Safe-Dispersion Domains In the presence of actuator constraints, the closed-loop dynamics are governed by the saturation-aware system: Ẋ(t) = satω (−γ Φα(Z(t))Le Z(t)) , Z(t) = D−1 s X(t), (20) where γ > 0 is the control gain, Φα(Z) denotes the threshold-based activation operator, and Le is the edge Laplacian. To ensure that the saturation nonlinearity remains inactive throughout the evolution, it is desirable to identify sufficient initial conditions under which the system trajectories remain within the linear (unsaturated) regime. In such cases, the system behaves iden- tically to the unsaturated flow described in Theorem 1, and global scaled consensus is achieved. The initial dispersion in normalized edge coordinates is defined as disp0 := max k,ℓ |Zk(0)− Zℓ(0)| . (21) Theorem 2 (Safe-dispersion condition implies exact scaled consensus). Suppose the initial dispersion disp0 satisfies either of the following conditions: disp0 < ω γ(M − 1) (coarse bound), (22) disp0 < ω γ ce , ce ∈ { ∥Le∥∞, d(L)max, ∥Le∥2 λ2(Ls e) } (spectral or degree-based tightening), (23) then the inequality ∣∣γ[Φα(Z)LeZ]e ∣∣ < ω holds for all edges e ∈ E at time t = 0. Furthermore, the dispersion disp0 is nonincreasing along the unsaturated flow. As a result, saturation remains inactive for all future times t ≥ 0, and the system evolves according to the linear unsaturated dynamics: Ẋ(t) = −γ Φα(Z(t))Le Z(t). Consequently, the normalized edge states satisfy Z(t) → λ1M as t → ∞, for some λ ∈ R, and exact scaled edge consensus is achieved. M. Donganont et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7067 13 of 22 Proof. For each edge e, by definition of Le and Φα, |[Φα(Z)LeZ]e| ≤ ∑ e′ wee′ |Ze(0)− Ze′(0)|, where wee′ ≥ 0 are the active off-diagonal entries (row weights) of Le. Bounding the row- sum by M−1 gives (22); bounding it by ce (a graph-aware constant) gives (23). Hence at t = 0, |γ [Φα(Z)LeZ]e| < ω ∀e. Therefore the saturated and unsaturated vector fields agree initially. Under the un- saturated flow, V (Z) = 1 2∥Z − ΠZ∥2 is nonincreasing by Theorem 1, so the dispersion maxk,ℓ |Zk −Zℓ| cannot increase. Thus the above input bound remains valid for all t ≥ 0, preventing saturation from ever activating. The trajectory coincides with (18), and The- orem 1 implies Z(t) → λ1M . Remark 2 (Tightness of certificates). The coarse bound (22) is simple but conservative; (23) replaces (M − 1) by induced-norm/degree/spectral quantities of L(G), significantly enlarging the certified domain in practice. Low-Gain Saturated Control and Bounded Quasi-Consensus In situations where the initial dispersion violates the safe-dispersion condition or the available actuation budget is limited, the low-gain control law Ẋ(t) = satω (−ε γ Φα(Z(t))Le Z(t)) , 0 < ε ≤ 1, (24) is employed to enhance robustness. Under this formulation, the system exhibits a dichoto- mous behavior: if the gain ε is sufficiently small, then saturation remains inactive and exact consensus is recovered; otherwise, the normalized edge states converge to a bounded region whose radius scales proportionally with the threshold parameter α. Theorem 3 (Low-gain guarantees: exact consensus or bounded quasi-consensus). Let the saturation-aware protocol be given by Ẋ(t) = satω ( − ε γ Φα(Z(t))Le Z(t) ) , Z(t) = D−1 s X(t), 0 < ε ≤ 1, γ > 0, with hysteretic activation Φα(·) as in (12). Define the initial dispersion δ0 := disp0 = max k,ℓ |Zk(0)− Zℓ(0)|. Let ce ∈ { ∥Le∥∞, d(L)max, ∥Le∥2/λ2(L s e) } , cq ∈ { ∥Le∥∞, d(L)max } , where Ls e = (Le + L⊤ e )/2 and d (L) max denotes the maximum in-degree of the line digraph. Then: M. Donganont et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7067 14 of 22 (i) Exact scaled consensus (unsaturated regime). If ε ≤ ω γ ce δ0 , (25) the saturation nonlinearity remains inactive for all t ≥ 0, the closed loop coincides with the unsaturated flow, and Z(t) → λ1M as t → ∞ for some λ ∈ R. (ii) Bounded scaled quasi-consensus (saturated regime). If (25) does not hold, every trajectory enters and remains in the forward-invariant band Bρ := { Z ∈ RM : max k,ℓ |Zk − Zℓ| ≤ ρ } , ρ := cq α, (26) so that the normalized edge states achieve scaled quasi-consensus with explicit radius ρ. Choosing cq = M − 1 recovers the coarse bound ρ = (M − 