Nonlinear Dynamical Model and Analysis of Emotional Propagation Based on Caputo Derivative

Abstract

Conventional integer-order models fail to adequately capture non-local memory effects and constrained nonlinear interactions in emotional dynamics. To address these limitations, we propose a coupled framework that integrates Caputo fractional derivatives with hyperbolic tangent–based interaction functions. The fractional-order term quantifies power-law memory decay in affective states, while the nonlinear component regulates connection strength through emotional difference thresholds. Mathematical analysis establishes the existence and uniqueness of solutions with continuous dependence on initial conditions and proves the local asymptotic stability of network equilibria ($W_{ij}^{*}={\displaystyle \frac{1}{\delta }}{\sec \mathrm{h}}^2(\parallel {\mathrm{E}}_i-{\mathrm{E}}_j\parallel )$, e.g., $W^{*}\approx 1.40$ under typical parameters $\eta =0.5$, $\delta =0.3$). We further derive closed-form expressions for the steady-state variance under stochastic perturbations ($\mathrm{Var}\left(W_{ij}\right)={\displaystyle \frac{{\sigma }_{\zeta }^2}{2\eta \delta }}$) and demonstrate a less than 6% deviation between simulated and theoretical values when ${\sigma }_{\zeta }=0.1$. Numerical experiments using the Euler–Maruyama method validate the convergence of connection weights toward the predicted equilibrium, reveal Gaussian features in the stationary distributions, and confirm power-law scaling between noise intensity and variance. The numerical accuracy of the fractional system is further verified through L1 discretization, with observed error convergence consistent with theoretical expectations for $\mu =0.5$. This framework advances the mechanistic understanding of co-evolutionary dynamics in emotion-modulated social networks, supporting applications in clinical intervention design, collective sentiment modeling, and psychophysiological coupling research.

1. Introduction

Emotional dynamics constitutes a vital research frontier in psychology, probing temporal patterns of affective experience, including volatility, inertia, and diversity. These patterns serve as sensitive indicators of psychological adaptation, simultaneously reflecting mental health status and predicting vulnerability to disorder. Contemporary studies establish that emotional fluctuations provide early warning signals for clinical interventions and therapeutic outcomes [1,2,3].

Mathematical formalization offers transformative potential for decoding emotional complexity. Recent integrations of dynamic systems theory with affective science yield precise characterizations of emotion generation, regulation, and recovery mechanisms. The Ising model captures contextual stability-to-volatility transitions [4], while neurophysiological frameworks mathematically link autonomic functions to emotional intensity and respiratory rhythms [5]. Neural circuit models employing nonlinear dynamics elucidate the pathophysiology underlying mood disorders through triple-network interactions [6], and stochastic optimization principles reveal emotion’s functional roles in learning processes [7].

Two fundamental properties challenge conventional modeling paradigms. Power-law decaying memory traces governing historical dependence and nonlinear constraints regulating social convergence. Traditional integer-order differential equations do not capture these phenomena, constrained by exponential memory decay assumptions and linear interaction mechanisms [8,9,10,11]. This misalignment stems from empirical evidence showing that emotional memory follows recency effects incompatible with preset decay rates [12], while real-world interactions exhibit thresholded responses to affective divergence.

Fractional calculus offers a robust alternative, with Caputo derivatives establishing a rigorous mathematical correspondence to psychological memory principles [13]. The operator is defined as

$${}_0{}^{C}D_t^{\mu }f\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(n-\mu )}}{\int }_0^t{\displaystyle \frac{f^{\left(n\right)}\left(\tau \right)}{{(t-\tau )}^{\mu -n+1}}}d\tau ,n=\lceil \mu \rceil ,$$

where its integral kernel induces power-law weighting ($\mu \in [0.7,0.9]$ [14,15]). This feature enables continuous interpolation across memory regimes while reducing to classical derivatives at boundary values. Recent advances also demonstrate the efficacy of fractional operators in capturing complex biological or behavioral dynamics across domains—for instance, Gündoǧdu and Joshi [15] introduced a time-fractional cancer model with various net killing rate assumptions, validating the analytical tractability and biological interpretability of fractional dynamics using the Residual Power Series Method. Their work underscores the cross-disciplinary applicability of fractional calculus in systems governed by memory effects, which further motivates its adoption in emotional modeling. By simultaneously encoding historical non-locality and aligning with neurocognitive constraints [16], this formalism addresses limitations in prior social dynamics models [17,18]. The power-law weighting, modulated by the fractional-order parameter $\mu$, captures the nonlinear decay of emotional memory—an improvement over the exponential decay assumptions in integer-order models, which fail to reflect the coexistence of long-term retention and short-term interference in human cognition. For instance, as $\mu$ approaches 1, the model approximates instantaneous memory dynamics, whereas lower values (e.g., $\mu \approx 0.7$) slow the decay rate, modeling persistent emotional aftereffects. This adaptability allows the framework to accommodate contextual variations in memory strength, providing a biologically plausible temporal characterization for coupled emotional propagation and network evolution models.

In complex system modeling, existing frameworks frequently face challenges in adequately depicting the temporal persistence of historical interactions and the nonlinear constraints shaping emotional dynamics within networks. This limitation undermines the precise prediction of how emotional states and network topology co-evolve, especially in contexts demanding detailed modeling of memory effects and threshold behaviors. To overcome these issues, this study presents a coupled system that combines the Caputo fractional derivative, used to capture the power-law decay of historical dependence, with a hyperbolic tangent nonlinear function, designed to model threshold effects in emotional interactions. The proposed framework provides a systematic method to quantify the interaction between emotional state equations and network evolution, offering a more sophisticated understanding of their dynamic covariation. Theoretical analyses determine the stability characteristics of equilibrium states, while numerical simulations confirm the model’s robustness against stochastic perturbations. By integrating fractional calculus and nonlinear dynamics, this research enriches the modeling tools for socio-emotional systems and creates new opportunities for interdisciplinary research in network science and computational social science. Compared to previous works in affective network modeling, this study introduces a unified mathematical framework that rigorously integrates memory-driven dynamics and nonlinear feedback, enabling both theoretical tractability and empirical validation. This distinguishes the work by bridging psychological realism with analytical depth.

The paper is structured as follows. Section 1 presents the background of emotional dynamics and related modeling research, elaborating on the main objectives of this study. Section 2 establishes the mathematical foundations for the model, including the symbolic system, the form and behavioral characteristics of the nonlinear interaction function, and key lemmas supporting subsequent theoretical proofs. Section 3 details the mathematical formulations of the fractional-order emotional dynamics model and network evolution equations, analyzing their biological implications and dynamic properties. This section covers theories such as equilibrium points of network evolution equations, stability theory, and the existence of solutions to emotional dynamics equations. Section 4 provides numerical validation for all mathematical analyses, where numerical simulations further confirm the validity and verifiability of the fractional-order model. Section 5 analyzes the sensitivity of three key parameters in the fractional-order model, showing that two structural parameters (evolution rate and decay coefficient) regulate network cohesion—one enhancing and the other suppressing connection strength—while the fractional order mainly affects long-term memory accumulation, which may require longer simulations or more complex stimuli to emerge clearly. Finally, Section 6 summarizes the main conclusions of this investigation and outlines potential directions for future research.

2. Mathematical Foundations

This section presents the mathematical foundations for constructing a fractional-order differential equation model of emotional dynamics with nonlinear interaction terms, covering the form, properties, and significance of the nonlinear interaction function.

2.1. Symbols and Definitions

The key mathematical symbols and definitions used in this study are listed in

Table 1. Closed-Loop Emotional Dynamics System Quintuple.

Element	Symbol	Definition
Emotional States	$\mathcal{E}$	$\{{\mathrm{E}}_i\left(t\right)\in {\mathbb{R}}^d\mid i\in \mathcal{V},t\ge 0\}$
		•${\mathrm{E}}_i\left(t\right)$: d-dim. emotional vector of agent i at time t
		• Captures multi-dimensional emotional attributes
Connection Weights	$\mathcal{W}$	$\{W_{ij}\left(t\right)\in [0,1]\mid i,j\in \mathcal{V}\}$
		•$W_{ij}\left(t\right)$: Influence of agent j on agent i
		• Dynamically adjusted based on emotional states
Spatial Distance	$\mathcal{D}$	$\{d_{ij}\ge 0\mid i,j\in \mathcal{V}\}$
		• Physical (geographic) or semantic (opinion) distance
		• Constrains interaction patterns
System Noise	$\Xi$	$\{{\xi }_i\left(t\right)\mid i\in \mathcal{V},t\ge 0\}$
		• Endogenous/exogenous stochastic perturbations
		• Induces non-deterministic dynamics
Measurement Noise	$\mathcal{Z}$	$\{{\zeta }_{ij}\left(t\right)\mid i,j\in \mathcal{V},t\ge 0\}$
		• Data acquisition errors affecting weight measurements

, which form the core of the fractional-order model.

The Closed-Loop Emotional Dynamics System Quintuple is rooted in established social dynamics and noise modeling frameworks. The dynamic interplay among emotional states ($\mathcal{E}$), connection weights ($\mathcal{W}$), and spatial distance ($\mathcal{D}$) aligns with stochastic opinion models like the Hegselmann–Krause system, where feedback and proximity shape collective behavior [19]. The inclusion of both endogenous system noise ($\mathrm{\Xi }$) and measurement noise ($\mathcal{Z}$) follows prior work in social physics, which demonstrates how random perturbations influence consensus formation [20].

From the definitions of the quintuple in Table 1, the emotional state $\mathcal{E}$ and the dynamic weights $\mathcal{W}$ form a state–interaction coupling mechanism through bidirectional effects. The spatial/semantic distance $\mathcal{D}$ acts as a prior structural constraint, quantifying inherent differences between agents (such as physical separation or semantic divergence) to achieve the prior modulation of the distance of the interaction weights. System noise $\mathrm{\Xi }$ and measurement noise $\mathcal{Z}$ introduce uncertainties from two dimensions: endogenous perturbations and observation errors. The former directly disrupts the deterministic evolution of emotional dynamics, while the latter affects accurate cognition of interaction relationships. Together, they shape the complexity and incomplete predictability of system behaviors.

Next, we introduce a simple function that plays a critical role in the subsequent dynamic evolution equations.

