Daley–Kendall Rumor Model Analysis

• The Daley–Kendall rumor model is a mathematical framework that partitions a closed population into ignorants, spreaders, and stiflers to study information dynamics.
• It uses quadratic interaction terms and deterministic ODE approximations to capture transitions, revealing key threshold phenomena and final size relations.
• Advanced extensions incorporate stochastic processes, non-Markovian delays, and network topology effects to refine predictions of rumor propagation.

The Daley–Kendall rumor model is a foundational mathematical framework for the analysis of information propagation dynamics in closed populations, structurally analogous to SIR epidemic models but characterized by stifling mechanisms unique to rumor processes. The model partitions individuals into ignorants, spreaders, and stiflers and employs quadratic interaction terms to capture the competition between rumor transmission and cessation. Advanced limit theorems, network generalizations, and non-Markovian extensions frame its contemporary treatment in the literature.

1. Model Definition and Governing Equations

The DK rumor model operates on a closed, homogeneously-mixed population of size $N$ (or $N+1$ in some conventions). Each individual occupies one of three mutually exclusive states:

• Ignorant ($I$): Has not heard the rumor.
• Spreader ($S$): Actively propagates the rumor.
• Stifler ($R$): Knows the rumor but refuses to spread it.

The discrete-time process is modeled by a continuous-time Markov chain, or, in the infinite-population limit, by deterministic ODEs. Let $I(t)$, $S(t)$, $R(t)$ denote the fractions (or counts) of ignorants, spreaders, and stiflers at time $t$; the normalization $I(t)+S(t)+R(t)=1$ applies throughout. The mean-field evolution is:

$$\begin{aligned} \frac{dI}{dt} &= -\lambda I S \ \frac{dS}{dt} &= \lambda I S - \alpha S(S + R) \ \frac{dR}{dt} &= \alpha S(S + R) \end{aligned}$$

where $\lambda$ is the transmission rate and $\alpha$ governs conversion of spreaders into stiflers via either $S$–$S$ or $S$–$R$ encounters (0807.1458, Lebensztayn et al., 2010). The quadratic terms $S(S+R)$ encapsulate the dual mechanism of stifling—spreaders cease spreading when encountering either other spreaders or stiflers.

2. Stochastic Formulation and Limit Theorems

On a finite population, the DK process is a density-dependent continuous-time Markov chain. Three fundamental transitions define its generator:

• $I+S \xrightarrow{\lambda} S+S$ (ignorant becomes spreader)
• $S+S \xrightarrow{\alpha} R+S$ (initiating spreader becomes stifler)
• $S+R \xrightarrow{\alpha} R+R$ (spreader becomes stifler)

The fluid limit as $N\to\infty$ leads to the deterministic ODE system above. Rigorous convergence results hold:

• Law of Large Numbers: The scaled process $(I^{(N)}(t), S^{(N)}(t), R^{(N)}(t))$ converges in probability to $(I(t), S(t), R(t))$ (solution of the DK ODE system) uniformly on compacts (Lebensztayn et al., 2010).
• Central Limit Theorem: The centered, rescaled fluctuations obey a linear SDE driven by transition-rate covariances, yielding explicit Gaussian process limits for fluctuations around the fluid trajectory (Lebensztayn et al., 2010, Coletti et al., 9 Jul 2024).

Extensions to non-Markovian delay mechanisms alter DK’s evolution from ODEs to Volterra integral equations, replacing memoryless transitions with general sojourn and incubation times. Functional LLN and CLT results obtain in this setting, with explicit characterizations of variance kernels and asymptotic behavior (Coletti et al., 9 Jul 2024).

3. Analytical Structure, First Integral, and Phase Portrait

The classical DK ODE admits both explicit and implicit solutions. Detailed phase-plane analysis yields:

• First Integral (Hamiltonian): $\mathcal H(I,R)=R/\rho_1 + \ln I/\rho_2 - I/\rho_2$ is invariant along trajectories, partitioning the phase triangle into non-crossing orbits (Ragagnin, 2016).
• General Solution: $R(t)$ is expressed implicitly as a function of $I(t)$ via $\mathcal H(I(t),R(t)) = k$ for some constant $k$ fixed by initial conditions. The passage of time parametrizes the trajectory in integral form.
• Final Size Relation: The terminal state $(I_\infty, R_\infty)$ solves the transcendental equation arising from the first integral, refining earlier asymptotic “final size” formulas and offering precise computation of outbreak impact (Ragagnin, 2016).

Stability analysis along the $I+R=1$ equilibrium segment reveals a threshold $\sigma=\rho_1/(\rho_1+\rho_2)$ demarcating Lyapunov-stable equilibria—solutions approach rest states along constant-level Hamiltonian contours.