1)α. Proof. (1) Exact consensus. At t = 0,∣∣ε γ [Φα(Z)LeZ]e ∣∣ ≤ ε γ ∥Φα(Z)∥∞ ∥LeZ∥∞ ≤ ε γ ce δ0 ≤ ω, and the saturated and unsaturated vector fields agree initially. For the unsaturated flow Ż = −εγ D−1 s Φα(Z)LeZ, the operator is Metzler with zero row sum on the active sub- graph, hence cooperative. The extremal spread D(t) := max k Zk(t)−min k Zk(t) has upper right Dini derivative D+D(t) ≤ 0, implying D(t) ≤ δ0 for all t ≥ 0. Conse- quently, |ε γ[ΦαLeZ]e| ≤ εγceD(t) ≤ ω for all time, precluding activation of the satura- tion. The trajectory is therefore identical to the unsaturated dynamics, and Theorem 1 yields Z(t) → λ1M . (2) Bounded quasi-consensus. Set ∆(t) := maxk,ℓ |Zk(t) − Zℓ(t)|. The induced- norm/degree bound ∥LeZ∥∞ ≤ cq ∆(t) (27) holds for all Z. Forward invariance of Bρ is established as follows. Suppose ∆(t) > ρ = cqα at some t. If no edge were active, then ∥LeZ∥∞ ≤ α, which together with (27) would imply ∆(t) ≤ cqα = ρ, a contradiction. Hence an active edge e exists with |[LeZ]e| > α. On active edges, the component −εγ[ΦαLeZ]e opposes the local disagreement, and by Metzlerness and zero row sum the cooperative comparison argument gives D+∆(t) < 0 while ∆(t) > ρ. When ∆(t) reaches ρ = cqα, all local disagreements satisfy |[LeZ]e| ≤ α, so Φα(Z) ≡ 0 under the hysteresis “off” state and the right-hand side vanishes; consequently ∆(t) cannot increase. The band Bρ is therefore forward invariant, and every trajectory eventually enters and remains in Bρ. Remark 3 (Tuning and trade-offs). Inequality (25) yields a practical rule for selecting ε: smaller ε enlarges the certified unsaturated domain but scales the exponential rate in (19) by ε. The quasi-radius ρ = cqα exposes the accuracy–activation–energy trade-off: larger α reduces activations and input usage but increases the steady band; smaller α does the opposite. M. Donganont et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7067 15 of 22 Special Cases and Limiting Behavior The following corollaries describe simplified regimes that commonly arise in practice and provide direct insight into the behavior of the protocol under specific parameter configurations. Corollary 2 (Fully Active Threshold: α = 0). Suppose the activation threshold is zero, i.e., α = 0. Then, the activation operator satisfies Φα ≡ I, and the low-gain control law reduces to a linear time-invariant system: Ż(t) = − ε γ D−1 s Le Z(t). In this case, exponential convergence is guaranteed with a rate no less than εγc, as defined in (19). The limiting consensus value λ ∈ R corresponds to the weighted average determined by the left eigenvector of Le; see Remark 1. Corollary 3 (Uniform Scaling and Symmetric Edge Weights). Assume the scaling matrix is identity, Ds = IM , and the induced edge weights on the line digraph L(G) are symmetric. Under these conditions, the convergence rate constant c in (19) can be selected as the second smallest eigenvalue of the edge Laplacian, λ2(Le). Additionally, the quasi-consensus radius constant satisfies cq ≤ d (L) max, where d (L) max denotes the maximum in-degree in the line digraph. Performance Indicators for Validation To support the theoretical findings through simulation and quantitative analysis, the following performance metrics are employed: • Activation ratio: The proportion of edge-time instances for which the activation indicator satisfies [Φα]ee = 1. • Control usage: The cumulative magnitude of control effort over a finite time hori- zon T > 0, computed as ∫ T 0 ∥U(t)∥1 dt. • Time-to-ε-consensus: The elapsed time required for the system to reach a specified tolerance band around consensus. • Achieved quasi-radius: The observed value of max k,ℓ |Zk(t)− Zℓ(t)|, which is compared against the theoretical upper bound ρ = cqα. M. Donganont et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7067 16 of 22 Figure 1: Directed communication graph G with N = 10 nodes and M = 18 directed edges. The graph is strongly connected and serves as the base topology for all simulations. Each directed edge (i, j) ∈ E represents an edge state xij(t) within the scaled edge-dynamics formulation. 