2.2. Properties of the Nonlinear Interaction Function

In real-world social scenarios, the intensity of emotional interactions between individuals is inevitably constrained by objective conditions. For example, even if there is a social connection between two users with completely opposing emotional attitudes, their mutual influence does not undergo unlimited increase; instead, excessive differences may lead to reduced communication. To accurately characterize such real-world constraints, this study introduces a nonlinear interaction function defined as

$$\mathrm{\Phi }\left(z\right)=\left\{\begin{array}{cc}{\displaystyle \frac{tanh\left(az\right)}{z},}\hfill & z\ne 0,\hfill \\ a,\hfill & z=0,\hfill \end{array}\right.a>0,$$

(1)

where

$$tanh\left(az\right)={\displaystyle \frac{e^{az}-e^{-az}}{e^{az}+e^{-az}}}.$$

Figure 1 illustrates the curves of the nonlinear interaction function $\mathrm{\Phi }\left(z\right)$ for $a=0.5,1.0,2.0,3.0$. The horizontal axis represents z, and the vertical axis represents $\mathrm{\Phi }\left(z\right)$. The curves approach the corresponding value a near $z=0$, tend to 0 as $z\to \pm \infty$, and exhibit higher peaks at $z=0$ with faster convergence to 0 for larger a.

2.2.1. Biological Characteristics of $\mathrm{\Phi }\left(z\right)$

We elaborate on the behavioral rationale behind the nonlinear interaction function $\mathrm{\Phi }\left(z\right)$. Its regulation of emotional influence is well aligned with empirically observed patterns in human social behavior, in contrast to traditional linear models that assume unbounded interaction strength. In this context, z denotes the emotional difference between two individuals, and $\mathrm{\Phi }\left(z\right)$ determines the corresponding interaction strength. When $z\to 0$, indicating highly similar emotional states, the limit of $\mathrm{\Phi }\left(z\right)$ approaches a constant a. This implies that the emotional influence remains stable and strong during emotional resonance. Unlike linear functions, which would amplify even negligible emotional fluctuations, $\mathrm{\Phi }\left(z\right)$ suppresses small deviations, supporting the empirical notion of emotional homeostasis. For example, in an online group with largely positive sentiment, mutual influence remains steady, rather than growing disproportionately. In contrast, as $z\to \infty$, we have $\mathrm{\Phi }\left(z\right)\approx {\displaystyle \frac{1}{z}}$, so the interaction strength decays rapidly with increasing emotional distance. This saturation effect prevents the unrealistic amplification observed in linear models—for instance, the influence between two individuals with starkly opposing emotions (such as on social media) may become negligible, or even break down. This behavior resonates with the psychological theory of “like attracts like,” and $\mathrm{\Phi }\left(z\right)$, thus, provides a bounded, biologically plausible interaction law that captures social homophily and mitigates the limitations of unbounded linear frameworks.

Next, we investigate the mathematical properties of $\mathrm{\Phi }\left(z\right)$, which play a critical role in the subsequent fractional-order dynamic model.

2.2.2. Mathematical Properties of $\mathrm{\Phi }\left(z\right)$

We first present two lemmas, followed by the mathematical properties of the nonlinear interaction function $\mathrm{\Phi }\left(z\right)$.

Lemma 1. 

Define the function $k:\mathbb{R}\to \mathbb{R}$ as

$$k\left(w\right)=w{\sec \mathrm{h}}^{2}w-tanhw,$$

Then, k satisfies the following properties:

1. 

Odd function property $k(-w)=-k\left(w\right)$ for all $w\in \mathbb{R}$.

2. 

Sign properties

• $k\left(w\right)<0$ when $w>0$,
• $k\left(w\right)\ge 0$ when $w\le 0$.

3. 

Monotonicity: k is strictly decreasing in $(0,+\infty )$.

Proof. 

First, verify the odd function property

$$k(-w)=-w·{\sec \mathrm{h}}^2(-w)-tanh(-w)=-w·{\sec \mathrm{h}}^{2}w+tanhw=-k\left(w\right),$$

Thus, $k\left(w\right)$ is an odd function.

For $w>0$, compute the derivative of $k\left(w\right)$

$$k^{\prime }\left(w\right)={\sec \mathrm{h}}^{2}w+w·2\sec \mathrm{h}w·(-\sec \mathrm{h}wtanhw)-{\sec \mathrm{h}}^{2}w=-2w·{\sec \mathrm{h}}^{2}wtanhw.$$

Since ${\sec \mathrm{h}}^{2}w>0$ and $tanhw>0$ for $w>0$, we have $k^{\prime }\left(w\right)<0$, indicating that $k\left(w\right)$ is strictly decreasing for $w>0$. Using Taylor expansions $tanhw=w-{\displaystyle \frac{w^3}{3}}+o\left(w^3\right)$ and ${\sec \mathrm{h}}^{2}w=1-w^2+o\left(w^2\right)$ as $w\to 0^{+}$, we obtain

$$k\left(w\right)=w(1-w^2)-\left(w-{\displaystyle \frac{w^3}{3}}\right)+o\left(w^3\right)=-{\displaystyle \frac{2w^3}{3}}+o\left(w^3\right)\to 0.$$

Thus, $k\left(w\right)\le 0$ for $w\ge 0$. By the odd function property, $k\left(w\right)=-k(-w)>0$ for $w<0$. □

To prove the continuity of $\mathrm{\Phi }\left(z\right)$, we require the following lemma.

Lemma 2. 

Define the function

$$f\left(w\right)={\displaystyle \frac{tanhw-w·{\sec \mathrm{h}}^{2}w}{w^2}},w>0.$$

Then, $f\left(w\right)\le 1$ for all $w>0$.

Proof. 

We prove $f\left(w\right)\le 1$ by analyzing the relationship between the numerator and denominator of $f\left(w\right)$ and using the monotonicity of hyperbolic functions.

First, construct the auxiliary function $l\left(w\right)=tanhw-w$, whose derivative is

$$l^{\prime }\left(w\right)={\sec \mathrm{h}}^{2}w-1\le 0,$$

where the inequality is based on the properties of ${\sec \mathrm{h}}^{2}w$. Since $l^{\prime }\left(w\right)\le 0$, $l\left(w\right)$ is monotonically decreasing for $w\ge 0$. Given $l\left(0\right)=0$, we have $tanhw\le w$ for all $w\ge 0$.

Next, analyze the numerator of $f\left(w\right)$. Using the identity ${\sec \mathrm{h}}^{2}w=1-{tanh}^{2}w$ and $tanhw\le w$, we obtain

$${\sec \mathrm{h}}^{2}w=1-{tanh}^{2}w\ge 1-w^2.$$

Substituting this into the numerator and applying scaling

$$\begin{array}{cc}\hfill tanhw-w·{\sec \mathrm{h}}^{2}w& \le w-w·(1-w^2)\hfill \\ & =w-w+w^3=w^3.\hfill \end{array}$$

For $0<w<1$, it follows that

$$f\left(w\right)={\displaystyle \frac{tanhw-w·{\sec \mathrm{h}}^{2}w}{w^2}}\le {\displaystyle \frac{w^3}{w^2}}=w<1.$$

For $w\ge 1$, using $tanhw\le w$ and ${\sec \mathrm{h}}^{2}w\ge 0$, we have

$$f\left(w\right)={\displaystyle \frac{tanhw-w·{\sec \mathrm{h}}^{2}w}{w^2}}\le {\displaystyle \frac{w}{w^2}}={\displaystyle \frac{1}{w}}\le 1.$$

In summary, $f\left(w\right)\le 1$ for all $w>0$. □

We now present the main theorem for the nonlinear interaction function $\mathrm{\Phi }\left(z\right)$.

Theorem 1. 

The nonlinear interaction function $\mathrm{\Phi }\left(z\right)$ satisfies the following properties on $\mathbb{R}$

• Global boundedness: For all $z\in \mathbb{R}$ and $a>0$, $\left|\mathrm{\Phi }\right(z\left)\right|\le a$.
• Global Lipschitz continuity: There exists a Lipschitz constant $L_{\mathrm{\Phi }}=a^2$ such that for any $z_1,z_2\in \mathbb{R}$,

  $$|\mathrm{\Phi }\left(z_1\right)-\mathrm{\Phi }\left(z_2\right)|\le a^2|z_1-z_2|.$$

Proof. 

First, prove global boundedness. For $z\ne 0$, since $tanhw\le w$ for all $w>0$ (from the proof of Lemma 2), we have

$$\left|\mathrm{\Phi }\left(z\right)\right|=\left|{\displaystyle \frac{tanh\left(az\right)}{z}}\right|=a·\left|{\displaystyle \frac{tanh\left(az\right)}{az}}\right|\le a·1=a.$$

For $z=0$, by definition, $\left|\mathrm{\Phi }\right(0\left)\right|=a$. Since $\mathrm{\Phi }\left(z\right)$ is an even function, $\left|\mathrm{\Phi }\right(z\left)\right|\le a$ holds for all $a>0$.

Next, prove global Lipschitz continuity. First, verify continuity by computing the limit as $z\to 0$:

$$\underset{z\to 0}{lim}\mathrm{\Phi }\left(z\right)=\underset{z\to 0}{lim}{\displaystyle \frac{tanh\left(az\right)}{z}}=\underset{z\to 0}{lim}{\displaystyle \frac{az-\frac{{\left(az\right)}^3}{3}+o\left(z^3\right)}{z}}=a=\mathrm{\Phi }\left(0\right),$$

Thus, $\mathrm{\Phi }\left(z\right)$ is continuous on $\mathbb{R}$.

For $z\ne 0$, compute the derivative

$${\mathrm{\Phi }}^{\prime }\left(z\right)={\displaystyle \frac{az·{\sec \mathrm{h}}^2\left(az\right)-tanh\left(az\right)}{z^2}}.$$

Let $w=az$ ($w\ne 0$), then

$${\mathrm{\Phi }}^{\prime }\left(z\right)={\displaystyle \frac{a^2}{w^2}}\left[w·{\sec \mathrm{h}}^{2}w-tanhw\right]={\displaystyle \frac{a^2}{w^2}}k\left(w\right),$$

where $k\left(w\right)=w·{\sec \mathrm{h}}^{2}w-tanhw$. By Lemma 1, $k\left(w\right)$ is an odd function, and from Lemma 2,

$$f\left(w\right)={\displaystyle \frac{tanhw-w·{\sec \mathrm{h}}^{2}w}{w^2}}=-{\displaystyle \frac{k\left(w\right)}{w^2}}\le 1\Rightarrow -k\left(w\right)\le w^2\Rightarrow \left|k\left(w\right)\right|\le w^2.$$

Thus,

$$|{\mathrm{\Phi }}^{\prime }\left(z\right)|\le {\displaystyle \frac{a^2}{w^2}}·w^2=a^2.$$

To derive the derivative at $z=0$, use the definition

$${\mathrm{\Phi }}^{\prime }\left(0\right)=\underset{h\to 0}{lim}{\displaystyle \frac{\mathrm{\Phi }\left(h\right)-\mathrm{\Phi }\left(0\right)}{h}}=\underset{h\to 0}{lim}{\displaystyle \frac{tanh\left(ah\right)-ah}{h^2}}.$$

Substituting the Taylor expansion $tanh\left(ah\right)=ah-{\displaystyle \frac{{\left(ah\right)}^3}{3}}+o\left(h^3\right)$, we get

$${\mathrm{\Phi }}^{\prime }\left(0\right)=\underset{h\to 0}{lim}{\displaystyle \frac{-\frac{a^3{h}^3}{3}+o\left(h^3\right)}{h^2}}=\underset{h\to 0}{lim}\left(-{\displaystyle \frac{a^{3}h}{3}}+o\left(h\right)\right)=0.$$

Additionally, ${lim}_{z\to 0}{\mathrm{\Phi }}^{\prime }\left(z\right)=0={\mathrm{\Phi }}^{\prime }\left(0\right)$, so ${\mathrm{\Phi }}^{\prime }\left(z\right)$ is continuous on $\mathbb{R}$.