4. Network Topology and Generalizations

The DK model’s homogeneous-mixing assumptions are subject to rigorous extension on complex networks:

• Degree-based Mean-field: For populations structured by degree $k$, state fractions $I_k, S_k, R_k$ follow block-wise ODEs accounting for neighbor distributions as $P(l | k) = l p(l)/\langle k \rangle$ (Naimi et al., 2013, 0807.1458).
• Generalized Stifling Rates: The DK framework admits splitting $\alpha$ into $\alpha^{(1)}$ (spreader–spreader stifling) and $\alpha^{(2)}$ (spreader–stifler stifling). This separation reveals distinct roles: $\alpha^{(1)}$ controls outbreak tempo/peak, $\alpha^{(2)}$ governs coverage/final reach (Naimi et al., 2013).
• Threshold Phenomena: In random graphs and homogeneous networks, the critical threshold for rumor propagation is $\lambda_c = 0$ (DK with $\delta=0$), but with forgetting ($\delta > 0$), it becomes $\lambda_c = \delta/\langle k \rangle$. On uncorrelated scale-free networks, $\lambda_c \rightarrow 0$ as $N \rightarrow \infty$ (0807.1458).
• Assortativity and Topology Effects: Assortative mixing (positive degree correlations) speeds up initial rumor spread but impacts final reach variably depending on $\lambda$; scale-free networks are characterized by super-spreader dynamics and nontrivial outbreak thresholds (0807.1458).

Generalizations incorporate heterogeneous subgroups (multiple spreader/stifler classes) with individualized transmission and stifling rates to reflect behavioral diversity in real-world informational epidemics (Isea et al., 2016).

5. Performance Metrics, Asymptotic Behavior, and Peak Characterization

Principal outcomes of the DK model include the quantification of reliability and efficiency:

• Reliability: Final fraction of stiflers $R_\infty$ is computed via degree distributions and auxiliary integrals (e.g., $\phi(t)=\int_0^t \langle k S(\tau) \rangle d\tau$). In homogeneous mixing, $R_\infty$ links directly to the final value of this accumulation (Naimi et al., 2013).
• Efficiency: Determined by the extinction time of spreaders, i.e., the time required for $S(t)$ to drop below threshold or $R(t)$ to saturate (Naimi et al., 2013).
• Maximum Spreaders/Peak: The maximum proportion of spreaders during the process, $S_N^* \to 1-\log 2 \approx 0.3069$ a.s. as $N \rightarrow \infty$ in the classical DK and Maki–Thompson models (Lebensztayn et al., 10 Jul 2025). Probabilistic variants (with parameters $\alpha$, $p$) yield modified peak heights, with higher stifling probability $\alpha$ or lower transmission probability $p$ reducing the peak magnitude.

A random time-change argument and density-dependent limit theorem (Ethier–Kurtz) underpin the almost-sure convergence of peak spreader proportions.

6. Quasi-Stationary and Pre-Absorption Dynamics

Absorption occurs when all spreaders vanish ($S=0$). Quasi-stationary distribution (QSD) analysis reveals:

• Classical QSD is trivial—all mass at $(0,1)$ (single ignorant left with one spreader)—but modified absorption conditioning (e.g., killing at $S=1$ on first return) admits nontrivial QSDs with explicit path-sum formulas for occupation densities (Ben-Ari et al., 28 Nov 2025).
• The ratio-of-expectations distribution (mean occupation measure before absorption) is a more informative pre-absorption stochastic descriptor, especially for transient analysis (Ben-Ari et al., 28 Nov 2025).

Large-deviation and saddle-point techniques approximate QSDs for large $N$, recovering deterministic DK trajectories as distribution peaks.

7. Structural Properties, Graph-Theoretic Classification, and Extensions

The DK model embeds naturally in broader interaction-driven ODE frameworks governed by the dependency graph of state interactions:

• Graph-theoretic Monotonicity: Transition graph as DAG; dependency graph conditions guarantee convergence to equilibrium $(y_1^*, 0, y_3^*)$, exponential decay of transient states, and finite total exposure (i.e., $\int_0^\infty y_2^*(t) dt < \infty$) (Chan et al., 4 Nov 2025).
• Robustness and Modifications: Removing edges in the dependency graph can transition decay rates from exponential to algebraic; small structural changes induce qualitative flips in asymptotic behavior.
• Heterogeneities and Couplings: DK generalizations accommodate multiple spreader/stifler subpopulations, time-varying rates, or coupling to epidemic processes (rumor-dependent SIR), reshaping equilibrium and outbreak characteristics (Isea et al., 2016, Chan et al., 4 Nov 2025).

These abstractions support systematic exploration of rumor dynamics in epidemiological, social, and network-theoretic contexts.

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Summary Table: Central Mechanisms and Equations in the DK Model

Mechanism/Parameter	DK Equation / Structure	Reference
Ignorant–Spreader (SI)	$dI/dt = -\lambda I S$	(0807.14581003.4995)
Spreader–Stifler (SR)	$dS/dt = \lambda I S - \alpha S(S+R)$	(0807.14581303.6120)
Stifler Formation	$dR/dt = \alpha S(S+R)$	(0807.14581003.4995)
Peak Spreaders	$S_N^* \to 1-\log2$	(Lebensztayn et al., 10 Jul 2025)
First Integral	$\mathcal H(I,R)$ (see section 3 above)	(Ragagnin, 2016)
Reliability, Efficiency	$R_\infty$, Extinction Time	(Naimi et al., 2013, 0807.1458)

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The Daley–Kendall rumor model remains central in rigorous studies of information propagation and stochastic rumor dynamics. Its analytic tractability, extensive theoretical underpinnings, and sensitivity to structural and probabilistic extensions make it a canonical tool for understanding spreading phenomena in both abstract and real-world network contexts.