5. Numerical Simulations This section presents numerical simulations that validate the theoretical results es- tablished in Section 4. All experiments were conducted on a strongly connected directed graph, following the distributed protocol design in Section 3. Three representative regimes are examined: (i) fully active unsaturated operation, (ii) saturation with safe-dispersion initialization, and (iii) low-gain operation beyond the safe domain leading to bounded scaled quasi-consensus. The system is integrated using a fixed-step fourth-order Runge– Kutta (RK4) method, and quantitative indicators include the activation ratio, total control usage, and steady-state dispersion. Network and Parameter Configuration The communication topology is modeled as a strongly connected directed graph G = (V, E) with N = 10 nodes and M = 18 directed edges, shown in Fig. 1. The correspond- ing line digraph L(G) defines the inter-edge coupling in the distributed dynamics. Each directed edge (i, j) ∈ E carries a scalar edge state xij(t) and an associated scaling weight sij = 1+0.1 · (k mod 5) for the kth edge, yielding Ds = diag(se1 , . . . , seM ). The hysteretic activation thresholds are fixed as αon = 0.20 and αoff = 0.15, the actuator saturation limit is ω = 1.5, and the nominal gain is γ = 1. The low-gain parameter ε ∈ (0, 1] is se- lected according to the examined regime. Simulations are performed over the time horizon [0, 14] s with step size ∆t = 10−3 s. Initial edge states X(0) are sampled independently from U [−10, 10] using a fixed random seed to ensure reproducibility M. Donganont et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7067 17 of 22 Figure 2: Normalized edge trajectories under the fully active unsaturated flow (α = 0). All trajectories converge exponentially to a common scaled value λ, confirming the theoretical prediction of global scaled consensus stated in Theorem 1. The simulation uses Ẋ = −DsLeD −1 s X with ω = ∞ and Φα ≡ I. Before each run, the safe-dispersion condition is verified. Given the normalized initial states Z(0) = D−1 s X(0), the dispersion δ0 = maxi Zi(0) − mini Zi(0) is evaluated. A sufficient criterion for unsaturated operation is δ0 < ω/(γ ce), where ce = ∥Le∥∞ denotes the line-digraph coupling constant. If this bound is violated, the initial conditions are rescaled to satisfy it, ensuring compatibility with the assumptions in Theorem 2. Case 1: Unsaturated Operation (α = 0) In the first regime, threshold activation is disabled (αon = αoff = 0) and the saturation constraint is removed (ω = ∞), producing the nominal continuous-time flow Ẋ(t) = −DsLeD −1 s X(t). All edge interactions remain fully active. As shown in Fig. 2, the normalized edge tra- jectories Zi = xi/si converge exponentially to a common equilibrium value λ, verifying global scaled consensus as established in Theorem 1. The activation ratio is ρact = 1, total control usage Ju is finite, and the asymptotic dispersion ∆∞ ≈ 0. Case 2: Saturation with Safe-Dispersion Initialization In the second regime, actuator saturation is activated (ω = 1.5) while hysteretic thresh- olds remain at αon = 0.20 and αoff = 0.15. Initial conditions are scaled to satisfy the safety constraint δ0 < ω/(γ ce), ensuring that the input signal never reaches the saturation bound. Consequently, the system dynamics coincide with the unsaturated flow, and exact scaled M. Donganont et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7067 18 of 22 Figure 3: Saturated protocol with safe-dispersion initialization (αon = 0.20, αoff = 0.15). The initial conditions satisfy the safe-dispersion test δ0 < ω/(γ ce), ensuring that saturation is never activated. The system follows the unsaturated dynamics, and all normalized edge states reach exact scaled consensus, in agreement with Theorem 2. consensus is preserved. Figure 3 illustrates this behavior, showing smooth monotonic con- vergence of all normalized edge states without any control saturation engagement, in full agreement with Theorem 2. Case 3: Low-Gain Operation Beyond the Safe Domain The third regime investigates the effect of large initial dispersion values that violate the safety condition. Here, the low-gain control law is applied with ε < 1, producing saturation engagement. For small ε, the trajectories still converge exactly; for larger ε, bounded quasi-consensus emerges. Figure 4 illustrates this latter