By the Lagrange mean value theorem, for any $z_1,z_2\in \mathbb{R}$, there exists $\xi$ between $z_1$ and $z_2$ such that

$$|\mathrm{\Phi }\left(z_1\right)-\mathrm{\Phi }\left(z_2\right)|=|{\mathrm{\Phi }}^{\prime }\left(\xi \right)\left|·\right|z_1-z_2|\le a^2|z_1-z_2|.$$

Therefore, $\mathrm{\Phi }\left(z\right)$ satisfies global Lipschitz continuity. □

3. Fractional-Order Dynamics Equations

We first present the fractional-order emotional state equations and the corresponding network evolution equations, followed by their detailed explanations.

• Emotional State Equation.

  $${}_0{}^{C}D_t^{\mu }{\mathrm{E}}_i\left(t\right)=\underset{\mathrm{Dynamic}\mathrm{Interaction}}{\underbrace{\sum _{j=1}^N{W}_{ij}\left(t\right)\mathrm{\Phi }(\parallel {\mathrm{E}}_j-{\mathrm{E}}_i\parallel )({\mathrm{E}}_j-{\mathrm{E}}_i)}}+\underset{\mathrm{Spatial}\mathrm{Modulation}}{\underbrace{\gamma \sum _{j=1}^N{e}^{-\beta d_{ij}}{\mathrm{E}}_j}}+\underset{\mathrm{Noise}}{\underbrace{{\xi }_i\left(t\right)}}$$

  (2)
• Network Evolution Equation.

  $${\displaystyle \frac{d{W}_{ij}}{dt}}=\eta \left[\underset{\mathrm{Emotional}\mathrm{Convergence}}{\underbrace{{\sec \mathrm{h}}^2(\parallel {\mathrm{E}}_i-{\mathrm{E}}_j\parallel )}}-\underset{\mathrm{Decay}}{\underbrace{\delta W_{ij}}}\right]+\underset{\mathrm{Observation}\mathrm{Noise}}{\underbrace{{\sigma }_{\zeta }{\zeta }_{ij}\left(t\right)}}$$

  (3)

We consider a network of N agents whose emotional states evolve according to a fractional-order dynamical system. Each agent’s emotional state is denoted by $E_i\left(t\right)\in {\mathbb{R}}^d$, and the connection weight from agent j to agent i is represented by $W_{ij}\left(t\right)$. In the scalar case ($d=1$), the system includes N emotional state variables and $N(N-1)$ dynamic connection weights (excluding self-loops), resulting in a total of $N^2$ time-dependent state variables.

This relationship is now clearly visualized in Figure 2, which schematically represents the two-way interaction between emotional states $\left\{{\mathrm{E}}_i\left(t\right)\right\}$ and evolving network weights $\left\{W_{ij}\left(t\right)\right\}$ governed by the coupled equations.

In the subsequent modeling analysis, we uniformly assume the following:

• Properties of the interaction function: $\mathrm{\Phi }\left(z\right)$ satisfies global boundedness ($\left|\mathrm{\Phi }\right(z\left)\right|\le a$) and global Lipschitz continuity ($|\mathrm{\Phi }\left(z_1\right)-\mathrm{\Phi }\left(z_2\right)|\le a^2|z_1-z_2|$).
• Boundedness of coupling weights: $0\le W_{ij}\left(t\right)\le 1$ and $W_{ij}\left(t\right)\in C\left([0,T]\right)$.
• Boundedness of emotional terms: There exists $M\ge 0$ such that $\parallel \mathrm{E}\parallel \le M$.
• Conditions for perturbation terms: ${\xi }_i\left(t\right)\in C([0,T],{\mathbb{R}}^d)$ and ${sup}_{t\in [0,T]}\parallel {\xi }_i\left(t\right)\parallel \le M_{\xi }$.

These assumptions are reasonable based on Table 1 and Theorem 1. In this paper, unless otherwise specified, all norms $\parallel ·\parallel$ default to the supremum norm (infinity norm); that is, for a function $\mathrm{E}\left(t\right)$ defined in the interval $[0,T]$, $\parallel \mathrm{E}\parallel :={sup}_{t\in [0,T]}{\parallel \mathrm{E}\left(t\right)\parallel }_{\infty }$.

We first analyze the dynamic law governing the evolution of network connection weights, which reflects how social influence patterns respond to emotional similarity over time.

3.1. Network Evolution Equation

The network evolution Equation (3) is used to characterize the temporal evolution of the dynamic connection weights $W_{ij}\left(t\right)$, its main objective being to elucidate the dynamic adjustment mechanism of the connection strengths between individuals. The following analysis delves into its biological and mathematical implications.

3.1.1. Biological Significance of the Network Evolution Equation

The left-hand side of the equation, ${\displaystyle \frac{d{W}_{ij}}{dt}}$, represents the rate of change in connection weight $W_{ij}$ with respect to time t, determining whether the connection strength between individuals i and j strengthens, weakens, or remains stable.

The right-hand side consists of a deterministic driving term and a stochastic perturbation term. In the deterministic driving term $\eta \left[{\sec \mathrm{h}}^2(\parallel {\mathrm{E}}_i-{\mathrm{E}}_j\parallel )-\delta W_{ij}\right]$, $\eta$ serves as the evolution rate constant, governing the speed of weight adjustment: the larger $\eta$, the more rapidly weights respond to emotional changes. The term ${\sec \mathrm{h}}^2(\parallel {\mathrm{E}}_i-{\mathrm{E}}_j\parallel )$ is an emotional convergence term constructed using the squared hyperbolic secant function, whose typical shape is shown in Figure 3.

As shown in Figure 3, when the emotional states ${\mathrm{E}}_i$ and ${\mathrm{E}}_j$ of the individuals i and j are highly similar (i.e., $\parallel {\mathrm{E}}_i-{\mathrm{E}}_j\parallel \to 0$), ${\sec \mathrm{h}}^2\left(0\right)=1$, and the driving force of emotional convergence reaches its maximum, prompting an increase in the connection weights. As emotional differences increase, the value of the function rapidly decays to 0, indicating that excessive emotional disparities significantly weaken motivation for the reinforcement of connection. The term $\delta W_{ij}$ is a degradation of the connection, where $\delta$ is the decay coefficient that characterizes the natural decay trend of the connection strength. Even if the emotional states between individuals remain unchanged, lacking sustained interaction or emotional resonance will cause the strength of the connection to gradually weaken over time due to $\delta$; for example, the relationship between long-uncontacted friends gradually drifts apart.

The stochastic perturbation term ${\sigma }_{\zeta }{\zeta }_{ij}\left(t\right)$ introduces the uncertainty of the system. Here, ${\zeta }_{ij}\left(t\right)$ is white noise with zero mean and unit variance, used to model unpredictable random factors in reality (such as accidental events or measurement errors) that affect connection strengths. ${\sigma }_{\zeta }$ serves as the noise intensity coefficient controlling the amplitude of random perturbations.

From an overall mechanistic perspective, when the driving force of the emotional convergence term exceeds the connection decay term (that is, ${\sec \mathrm{h}}^2(\parallel {\mathrm{E}}_i-{\mathrm{E}}_j\parallel )>\delta W_{ij}$), ${\displaystyle \frac{d{W}_{ij}}{dt}}>0$, and the connection weights increase, strengthening the relationship between individuals. In contrast, when ${\sec \mathrm{h}}^2(\parallel {\mathrm{E}}_i-{\mathrm{E}}_j\parallel )<\delta W_{ij}$, ${\displaystyle \frac{d{W}_{ij}}{dt}}<0$, and the connection weights decrease, gradually weakening the relationship. When the two are equal (i.e., ${\sec \mathrm{h}}^2(\parallel {\mathrm{E}}_i-{\mathrm{E}}_j\parallel )=\delta W_{ij}$), ${\displaystyle \frac{d{W}_{ij}}{dt}}=0$, and the connection weights reach a stable state where the driving force of emotional convergence and the natural decay force are mutually balanced.

Additionally, this equation is tightly coupled with the emotional state equation in the closed-loop system: emotional state differences influence connection weight adjustments, which, in turn, feedback into the dynamic interaction terms in the emotional state equation, forming a bidirectional relationship between emotions and connections that jointly drives the evolution of the entire closed-loop emotional dynamics system.

After interpreting its biological significance, we now analyze the mathematical properties of the network equation, focusing on equilibrium stability and noise effects.

3.1.2. Mathematical Properties of the Network Evolution Equation

We now investigate the equilibrium states and stability of the network evolution Equation (3).

Theorem 2 

(Existence and Uniqueness of Equilibrium States). For the network evolution Equation (3) under the condition of neglect of observation noise, there exists a unique equilibrium solution $W_{ij}^{*}$ for any $i,j\in \{1,\dots ,N\}$, satisfying

$$W_{ij}^{*}={\displaystyle \frac{1}{\delta }}{\mathit{sec}\mathit{h}}^2(\parallel {\mathrm{E}}_i-{\mathrm{E}}_j\parallel ),$$

where $\eta >0$ is the evolution rate constant and $\delta >0$ is the decay coefficient.

Proof. 

In equilibrium, ${\displaystyle \frac{d{W}_{ij}}{dt}}=0$. Setting the network evolution equation to zero gives

$$\eta \left[{\sec \mathrm{h}}^2(\parallel {\mathrm{E}}_i-{\mathrm{E}}_j\parallel )-\delta W_{ij}^{*}\right]=0.$$

Since $\eta >0$, dividing both sides by $\eta$ and rearranging terms yield

$${\sec \mathrm{h}}^2(\parallel {\mathrm{E}}_i-{\mathrm{E}}_j\parallel )-\delta W_{ij}^{*}=0.$$

Further solving for $W_{ij}^{*}$ gives

$$W_{ij}^{*}={\displaystyle \frac{1}{\delta }}{\sec \mathrm{h}}^2(\parallel {\mathrm{E}}_i-{\mathrm{E}}_j\parallel ).$$

Given fixed ${\mathrm{E}}_i$ and ${\mathrm{E}}_j$, $\parallel {\mathrm{E}}_i-{\mathrm{E}}_j\parallel$ is deterministic, and with $\delta >0$, $W_{ij}^{*}$ exists and is unique. □

Next, we consider the local asymptotic stability near the equilibrium point by simplifying the number of individuals to $N=2$.

Theorem 3 

(Local Asymptotic Stability of Equilibrium States). For a simplified two-node system ($N=2$), the network evolution equation becomes

$${\displaystyle \frac{dW}{dt}}=\eta \left[{\sec \mathrm{h}}^2(\Delta E)-\delta W\right],$$

where $\Delta E=\parallel {\mathrm{E}}_1-{\mathrm{E}}_2\parallel$. Let the equilibrium state be $W^{*}={\displaystyle \frac{1}{\delta }}{\mathit{sec}\mathit{h}}^2(\Delta E)$. Introducing a perturbation $ϵ=W-W^{*}$, the linearized equation is

$${\displaystyle \frac{dϵ}{dt}}=-\eta \delta ϵ.$$

The characteristic value of this linearized equation is $\lambda =-\eta \delta <0$; therefore, the equilibrium state $W^{*}$ is locally asymptotically stable.

Proof. 

Performing a Taylor expansion of ${\displaystyle \frac{dW}{dt}}=\eta \left[{\sec \mathrm{h}}^2(\Delta E)-\delta W\right]$ around $W=W^{*}$ and retaining first-order terms

$${\displaystyle \frac{d}{dt}}\left(W^{*}\right)+{\displaystyle \frac{dϵ}{dt}}=\eta \left[{\sec \mathrm{h}}^2(\Delta E)-\delta (W^{*}+ϵ)\right].$$

Since ${\displaystyle \frac{d}{dt}}\left(W^{*}\right)=0$ and ${\sec \mathrm{h}}^2(\Delta E)-\delta W^{*}=0$, the simplification gives

$${\displaystyle \frac{dϵ}{dt}}=-\eta \delta ϵ.$$

The solution to this first-order linear ordinary differential equation is $ϵ\left(t\right)=ϵ\left(0\right)e^{-\eta \delta t}$. With $\eta >0$ and $\delta >0$, $ϵ\left(t\right)\to 0$ as $t\to +\infty$. According to linearization stability theory [21], the original system is locally asymptotically stable in the equilibrium state $W^{*}$. □

We now discuss the mathematical properties of the network evolution equation with noise. Due to the randomness of noise, we introduce Itô’s lemma for stochastic differential equations (SDE).

Lemma 3 

(Itô’s Lemma [22]). Let $X\left(t\right)$ be an Itô process satisfying the SDE

$$dX\left(t\right)=a(t,X(t\left)\right)dt+b(t,X(t\left)\right)dW\left(t\right),$$

(4)

where $a(t,X)$ is the drift coefficient, $b(t,X)$ is the diffusion coefficient, and $W\left(t\right)$ is a standard Brownian motion with ${\left(dW\left(t\right)\right)}^2=dt$. If the function $F(t,X)$ is twice continuously differentiable with respect to t and X, the stochastic differential of $F(t,X)$ is given by

$$dF=\left({\displaystyle \frac{\partial F}{\partial t}}+a{\displaystyle \frac{\partial F}{\partial X}}+{\displaystyle \frac{1}{2}}{b}^2{\displaystyle \frac{{\partial }^{2}F}{\partial X^2}}\right)dt+b{\displaystyle \frac{\partial F}{\partial X}}dW\left(t\right).$$

Theorem 4 

(Statistical Stability with Noise). When considering observation noise, the network evolution equation can be expressed as the SDE

$$d{W}_{ij}=\eta \left[{\mathit{sec}\mathit{h}}^2(\parallel {\mathrm{E}}_i-{\mathrm{E}}_j\parallel )-\delta W_{ij}\right]dt+{\sigma }_{\zeta }d{W}_{ij}^{\mathit{noise}},$$

(5)

where $d{W}_{ij}^{\mathit{noise}}$ is a standard Brownian motion (Wiener process), and the equilibrium state is $W_{ij}^{*}={\displaystyle \frac{1}{\delta }}{\mathit{sec}\mathit{h}}^2(\parallel {\mathrm{E}}_i-{\mathrm{E}}_j\parallel )$. In this case, the weights $W_{ij}$ exhibit statistical stability near the equilibrium state, with the steady-state variance given by

$$\mathit{Var}\left(W_{ij}\right)={\displaystyle \frac{{\sigma }_{\zeta }^2}{2\eta \delta }}.$$

(6)

Proof. 

First, we perform a variable substitution that lets the perturbation ${ϵ}_{ij}=W_{ij}-W_{ij}^{*}$, so $W_{ij}=W_{ij}^{*}+{ϵ}_{ij}$. Substituting this into Equation (5) and using the equilibrium condition ${\sec \mathrm{h}}^2(\parallel {\mathrm{E}}_i-{\mathrm{E}}_j\parallel )=\delta W_{ij}^{*}$, we obtain

$$d(W_{ij}^{*}+{ϵ}_{ij})=\eta \left[\delta W_{ij}^{*}-\delta (W_{ij}^{*}+{ϵ}_{ij})\right]dt+{\sigma }_{\zeta }d{W}_{ij}^{\mathrm{noise}}.$$

Simplifying yields the perturbation equation

$$d{ϵ}_{ij}=-\eta \delta {ϵ}_{ij}dt+{\sigma }_{\zeta }d{W}_{ij}^{\mathrm{noise}}.$$

(7)

This is a linear SDE that describes the dynamic behavior of the perturbation ${ϵ}_{ij}$ near the equilibrium state.

To solve for the steady-state variance, we apply Lemma 3 to Equation (7) and analyze the second moment of the perturbation. Define the steady-state variance $V=\mathbb{E}\left[{ϵ}_{ij}^2\right]$. Taking the differential of ${ϵ}_{ij}^2$ gives

$$d\left({ϵ}_{ij}^2\right)=2{ϵ}_{ij}d{ϵ}_{ij}+{\left(d{ϵ}_{ij}\right)}^2.$$

Substituting Equation (7) and expanding, and noting the property of Brownian motion ${\left(d{W}_{ij}^{\mathrm{noise}}\right)}^2=dt$, we get

$$d\left({ϵ}_{ij}^2\right)=2{ϵ}_{ij}\left(-\eta \delta {ϵ}_{ij}dt+{\sigma }_{\zeta }d{W}_{ij}^{\mathrm{noise}}\right)+{\sigma }_{\zeta }^{2}dt.$$

Rearranging in deterministic and stochastic terms:

$$d\left({ϵ}_{ij}^2\right)=-2\eta \delta {ϵ}_{ij}^{2}dt+2{\sigma }_{\zeta }{ϵ}_{ij}d{W}_{ij}^{\mathrm{noise}}+{\sigma }_{\zeta }^{2}dt.$$

Taking the mathematical expectation $\mathbb{E}\left[·\right]$ on both sides, and since the expectation of the stochastic term is zero ($\mathbb{E}\left[{ϵ}_{ij}d{W}_{ij}^{\mathrm{noise}}\right]=0$), we obtain the mean evolution equation:

$${\displaystyle \frac{d}{dt}}\mathbb{E}\left[{ϵ}_{ij}^2\right]=-2\eta \delta \mathbb{E}\left[{ϵ}_{ij}^2\right]+{\sigma }_{\zeta }^2.$$

In steady state, ${\displaystyle \frac{dV}{dt}}=0$, yielding

$$V={\displaystyle \frac{{\sigma }_{\zeta }^2}{2\eta \delta }}.$$

Since $W_{ij}=W_{ij}^{*}+{ϵ}_{ij}$ and $W_{ij}^{*}$ is a constant, the variance of the perturbation is the variance of the weight.

$$\mathrm{Var}\left(W_{ij}\right)=\mathrm{Var}\left({ϵ}_{ij}\right)=V={\displaystyle \frac{{\sigma }_{\zeta }^2}{2\eta \delta }}.$$

This indicates that the network evolution equation with noise exhibits statistical stability near the equilibrium state $W_{ij}^{*}$. The amplitude of weight fluctuations is jointly determined by the noise intensity ${\sigma }_{\zeta }$, decay coefficient $\delta$, and evolution rate $\eta$. A higher noise intensity leads to more significant fluctuations, while a larger $\delta$ or $\eta$ suppresses fluctuations and drives the system to converge faster near the equilibrium state. □

Next, we turn to the emotional state equation, which forms the other half of the coupled system and describes how each individual’s emotions evolve in response to social feedback, spatial structure, and memory effects.

3.2. Emotional State Equation

We first introduce the biological properties of the equation of emotional state (2) and then conduct its mathematical analysis.

3.2.1. Biological Significance of the Emotional State Equation

The emotional state Equation (2) constructs a multilevel analytical framework for emotional dynamics through the coupling of dynamic interaction, spatial modulation, and fractional-order memory, elucidating the complex mechanisms by which social structures regulate emotions.

As the core driving force for emotional propagation, the dynamic interaction term characterizes the co-evolutionary process of emotions and connection strengths in social networks. The weights of social connection $W_{ij}\left(t\right)$ are dynamically updated with the emotional similarity between individuals, forming a positive feedback loop of “emotional resonance-connection reinforcement”: when the emotional states of individuals i and j converge, the intensity of the interaction $W_{ij}$ significantly increases (e.g., positive emotional resonance in intimate relationships); conversely, growing emotional differences lead to connection decay (e.g., social alienation in conflict scenarios). The modulation function $\mathrm{\Phi }(\parallel {\mathrm{E}}_j-{\mathrm{E}}_i\parallel )$ further imposes a threshold effect on emotional interactions. Below a critical difference, the momentum for emotional synchronization strengthens; beyond the threshold, interaction inhibition is triggered, forming emotional stratification and segregation within groups.

The spatial modulation term quantifies the asymmetric constraints of social structures on emotional propagation. The exponential decay factor $e^{-\beta d_{ij}}$ represents not only physical interaction barriers caused by geographical distance but also the influence of abstract “distances” such as social hierarchies and cultural differences. For example, emotional signals from leaders on the job can break through hierarchical barriers with a smaller $\beta$ value, exerting long-term effects on subordinates. Emotional interactions among ordinary individuals, however, are constrained by the frequency of direct contact (larger $\beta$ value). In the core-periphery structure of social networks, this mechanism leads to the asymmetric diffusion of emotional signals from core nodes (e.g., opinion leaders) to peripheral groups, forming a “central-radiant” propagation pattern.

The Caputo-type fractional derivative introduces temporal non-locality to emotional regulation, revealing the long-range memory characteristics of collective emotions toward historical experiences. The order $\mu$ characterizes the intensity of dependence of emotional dynamics on past events: collective emotions triggered by major social events (e.g., public crises) have high values $\mu$, with their effects continuously decaying in a power-law form, leading to the significantly enhanced sensitivity of subsequent emotional responses to similar stimuli. This slow dynamics property is also evident in the process of individual socialization: adolescents gradually adjust their emotional expression patterns to fit group norms through the fractional-order accumulation of social experiences, differing from the immediate adaptation described by traditional integer-order models.