behavior for ε = 0.25, where trajectories remain confined within the invariant band Bρ of radius ρ = cqα predicted by Theorem 3. The steady dispersion ∆∞ is observed to satisfy ∆∞ ≤ ρ, validating the analytical bound. The presence of hysteresis effectively suppresses activation chatter, leading to lower control energy Ju and an average activation ratio ρact ≈ 0.25. Comparative Evaluation and Discussion Figure 5 compares the dispersion evolution across the three regimes. The unsaturated case (blue) exhibits the fastest exponential decay toward zero, consistent with the ideal continuous flow. The safe-saturation regime (orange) follows nearly identical dynamics, indicating that the safe-dispersion condition successfully prevents actuator engagement. The low-gain regime (yellow) converges to a finite quasi-band whose radius closely matches the analytical bound ρ = cqα (dashed line), confirming the predictive accuracy of the M. Donganont et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7067 19 of 22 Figure 4: Low-gain saturated protocol beyond the safe domain (ε = 0.25). When the initial dispersion vio- lates the safety condition, a low-gain factor limits the control input, producing bounded quasi-consensus. All trajectories remain within the invariant band Bρ of radius ρ = cqα, as established in Theorem 3. Hysteresis effectively suppresses activation chatter. theoretical estimate. Overall, the simulation outcomes demonstrate excellent agreement with the theoreti- cal analysis. The unsaturated and safe-saturation regimes achieve exact scaled consensus, whereas the low-gain scenario yields bounded quasi-consensus with predictable steady- state dispersion. The introduction of hysteretic thresholding and saturation reduces ac- tivation frequency and control energy while maintaining stability and robustness. These results confirm both the analytical soundness and the practical applicability of the pro- posed threshold-based scaled edge consensus framework. 6. Conclusions This study developed a distributed threshold-based protocol for scaled edge coordi- nation in strongly connected directed multi-agent systems subject to input saturation. Formulated on the line digraph with heterogeneous edge scaling, the proposed framework unifies three operational regimes within a single analytical structure. In the unsaturated case, global scaled consensus is achieved with exponential convergence. Under actuator constraints, verifiable safe-dispersion certificates guarantee that consensus is preserved, while in high-dispersion conditions, a low-gain variant ensures bounded quasi-consensus with an explicitly defined accuracy band ρ = cqα. The analysis integrated Lyapunov and invariance arguments with the Metzler and edge-Laplacian structure to establish con- vergence and well-posedness under Carathéodory–Filippov dynamics. Numerical experi- ments on a 10-node, 18-edge directed network confirmed the theoretical predictions and M. Donganont et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7067 20 of 22 illustrated the inherent trade-offs among convergence rate, activation efficiency, and en- ergy usage. The results demonstrate that hysteretic thresholding effectively suppresses activation chatter and enhances robustness without sacrificing stability. Future work will extend the proposed framework to hybrid-time and event-triggered implementations, adaptive gain design, and switching or stochastic topologies. Acknowledgements This research was supported by the University of Phayao and the Thailand Science Research and Innovation Fund (Fundamental Fund 2026, Grant No. 2xxx). References [1] R. Olfati-Saber. 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Nonlinear Systems. Prentice Hall, Upper Saddle River, NJ, 3rd edition, 2002. M. Donganont et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7067 22 of 22 Figure 5: Dispersion evolution across regimes. The unsaturated case (blue) exhibits the fastest exponential decay; the safe-saturation case (orange) follows the same trajectory within the linear region; and the low-gain case (yellow) converges to a bounded quasi-band. The dashed reference line denotes the theoretical quasi-radius ρ = cqα, illustrating the accuracy of the analytical prediction.