The noise term characterizes the dynamic response of the emotional system to social uncertainties. Sudden social events (e.g., public opinion reversals, policy changes) as random perturbations may break through emotional steady-state thresholds, triggering extreme states such as group anxiety or euphoria. However, moderate noise maintains emotional diversity within groups by introducing fluctuations, inhibiting the innovative rigidity caused by “groupthink.” This “perturbation-equilibrium” mechanism is particularly critical in creative communities, providing a dynamical basis for the emergence of breakthrough emotional patterns through noise-driven retention of individual differences.

By integrating social connection dynamics, structural constraints, historical memory, and random perturbations, this equation constructs a unified dynamical model for the propagation and evolution of emotions in social systems, offering cross-scale analytical tools for understanding group emotion synchronization, social norm formation, and response to emergency public opinion.

We now shift from qualitative understanding to mathematical analysis, establishing the continuity, uniqueness, and stability of the emotional dynamics.

3.2.2. Mathematical Properties of the Emotional State Equation

To discuss the properties of the emotional state equation, we first investigate its nonlinear term ${\sum }_{j=1}^N{W}_{ij}\left(t\right)\mathrm{\Phi }(\parallel {\mathrm{E}}_j-{\mathrm{E}}_i\parallel )({\mathrm{E}}_j-{\mathrm{E}}_i)$.

Define

$$F_i(t,\mathrm{E})=\sum _{j=1}^N{W}_{ij}\left(t\right)\mathrm{\Phi }(\parallel {\mathrm{E}}_j-{\mathrm{E}}_i\parallel )({\mathrm{E}}_j-{\mathrm{E}}_i).$$

(8)

We now prove the continuity of $F_i(t,\mathrm{E})$.

Lemma 4 

(Lipschitz Continuity of Nonlinear Term $F_i$). Assume that the system states ${\mathrm{E}}_i\left(t\right)$ are uniformly bounded over the finite time interval $[0,T]$: there exists a constant $M>0$ such that $\parallel {\mathrm{E}}_i\left(t\right)\parallel \le M$ for all nodes i and times $t\in [0,T]$. Then, the nonlinear function $F_i(t,\mathrm{E})$ satisfies a Lipschitz condition with respect to $\mathrm{E}$; that is, there exists a constant $L_F$ such that for all $\mathrm{E},{\mathrm{E}}^{\prime }\in {\mathbb{R}}^N$ and $t\in [0,T]$,

$$\parallel F_i(t,\mathrm{E})-F_i(t,{\mathrm{E}}^{\prime })\parallel \le L_F\parallel \mathrm{E}-{\mathrm{E}}^{\prime }\parallel ,$$

where $L_F=2N(a+2a^{2}M)$.

Proof. 

We first compute the function difference:

$$F_i(t,\mathrm{E})-F_i(t,{\mathrm{E}}^{\prime })=\sum _{j=1}^N{W}_{ij}\left(t\right)\left[\mathrm{\Phi }(\parallel {\mathrm{E}}_j-{\mathrm{E}}_i\parallel )({\mathrm{E}}_j-{\mathrm{E}}_i)-\mathrm{\Phi }(\parallel {\mathrm{E}}_j^{\prime }-{\mathrm{E}}_i^{\prime }\parallel )({\mathrm{E}}_j^{\prime }-{\mathrm{E}}_i^{\prime })\right].$$

Applying the norm inequality:

$$\parallel F_i(t,\mathrm{E})-F_i(t,{\mathrm{E}}^{\prime })\parallel \le \sum _{j=1}^N{W}_{ij}\left(t\right)∥\mathrm{\Phi }(\parallel {\mathrm{E}}_j-{\mathrm{E}}_i\parallel )({\mathrm{E}}_j-{\mathrm{E}}_i)-\mathrm{\Phi }(\parallel {\mathrm{E}}_j^{\prime }-{\mathrm{E}}_i^{\prime }\parallel )({\mathrm{E}}_j^{\prime }-{\mathrm{E}}_i^{\prime })∥.$$

Decompose the key term: let $\mathrm{u}={\mathrm{E}}_j-{\mathrm{E}}_i$ and $\mathrm{v}={\mathrm{E}}_j^{\prime }-{\mathrm{E}}_i^{\prime }$. Then,

$$∥\mathrm{\Phi }(\parallel \mathrm{u}\parallel )\mathrm{u}-\mathrm{\Phi }(\parallel \mathrm{v}\parallel )\mathrm{v}∥\le \underset{\left(A\right)}{\underbrace{∥\mathrm{\Phi }(\parallel \mathrm{u}\parallel )(\mathrm{u}-\mathrm{v})∥}}+\underset{\left(B\right)}{\underbrace{∥\left[\mathrm{\Phi }\right(\parallel \mathrm{u}\parallel )-\mathrm{\Phi }(\parallel \mathrm{v}\parallel \left)\right]\mathrm{v}∥}}.$$

For term (A),

$$∥\mathrm{\Phi }(\parallel \mathrm{u}\parallel )(\mathrm{u}-\mathrm{v})∥=\left|\mathrm{\Phi }\right(\parallel \mathrm{u}\parallel \left)\right|·\parallel \mathrm{u}-\mathrm{v}\parallel \le a\parallel \mathrm{u}-\mathrm{v}\parallel .$$

For term (B),

$$∥\left[\mathrm{\Phi }\right(\parallel \mathrm{u}\parallel )-\mathrm{\Phi }(\parallel \mathrm{v}\parallel \left)\right]\mathrm{v}∥\le \left|\mathrm{\Phi }\right(\parallel \mathrm{u}\parallel )-\mathrm{\Phi }(\parallel \mathrm{v}\parallel )|·\parallel \mathrm{v}\parallel \le a^2\left|\parallel \mathrm{u}\parallel -\parallel \mathrm{v}\parallel \right|·\parallel \mathrm{v}\parallel \le a^2\parallel \mathrm{u}-\mathrm{v}\parallel ·2M,$$

where the last inequality follows from $\parallel \mathrm{v}\parallel =\parallel {\mathrm{E}}_j^{\prime }-{\mathrm{E}}_i^{\prime }\parallel \le \parallel {\mathrm{E}}_j^{\prime }\parallel +\parallel {\mathrm{E}}_i^{\prime }\parallel \le 2M$.

Combining the estimates,

$$∥\mathrm{\Phi }(\parallel \mathrm{u}\parallel )\mathrm{u}-\mathrm{\Phi }(\parallel \mathrm{v}\parallel )\mathrm{v}∥\le (a+2a^{2}M)\parallel \mathrm{u}-\mathrm{v}\parallel .$$

Note that $\parallel \mathrm{u}-\mathrm{v}\parallel \le \parallel {\mathrm{E}}_j-{\mathrm{E}}_j^{\prime }\parallel +\parallel {\mathrm{E}}_i-{\mathrm{E}}_i^{\prime }\parallel \le 2\parallel \mathrm{E}-{\mathrm{E}}^{\prime }\parallel$.

Summing over all j and using $W_{ij}\left(t\right)\le 1$, we obtain

$$\parallel F_i(t,\mathrm{E})-F_i(t,{\mathrm{E}}^{\prime })\parallel \le \sum _{j=1}^N(a+2a^{2}M)\left(\parallel {\mathrm{E}}_j-{\mathrm{E}}_j^{\prime }\parallel +\parallel {\mathrm{E}}_i-{\mathrm{E}}_i^{\prime }\parallel \right)\le 2N(a+2a^{2}M)\parallel \mathrm{E}-{\mathrm{E}}^{\prime }\parallel .$$

Thus, taking $L_F=2N(a+2a^{2}M)$ completes the proof. □

To prove the solution properties of the fractional-order emotional dynamics equation, we first introduce a lemma.

Lemma 5 

(Fractional Gronwall Inequality [23]). Let $\mu \in (0,1)$, and let constants $C\ge 0$ and $\lambda >0$. If a non-negative function $u\left(t\right)\in C\left(\right[0,T\left]\right)$ satisfies the inequality

$$u\left(t\right)\le C+{\displaystyle \frac{\lambda }{\mathrm{\Gamma }\left(\mu \right)}}{\int }_0^t{(t-s)}^{\mu -1}u\left(s\right)ds\forall t\in [0,T],$$

then for any $t\in [0,T]$,

$$u\left(t\right)\le C{E}_{\mu }\left(\lambda t^{\mu }\right),$$

where $E_{\mu }\left(z\right)$ is the Mittag-Leffler function defined by

$$E_{\mu }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\mathrm{\Gamma }(\mu k+1)}}.$$

Moreover, $E_{\mu }\left(\lambda T^{\mu }\right)$ is bounded over a finite interval $[0,T]$.

We now derive the fractional-order emotional dynamics equation.

Theorem 5 

(Existence and Uniqueness of Solutions to the Fractional Master Equation). Consider the fractional master equation

$${}_0{}^{C}D_t^{\mu }{\mathrm{E}}_i\left(t\right)=\sum _{j=1}^N{W}_{ij}\left(t\right)\mathrm{\Phi }(\parallel {\mathrm{E}}_j-{\mathrm{E}}_i\parallel )({\mathrm{E}}_j-{\mathrm{E}}_i)+\gamma \sum _{j=1}^N{e}^{-\beta d_{ij}}{\mathrm{E}}_j+{\xi }_i\left(t\right),$$

(9)

where $\mu \in (0,1]$ and the initial condition is ${\mathrm{E}}_i\left(0\right)={\mathrm{E}}_i^0\in {\mathbb{R}}^d$. Then, Equation (9) has a unique solution in the space $C([0,T],{\left({\mathbb{R}}^d\right)}^N)$, and the solution continuously depends on the initial conditions.

Proof. 

We first transform Equation (9) into an integral equation. By the definition of the Caputo fractional derivative, Equation (9) is equivalent to the Volterra integral equation:

$${\mathrm{E}}_i\left(t\right)={\mathrm{E}}_i^0+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mu \right)}}{\int }_0^t{(t-s)}^{\mu -1}\left[F_i(t,\mathrm{E})+\gamma \sum _{j=1}^N{e}^{-\beta d_{ij}}{\mathrm{E}}_j+{\xi }_i\left(t\right)\right]ds,$$

(10)

where the nonlinear term $F_i(t,\mathrm{E})$ is defined by

$$F_i(t,\mathrm{E})=\sum _{j=1}^N{W}_{ij}\left(t\right)\mathrm{\Phi }(\parallel {\mathrm{E}}_j-{\mathrm{E}}_i\parallel )({\mathrm{E}}_j-{\mathrm{E}}_i).$$

(11)

By Lemma 1 and Lemma 4, for any $\mathrm{E},{\mathrm{E}}^{\prime }\in C([0,T],{\left({\mathbb{R}}^d\right)}^N)$ with uniformly bounded states ($\parallel {\mathrm{E}}_i\left(t\right)\parallel \le M$), we have

$$\parallel F_i(t,\mathrm{E})-F_i(t,{\mathrm{E}}^{\prime })\parallel \le L_F\parallel \mathrm{E}-{\mathrm{E}}^{\prime }\parallel ,$$

(12)

where the Lipschitz constant $L_F=2N(a+2a^{2}M)$, and $\parallel ·\parallel$ denotes the norm in $C([0,T],{\left({\mathbb{R}}^d\right)}^N)$.

The spatial modulation term $\gamma {\sum }_{j=1}^N{e}^{-\beta d_{ij}}{\mathrm{E}}_j$ is linear in $\mathrm{E}$. Denoting it as $G_i\left(\mathrm{E}\right)$, we have

$$\begin{array}{cc}\hfill \parallel G_i\left(\mathrm{E}\right)-G_i\left({\mathrm{E}}^{\prime }\right)\parallel & \le \gamma \sum _{j=1}^N{e}^{-\beta d_{ij}}\parallel {\mathrm{E}}_j-{\mathrm{E}}_j^{\prime }\parallel \hfill \\ \hfill & \le \gamma N\underset{1\le j\le N}{sup}\parallel {\mathrm{E}}_j-{\mathrm{E}}_j^{\prime }\parallel \hfill \\ \hfill & \le \gamma N\parallel \mathrm{E}-{\mathrm{E}}^{\prime }\parallel .\hfill \end{array}$$

(13)

Let $L_G=\gamma N$, so the Lipschitz constant for the linear term is $L_G$.

Combining the nonlinear (12) and linear (13) terms, the overall Lipschitz constant for the equation’s right-hand side is

$$L=L_F+L_G=2N(a+2a^{2}M)+\gamma N.$$

(14)

Define the solution operator $\mathcal{T}:\mathcal{C}\to \mathcal{C}$ by

$${\left(\mathcal{T}\mathrm{E}\right)}_i\left(t\right)={\mathrm{E}}_i^0+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mu \right)}}{\int }_0^t{(t-s)}^{\mu -1}\left[F_i(s,\mathrm{E})+G_i\left(\mathrm{E}\right)+{\xi }_i\left(s\right)\right]ds.$$

(15)

For any $\mathrm{E},{\mathrm{E}}^{\prime }\in \mathcal{C}$, the difference under the operator $\mathcal{T}$ satisfies

$$\begin{array}{cc}\hfill {\parallel \left(\mathcal{T}\mathrm{E}\right)}_i\left(t\right)-{\left(\mathcal{T}{\mathrm{E}}^{\prime }\right)}_i\left(t\right)\parallel & \le {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mu \right)}}{\int }_0^t{(t-s)}^{\mu -1}∥F_i(s,\mathrm{E})-F_i(s,{\mathrm{E}}^{\prime })+G_i\left(\mathrm{E}\right)-G_i\left({\mathrm{E}}^{\prime }\right)∥ds\hfill \\ \hfill & \le {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mu \right)}}{\int }_0^t{(t-s)}^{\mu -1}\left(L_F+L_G\right)\parallel \mathrm{E}-{\mathrm{E}}^{\prime }\parallel ds\hfill \\ \hfill & ={\displaystyle \frac{L}{\mathrm{\Gamma }\left(\mu \right)}}{\int }_0^t{(t-s)}^{\mu -1}ds·\parallel \mathrm{E}-{\mathrm{E}}^{\prime }\parallel .\hfill \end{array}$$

(16)

Evaluating the integral

$${\int }_0^t{(t-s)}^{\mu -1}ds={\left[-{\displaystyle \frac{{(t-s)}^{\mu }}{\mu }}\right]}_0^t={\displaystyle \frac{t^{\mu }}{\mu }}.$$

(17)

Substituting into (16) gives

$${\parallel \left(\mathcal{T}\mathrm{E}\right)}_i\left(t\right)-{\left(\mathcal{T}{\mathrm{E}}^{\prime }\right)}_i\left(t\right)\parallel \le {\displaystyle \frac{L{t}^{\mu }}{\mathrm{\Gamma }(\mu +1)}}\parallel \mathrm{E}-{\mathrm{E}}^{\prime }\parallel .$$

(18)

Summing over all i and taking the supremum,

$$\parallel \mathcal{T}\mathrm{E}-\mathcal{T}{\mathrm{E}}^{\prime }\parallel \le {\displaystyle \frac{L{T}^{\mu }}{\mathrm{\Gamma }(\mu +1)}}\parallel \mathrm{E}-{\mathrm{E}}^{\prime }\parallel .$$

(19)

Choose $T_0=min\left\{T,{\left({\displaystyle \frac{\mathrm{\Gamma }(\mu +1)}{2L}}\right)}^{1/\mu }\right\}$ such that

$${\displaystyle \frac{L{T}_0^{\mu }}{\mathrm{\Gamma }(\mu +1)}}\le {\displaystyle \frac{1}{2}}.$$

(20)

Then, $\mathcal{T}$ is a contraction mapping on $[0,T_0]$.

By the Banach fixed-point theorem, $\mathcal{T}$ has a unique fixed point on $[0,T_0]$; i.e., Equation (10) has a unique solution on $[0,T_0]$. Divide $[0,T]$ into $k=\lceil T/T_0\rceil$ subintervals $[0,T_0],[T_0,2T_0],\dots ,[(k-1)T_0,k{T}_0]$. On each subinterval, repeat the contraction mapping argument:

On the first subinterval $[0,T_0]$, a unique solution ${\mathrm{E}}^{\left(0\right)}\left(t\right)$ exists. On the second subinterval $[T_0,2T_0]$, define the initial condition ${\mathrm{E}}_i^{\left(1\right)}\left(T_0\right)={\mathrm{E}}_i^{\left(0\right)}\left(T_0\right)$ and consider the integral equation

$${\mathrm{E}}_i^{\left(1\right)}\left(t\right)={\mathrm{E}}_i^{\left(1\right)}\left(T_0\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mu \right)}}{\int }_{T_0}^t{(t-s)}^{\mu -1}\left[F_i(s,{\mathrm{E}}^{\left(1\right)})+G_i\left({\mathrm{E}}^{\left(1\right)}\right)+{\xi }_i\left(s\right)\right]ds.$$

Since $F_i$ and $G_i$ satisfy Lipschitz conditions on $[T_0,2T_0]$, a unique solution ${\mathrm{E}}^{\left(1\right)}\left(t\right)$ exists. By induction, on each subinterval $[m{T}_0,(m+1)T_0]$, a unique solution ${\mathrm{E}}^{\left(m\right)}\left(t\right)$ exists by setting the initial condition as the solution at the previous subinterval’s endpoint.

Due to the continuity of solutions at subinterval endpoints, the pieced-together function $\mathrm{E}\left(t\right)={\mathrm{E}}^{\left(m\right)}\left(t\right)$ for $t\in [m{T}_0,(m+1)T_0]$ is the unique solution on $[0,T]$.

Let $\mathrm{E}$ and $\tilde{\mathrm{E}}$ be solutions with initial conditions ${\mathrm{E}}^0$ and ${\tilde{\mathrm{E}}}^0$. Define the error function $u\left(t\right)=\parallel \mathrm{E}\left(t\right)-\tilde{\mathrm{E}}\left(t\right)\parallel$. From integral Equation (10), we have

$$\begin{array}{cc}\hfill u\left(t\right)& \le \parallel {\mathrm{E}}^0-{\tilde{\mathrm{E}}}^0\parallel +{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mu \right)}}{\int }_0^t{(t-s)}^{\mu -1}∥F_i(s,\mathrm{E})-F_i(s,\tilde{\mathrm{E}})+G_i\left(\mathrm{E}\right)-G_i\left(\tilde{\mathrm{E}}\right)∥ds\hfill \\ \hfill & \le \parallel {\mathrm{E}}^0-{\tilde{\mathrm{E}}}^0\parallel +{\displaystyle \frac{L}{\mathrm{\Gamma }\left(\mu \right)}}{\int }_0^t{(t-s)}^{\mu -1}u\left(s\right)ds.\hfill \end{array}$$

(21)

Applying the fractional Gronwall inequality (Lemma 5), we obtain

$$u\left(t\right)\le \parallel {\mathrm{E}}^0-{\tilde{\mathrm{E}}}^0\parallel E_{\mu }\left(L{t}^{\mu }\right),$$

(22)

where $E_{\mu }\left(z\right)$ is the Mittag-Leffler function [24]. Since $E_{\mu }\left(L{T}^{\mu }\right)$ is bounded on $[0,T]$, the solution depends continuously on the initial conditions. □

4. Numerical Simulations

This study conducted numerical validations for the main conclusions of the network evolution equation and the emotional state equation, respectively. All computations were implemented in a Python 3.9 environment, relying on core tools including Matplotlib 3.8.4, NumPy 1.26.4, Pandas 2.2.3, and SciPy 1.13.0. Simulations were performed on a personal laptop equipped with an Intel Core i9 processor and 32 GB of RAM.

4.1. Numerical Validation of Network Evolution Equation

The stability of the network evolution Equation (3) is numerically validated under noise-free and noisy conditions, both using the same simulation parameters listed in

Table 2. List of stability parameters.

Symbol	Name	Physical Meaning	Value
$\eta$	Evolution rate	State change rate	$0.5$
$\delta$	Decay coefficient	State decay intensity	$0.3$
$\Delta E$	Emotional fifference	Emotional discrepancy	$1.0$

.

These parameter settings are selected to guarantee analytical solvability and stable numerical integration under biologically meaningful conditions.

4.1.1. Numerical Validation of Noise-Free Equation Stability

For the noise-free network evolution Equation (3), specific parameters are selected according to

Table 3. List of noise-free simulation parameters.

Symbol	Name	Meaning	Value
$W_0$	Initial Weight	Initial state value	$[0.0,0.5,1.0,1.5,2.0]$
$t_{\mathrm{span}}$	Time Range	Simulation duration	$(0,20)$
$t_{\mathrm{eval}}$	Samples	Discrete time points	1000

, and the explicit fourth-order Runge–Kutta method (RK45) is used to solve it, with simulations conducted for different initial conditions $W_0$.

Figure 4 verifies the existence, uniqueness, and local asymptotic stability of the equilibrium state for the network weight evolution model through numerical simulations and phase diagram analysis. The main plot on the left shows that trajectories with different initial conditions $W\left(0\right)$ all converge to the equilibrium state $W^{*}=1.400$, exhibiting global attractivity: when the initial value is below $W^{*}$, weights increase over time toward the equilibrium state; when the initial value is above $W^{*}$, weights decrease and converge to $W^{*}$; intermediate initial values also converge to the equilibrium state, visually confirming the convergence of all trajectories. In the right-phase diagram, the curve of ${\displaystyle \frac{dW}{dt}}$ versus W shows that ${\displaystyle \frac{dW}{dt}}>0$ when $W<W^{*}$ and ${\displaystyle \frac{dW}{dt}}<0$ when $W>W^{*}$. The arrow directions clarify the system’s trend toward the equilibrium state, further supporting the attractivity of the equilibrium.

4.1.2. Numerical Validation of Statistical Stability for Noisy Equations

For the network evolution Equation (3) with noise, we set the simulation parameters as shown in

Table 4. Parameters for noisy simulations.

Symbol	Name	Description	Value
${\sigma }_{\zeta }$	Noise Intensity	Std. dev. of perturbations	0.1
T	Total Time	Simulation duration (s)	50
$dt$	Time Step	Integration interval	0.01
$N_{\mathrm{paths}}$	Sample Paths	Monte Carlo realizations	500

. The network evolution Equation (3) is an SDE composed of a drift term and a diffusion term. The drift term $\mu (W,t)=\eta \left({\sec \mathrm{h}}^2\left({\Delta }_E\right)-\delta W\right)$ characterizes the deterministic trend of the system evolving towards the equilibrium state $W^{*}={\displaystyle \frac{{\sec \mathrm{h}}^2\left({\Delta }_E\right)}{\delta }}$, while the diffusion term $\sigma (W,t)={\sigma }_{\zeta }$ represents additive Gaussian white noise with intensity ${\sigma }_{\zeta }$. For this SDE model, the Euler–Maruyama method [25] is used for the numerical solution, with the discretized system state update achieved using the iterative formula $W_{n+1}=W_n+\mu (W_n,t_n)dt+\sigma (W_n,t_n)\sqrt{dt}\mathcal{N}(0,1)$. Theoretical analysis shows that the steady-state variance of the system is $\mathrm{Var}\left(W\right)={\displaystyle \frac{{\sigma }_{\zeta }^2}{2\eta \delta }}$. We also analyze the convergence process of variance over time by calculating the rolling variance $\mathrm{Rolling}\mathrm{Variance}={\displaystyle \frac{1}{M}}{\sum }_{j=i}^{i+M-1}{\left(W_j-\overline{W}\right)}^2$ (where $M={\displaystyle \frac{1}{dt}}$) and conduct sensitivity analysis by varying the noise intensity ${\sigma }_{\zeta }$, aiming to verify the proportional relationship $\mathrm{Var}\left(W\right)\propto {\sigma }_{\zeta }^2$.

The numerical validation results are presented in

Table 5. Comparison between theoretical and simulated values.

Metric	Theoretical Value	Simulated Value	Error (%)
Equilibrium Value ($W^{*}$)	1.3999	1.3881	0.84
Steady-State Variance ($\mathrm{Var}\left(W\right)$)	0.033333	0.035020	5.06

Note: The percentage error is computed as $\left|{\displaystyle \frac{\mathrm{Simulated}-\mathrm{Theoretical}}{\mathrm{Theoretical}}}\right|\times 100\%$.

, with graphical outputs shown in Figure 5.

Table 5 shows the error comparison between theoretical and simulated values. Under the given noise intensity of 0.1, the simulation error for the equilibrium state is only 0.84%, and the steady-state variance error is 5.06%. These results demonstrate that the network evolution Equation (3) with noise is statistically stable in the presence of noise.

Figure 5 verifies the dynamic evolution and steady-state statistical properties of network weights in a noisy environment through numerical simulations of the SDE model. In panel (a), the mean trajectory (blue) converges to the theoretical equilibrium state $W^{*}=1.400$, with the $\pm 1$ standard deviation range fluctuating around the equilibrium, reflecting the combined effect of the deterministic drift term and random diffusion term. The steady-state distribution in panel (b) closely matches the theoretical normal distribution, with the simulated mean (1.4035) approaching the equilibrium state, validating the steady-state statistical behavior of the SDE model. In panel (c), the rolling variance (green) converges to the theoretical variance of 0.0333 (red dashed line) over time, quantifying the variance convergence process and supporting the steady-state variance formula $\mathrm{Var}\left(W\right)={\displaystyle \frac{{\sigma }_{\zeta }^2}{2\eta \delta }}$. Panel (d) shows that by varying the noise intensity ${\sigma }_{\zeta }$, the simulation results are consistent with the theoretical relationship, indicating that steady-state variance is proportional to ${\sigma }_{\zeta }^2$, thus verifying the quantitative impact of the diffusion term on system variance. The subfigures comprehensively demonstrate the theoretical and numerical consistency of the model from the perspectives of dynamic evolution, steady-state distribution, variance convergence, and noise sensitivity: the mean converges to the equilibrium state dominated by the drift term, the variance is controlled by the diffusion term and proportional to the square of noise intensity, and the steady-state distribution conforms to the normal assumption. These analyses provide a rigorous numerical validation of the system’s statistical properties in random environments, enhancing the theoretical credibility of the SDE model and supporting its application rationale in noise-driven systems.

4.2. Numerical Validation of the Emotional State Equation

We conduct numerical simulations for the fractional-order differential Equation (2) to illustrate the existence, uniqueness, and continuous dependence of its solutions.

4.2.1. Numerical Solution for Fractional-Order Equations

The L1 method [26] is employed for the numerical discretization of the fractional derivative in (2), constructing a fractional-order complex network system with time-delay dependence. For the Caputo fractional derivative $D_t^{\mu }E\left(t\right)$, its L1 approximation at time $t=t_n$ is given by

$$D_t^{\mu }E\left(t_n\right)\approx {\displaystyle \frac{{\left(dt\right)}^{-\mu }}{\mathrm{\Gamma }(2-\mu )}}\sum _{k=0}^n{c}_k\Delta E_{n-k},$$

where $\Delta E_j=E\left(t_j\right)-E\left(t_{j-1}\right)$, and the coefficients $c_k$ are generated through the recurrence relation

$$c_0=1,c_k=\left(1-{\displaystyle \frac{1+\mu }{k}}\right)c_{k-1},k=1,2,\dots ,n.$$

The numerical solution for system state evolution is obtained through stepwise integration, with the specific expression

$$E_i\left(t_{n+1}\right)=E_i\left(t_0\right)+{\displaystyle \frac{{\left(dt\right)}^{\mu }}{\mathrm{\Gamma }(\mu +1)}}\sum _{k=0}^n{c}_{n-k}{f}_i\left(t_k\right),$$

where

$$f_i\left(t_k\right)=F_i(E\left(t_k\right),W\left(t_k\right))+G_i\left(E\left(t_k\right)\right)+{\xi }_i\left(t_k\right).$$

The entire numerical method is implemented via time-step iteration. At each time step n, the weight matrix $W\left(t_n\right)$ is first updated based on the current state, followed by the computation of the interaction term $F_i$ and the diffusion term $G_i$. Combining with the noise term ${\xi }_i$, the system state $E\left(t_{n+1}\right)$ is ultimately updated. The algorithm clips the state at each time step to ensure physical feasibility and improves computational accuracy through adaptive step-size control and error estimation.

4.2.2. Parameter Settings

Uniform parameter settings for the numerical validation of the emotional state equation are listed in

Table 6. Numerical validation parameter details.

Symbol	Default Value	Description
N	3	Network node quantity
d	1	State dimension per agent
T	3.0	Total simulation time (seconds)
${\mu }_{\mathrm{range}}$	$(0.3,0.7)$	Order range of fractional derivatives
$\mathcal{D}$	$\{0.1,0.05,0.025,0.0125\}$	Decreasing time step sequence
$d{t}_{\mathrm{ref}}$	$min\left(\mathcal{D}\right)/4$	High-precision reference step size
${\beta }_{\mathrm{tol}}$	$[0.7,1.3]$	Acceptable convergence slope range
$n_{\mathrm{trials}}$	10	Number of independent simulations
$ϵ$	0.001	System perturbation amplitude
${\Delta }_{\mathrm{th}}$	0.1	Upper bound of state difference median
$\mathcal{E}$	$\{{10}^{-4},5\times {10}^{-4},{10}^{-3},5\times {10}^{-3}\}$	Initial perturbation amplitudes
${\gamma }_{\mathrm{tol}}$	$[0.9,1.1]$	Continuous dependence slope range

.

4.2.3. Verification of Solution Existence

To examine the convergence of numerical solutions, we designed a set of decreasing time steps $\mathcal{D}=\{0.1,0.05,0.025,0.0125\}$, conducting system simulations for each $dt\in \mathcal{D}$. Meanwhile, a high-precision reference solution $E_{\mathrm{ref}}$ (using $d{t}_{\mathrm{ref}}=min\left(\mathcal{D}\right)/4$) was computed as a benchmark. For each time step $dt$, the average Frobenius norm error between its solution and the reference solution over the entire time interval was calculated:

$$\mathrm{Error}\left(dt\right)={\displaystyle \frac{1}{T}}{\int }_0^T{\parallel E(t;dt)-E_{\mathrm{ref}}\left(t\right)\parallel }_{\mathrm{F}}dt.$$

According to numerical analysis theory, if the system has a unique solution, the error should satisfy $\mathrm{Error}\left(dt\right)\propto d{t}^{\alpha }$, where $\alpha \approx \mu$ is the theoretical convergence order. By performing linear regression on $log\left(\mathrm{Error}\right)$ and $log\left(dt\right)$ as $log\left(\mathrm{Error}\right)=\beta log\left(dt\right)+C$, the slope $\beta$ was estimated. In the experiment, if the estimated convergence order $\beta$ satisfies $0.7\le \beta \le 1.3$, the numerical solution is considered to converge to the true solution, thereby verifying the existence of solutions for the original system under the given parameters. This verification method strictly follows the best practices of numerical analysis for fractional-order differential equations [27], ensuring the reliability of the results.

Figure 6 validates the existence of the system’s solution by analyzing the variation of numerical solution errors with time step $dt$. Specifically, a set of decreasing time steps $\mathcal{D}$ was designed, and the error $\mathrm{Error}\left(dt\right)$ for each $dt$ was calculated and compared with the theoretical convergence order $O\left(d{t}^{\alpha }\right)$ ($\alpha \approx \mu =0.5$). The blue solid line (observed convergence) in the figure aligns with the orange dashed line (theoretical $O\left(d{t}^{0.5}\right)$), indicating that the error growth matches theoretical expectations. Further estimation of the convergence order $\beta$ through linear regression, which falls within the range $[0.7,1.3]$, confirms that the numerical solution converges to the true solution, indirectly proving the existence of a unique solution for the original system. The figure shows that errors increase with larger values of $dt$ and align with the theoretical convergence order, visually demonstrating the convergence of numerical solutions and providing empirical support for solution’s existence.

4.2.4. Verification of Solution Uniqueness

Based on the core idea of the Picard–Lindelöf theorem [28], we provide empirical support for uniqueness through numerical experiments quantifying the sensitivity of solutions to initial conditions, specifically via analysis of solution differences under multiple sets of independent random perturbations. Specifically, with fixed initial conditions $E_i\left(0\right)$ and fractional-order parameter $\mu$, $n_{\mathrm{trials}}=10$ independent simulations were conducted at the same time step $dt=0.01$, each introducing minute random perturbations. For each simulation result, the state vector $E_i\left(T\right)$ at the final time $t=T$ was extracted, and the Frobenius norm differences between all pairwise solutions were calculated:

$${\Delta }_{ij}={\parallel E^{\left(i\right)}\left(T\right)-E^{\left(j\right)}\left(T\right)\parallel }_{\mathrm{F}},i<j,$$

where $E^{\left(i\right)}\left(T\right)$ and $E^{\left(j\right)}\left(T\right)$ denote the final states of the i-th and j-th independent simulations. The uniqueness of solutions was evaluated by statistically analyzing the median $\mathrm{Median}(\Delta )$ of the difference sequence $\left\{{\Delta }_{ij}\right\}$: if $\mathrm{Median}(\Delta )<0.1$, the solutions under different initial perturbations were considered to converge to the same attractor, verifying the uniqueness of the system’s solution.

In the uniqueness verification, we consider the solution to be numerically unique if the median pairwise Frobenius norm difference between final states across 10 trials is less than 0.1. This threshold reflects a less than 5% deviation relative to the bounded emotional state range $[-1,1]$, and it was confirmed empirically to be stable across repeated tests.

In Figure 7, the left histogram displays the frequency distribution of difference values, while the right box plot presents the statistical characteristics of difference magnitudes: the median is approximately 0.010 (well below the threshold of 0.1), the interquartile range is 0.005–0.015, and over 90% of differences are less than 0.025. These data indicate that solution differences under minute perturbations are highly concentrated, verifying that solutions under different initial perturbations converge to the same attractor. The statistical characteristics of solution differences (small median, concentrated distribution) visually reflect the numerical realization of “continuous dependence of solutions on initial conditions” in the Picard–Lindelöf theorem. The quantitative analysis ($\mathrm{median}<0.1$) strictly supports the uniqueness of system solutions, providing empirical evidence for the theoretical conclusions.

4.2.5. Verification of Continuous Dependence

The verification of continuous dependence of solutions on initial conditions is achieved through quantitative relationship analysis between perturbation amplitudes and state differences. The specific steps are as follows: First, fix the fractional-order parameter $\mu$ and time step $dt=0.01$, and set a set of initial perturbation amplitudes $\mathcal{E}=\{{10}^{-4},5\times {10}^{-4},{10}^{-3},5\times {10}^{-3}\}$. For each perturbation amplitude $ϵ\in \mathcal{E}$, generate perturbed initial conditions $E_i^{ϵ}\left(0\right)=E_i\left(0\right)+ϵ·{\eta }_i$, where ${\eta }_i$ is a standard normal random vector. Using the simulation result $E_{\mathrm{ref}}\left(t\right)$ with unperturbed initial conditions $E_i\left(0\right)$ as a benchmark, calculate the maximum state difference between the perturbed solution and the benchmark solution over the entire time interval:

$$\Delta \left(ϵ\right)=\underset{t\in [0,T]}{max}{\parallel E^{ϵ}\left(t\right)-E_{\mathrm{ref}}\left(t\right)\parallel }_{\mathrm{F}}.$$

According to the theory of continuous dependence, if the system satisfies the Lipschitz condition, $\Delta \left(ϵ\right)$ should exhibit a linear relationship with $ϵ$, i.e., $\Delta \left(ϵ\right)\propto {ϵ}^{\gamma }$ with $\gamma \approx 1$. By performing linear regression on $log(\Delta )$ and $log\left(ϵ\right)$ as $log(\Delta )=\beta log\left(ϵ\right)+C$, the slope $\beta$ is estimated. In the experiment, if the estimated slope satisfies $0.9\le \beta \le 1.1$, it verifies the continuous dependence of system solutions on initial conditions, indicating that small initial perturbations only cause minor changes in solutions. This method provides rigorous numerical evidence for solution stability analysis by quantifying the sensitivity index.

In Figure 8, both the maximum state difference curve and the relative difference curve exhibit an approximately linear growth with increasing initial perturbation $ϵ$, reflecting $\Delta \left(ϵ\right)\propto {ϵ}^{\gamma }$ where $\gamma \approx 1$. The slope $\beta$ estimated through linear regression satisfies $0.9\le \beta \le 1.1$, with the curve trend indicating $\beta \approx 1$—consistent with continuous dependence theory under Lipschitz conditions. This demonstrates that small initial perturbations induce only minor solution changes, while the linear growth relationship visually verifies the continuous dependence of solutions on initial conditions.

The left curve shows nearly linear amplification of $\Delta \left(ϵ\right)$ as $ϵ$ increases from ${10}^{-4}$ to $5\times {10}^{-3}$, while the right relative difference curve reflects proportional perturbation influence. Together, they quantify the solution sensitivity index and provide rigorous numerical evidence for stability analysis. The curve trends and theoretical regression strictly follow the continuous dependence verification framework, ensuring solution stability under small perturbations. These results further support solution existence and uniqueness of solutions, enhancing the theory’s empirical credibility.

5. Sensitivity Analysis of Key Parameters

To assess the robustness and responsiveness of the proposed emotional dynamics model, we conducted a comprehensive sensitivity analysis on three core parameters: the fractional order $\mu$, the evolution rate $\eta$, and the decay coefficient of social ties $\delta$. These parameters respectively control the system’s memory depth, the adaptivity of social ties to emotional differences, and the natural fading tendency of interpersonal influence. Their joint effects shape the network co-evolution of emotion and structure.

5.1. Parameter Settings

The system is simulated using a minimal network of $N=3$ agents, each with scalar emotional states. The time horizon is set to $T=10$ with a fixed step size $\Delta t=0.02$, resulting in 500 discrete time steps per simulation.

The sensitivity analysis scans across different values of $\mu$, $\eta$, and $\delta$ while keeping other parameters fixed.

Table 7. Ranges of parameters used in sensitivity analysis.

Parameter	Meaning	Scanned Values
$\mu$	Fractional order (memory)	0.5, 0.7, 0.9
$\eta$	Weight evolution rate	0.1, 0.3, 0.5, 0.7, 1.0
$\delta$	Connection decay coefficient	0.1, 0.2, 0.3, 0.4, 0.5

summarizes the parameter ranges used in the simulation.

For each $(\mu ,\eta ,\delta )$ combination, the system is simulated once, and the following quantities of interest (QoIs) are extracted from the final 50 time steps:

• The variance of emotional states $Var\left(E\right)$, representing internal emotional fluctuation;
• The average connection strength $\overline{W}$, indicating group cohesion level.

5.2. Results and Interpretation

Figure 9 presents the effect of each parameter on system output by plotting one-dimensional slices with other parameters fixed ($\eta =0.5$, $\delta =0.3$, $\mu =0.7$, respectively).

In the left panel, we observe that varying the fractional order $\mu$ has minimal influence on $Var\left(E\right)$. This indicates that within the short simulation horizon, the memory effect embedded in the fractional operator does not significantly accumulate to alter emotional variance. The system shows temporal robustness to $\mu$ in this setting.

The middle panel reveals a strong sensitivity of the average connection strength $\overline{W}$ to the parameter $\eta$. As $\eta$ increases from 0.1 to 0.5, $\overline{W}$ rises sharply, indicating more responsive adaptation of social ties. However, the effect saturates beyond $\eta =0.5$, suggesting diminishing returns for increasing update rates.

In the right panel, we examine the influence of $\delta$. A monotonic decrease in $\overline{W}$ is observed as $\delta$ increases. This confirms that higher decay rates cause rapid weakening of interpersonal ties, reducing overall connectivity. The dependence appears approximately linear, indicating predictable degradation of network cohesion under strong fading mechanisms.

This sensitivity analysis reveals distinct roles of the three parameters. Structural parameters $\eta$ and $\delta$ directly shape network cohesion, with $\eta$ enhancing and $\delta$ suppressing connection strength. In contrast, $\mu$ predominantly affects long-term memory accumulation, which may require longer simulations or more complex stimuli to manifest strongly.

In practical modeling and control of emotional collective behavior, $\eta$ and $\delta$ can serve as tuning knobs to regulate responsiveness and cohesion, while $\mu$ may be more relevant for applications involving history-dependent behavior or long-memory learning.

6. Conclusions

This research presents a coupled dynamics model integrating Caputo fractional derivatives and hyperbolic tangent nonlinear functions to overcome limitations in traditional models, specifically the insufficient representation of emotional memory’s non-local properties and the absence of nonlinear interaction mechanisms. The power-law integral kernel of the fractional derivative enables the continuous adjustment of emotional influence with time, reflecting the psychological recency effect, where recent emotional experiences have stronger residual impacts. The nonlinear function characterizes how emotional differences impose threshold effects on interaction intensities, with its global Lipschitz continuity ensuring solution uniqueness.

Theoretical analyses demonstrate that the network evolution equation possesses a unique equilibrium state, whose local asymptotic stability is confirmed through eigenvalue analysis. For stochastic scenarios, a steady-state variance formula quantifies the relationship between noise intensity and system fluctuations, revealing that variance scales with the square of noise intensity. Numerical simulations across multiple scales validate these theoretical predictions: mean connection weights converge to equilibrium states, stationary distributions exhibit Gaussian characteristics, and variance dynamics align with theoretical power-law relationships.

This framework provides interdisciplinary analytical tools for studying emotional dynamics. In psychology, it can quantify the effectiveness of emotional interventions and clarify links between depressive symptoms and emotional volatility. In social network research, the spatial modulation term simulates how asymmetric social structures constrain emotional propagation. Future directions may include extending the model to multilayer network coupling, simulating extreme events under multiplicative noise, calibrating parameters via Bayesian methods, and validating power-law memory decay through virtual social experiments. Additionally, integrating multidimensional emotional states with neuroscience data could enhance the model’s applications in mental health early warning and social governance, further bridging theoretical advancements with real-world complexity.

Future work may incorporate empirical emotional datasets (e.g., from social media platforms) to validate the model and calibrate its parameters, especially in applications at the interface of computational sociology and emotional dynamics. Further investigation may also explore qualitative transitions in emotional behavior (e.g., steady-state versus oscillatory patterns) as key parameters such as $\mu$, $\delta$, or ${\sigma }_{\xi }$ vary, using tools from nonlinear dynamics and bifurcation theory. In addition, future extensions may consider heterogeneous network topologies (e.g., scale-free or small-world graphs) to examine how structural diversity influences emotional regulation, resonance formation, and the emergence of equilibrium states.