Forgetting Makes Rumors Live Longer — A Surprising Twist in How Information Spreads

On a one-dimensional street where everyone talks only to their neighbors, a rumor should die out quickly. After all, once people lose interest, they stop spreading it — and with no one left to carry the message, the rumor fades. That’s what classical models predict. But a new mathematical study reveals a counterintuitive truth: if people can forget the rumor, it might survive indefinitely.

In fact, the researchers found that forgetting doesn’t just prolong a rumor — it creates a tipping point. Below a certain transmission rate, the rumor dies. Above it, the rumor spreads forever, reaching an infinite number of people over time. And this threshold isn’t fixed: it depends on how quickly people lose interest. For one model, the critical transmission rate grows quadratically with the stifling rate — meaning that as people become more skeptical, you need dramatically more contagious information to sustain spread.

This isn’t about gossip on a city block. It’s a fundamental insight into how information behaves in structured networks — from social media echo chambers to disease outbreaks and financial panic. The result challenges decades of intuition: mechanisms that seem to suppress spread can, under the right conditions, enable persistence.

The Science

The paper by Nevena Marić, Aleksandar Minić, and Pablo M. Rodriguez (2026) examines three variations of the Maki–Thompson rumor model on the one-dimensional integer lattice $\mathbb{Z}$ — essentially, an infinite line of individuals, each connected only to their immediate neighbors. These models are formulated as continuous-time Markov processes, where each person exists in one of three states:

• Ignorant (0): doesn’t know the rumor
• Spreader (1): knows and actively shares it
• Stifler (2): knows but no longer spreads it

The classical model, called MT0, follows strict rules:

• A spreader transmits the rumor to an ignorant neighbor at rate $\lambda$.
• When a spreader contacts any informed individual (spreader or stifler), they become a stifler — modeling loss of interest due to redundancy.
• Once someone becomes a stifler, they stay that way forever.

This last assumption is key: stiflers are permanent. There’s no way back to being ignorant. In real life, of course, people forget. So the authors introduce two variants that include forgetting:

1. CRS model (Coletti–Rodriguez–Schinazi): Stifling occurs only when two spreaders interact (at rate $\alpha$), and both spreaders and stiflers forget the rumor at rate 1.
2. MT model (spatial Maki–Thompson): Stifling occurs when a spreader contacts any informed individual (at rate $\alpha$), and both spreaders and stiflers forget at rate 1.

All models start with a single spreader at position 0 and all others ignorant — the “standard initial configuration.” The central question is: does the rumor die out almost surely, or does it survive with positive probability?

The analysis combines rigorous mathematical proofs with large-scale numerical simulations using the Gillespie algorithm — a method for exact stochastic simulation of continuous-time processes. To handle the infinite lattice, the authors use dynamically expanding domains, allowing the rumor front to propagate without boundary effects.

What They Found

The results are striking — and sharply divergent across models.

MT0: Inevitable Extinction

For the classical model with permanent stiflers, the rumor always dies out, no matter how high the transmission rate $\lambda$. This isn’t just a simulation observation — it’s a theorem.

Theorem 1 (Marić et al., 2026): For all $\lambda > 0$, the probability of survival is zero. Moreover, the expected number of people who ever hear the rumor — the total range $N(\tau_\infty)$ — is exactly

$$\mathbb{E}[N({\tau }_{\infty })]=2(\lambda +1)$$

This means the rumor spreads linearly with $\lambda$, but never escapes finite reach. Even if it’s extremely contagious, it eventually hits a wall of stiflers and collapses.

Why? Because every spreader eventually becomes a stifler, and stiflers never forget. So the rumor leaves behind a permanent “wake” of immune individuals. On a one-dimensional lattice, this wake eventually blocks both directions. It’s like a wildfire burning through a narrow canyon — no matter how fast it spreads, it can’t jump the walls.

CRS and MT: Phase Transitions Enabled by Forgetting

Now consider the models with forgetting. Here, stiflers can revert to ignorants. This seems like it should make extinction more likely — after all, you’re removing informed individuals from the system.

But the opposite happens.

Both the CRS and MT models exhibit nontrivial phase transitions: there exists a critical transmission rate $\lambda_c(\alpha)$ such that

• If $\lambda < \lambda_c(\alpha)$, the rumor dies out almost surely.
• If $\lambda > \lambda_c(\alpha)$, the rumor survives with positive probability.

This is a qualitative shift. The system goes from certain death to potential immortality — all because people can forget.

Using simulations over $\alpha \in [0, 20]$, the authors estimate the critical curve $\lambda_c(\alpha)$ for both models.

For the CRS model, the data fit a quadratic law:

$${\lambda }_c(\alpha )\approx 1.65+7{\alpha }^2$$

At $\alpha = 0$, this recovers the contact process threshold $\lambda_c^{cp} \approx 1.6489$. As $\alpha$ increases, the required $\lambda$ grows rapidly — suggesting that high skepticism demands much higher contagiousness for sustained spread.

For the MT model, the fit is steeper:

$${\lambda }_c(\alpha )\approx 1.65+\frac{2.8\alpha }{(1-\alpha {)}^{2.65}}$$

This function blows up as $\alpha \uparrow 1$, indicating a sharp divergence in the critical threshold near maximum stifling.

Critical transmission rate vs. stifling rate in the CRS model

Label	Value
α = 0	1.65
α = 5	176.65
α = 10	701.65
α = 15	1,576.65
α = 20	2,801.65

Fit: λ_c(α) ≈ 1.65 + 7α². At α=0, recovers contact process threshold.

Critical transmission rate vs. stifling rate in the MT model

Label	Value
α = 0.1	1.68
α = 0.5	3.05
α = 0.7	6.12
α = 0.9	20.45
α = 0.95	82.3

Fit: λ_c(α) ≈ 1.65 + 2.8α/(1−α)^2.65. Diverges as α ↑ 1.

These aren’t just curves — they represent phase diagrams in the $(\lambda, \alpha)$ plane. Each point tells you whether a rumor lives or dies. The existence of such a diagram is itself a discovery: prior work had not established critical thresholds for spatial rumor models with forgetting.

The authors also provide a heuristic explanation for the quadratic scaling in the CRS model. At the rumor front, new spreaders are created just behind the edge. If two such “parent” and “child” spreaders interact, both can be stifled — a mutual cancellation that suppresses growth. The rate of such events scales with $\alpha$, and their cumulative effect requires $\lambda$ to grow quadratically to compensate.

Why This Changes Things

At first glance, this is abstract mathematics: particles on a lattice, phase transitions, critical exponents. But the implications ripple far beyond.

1. Forgetting Is Not Just Noise — It’s a Structural Enabler

Traditionally, forgetting has been treated as a damping mechanism — a way to slow or stop information spread. Public health campaigns assume that waning immunity reduces outbreak risk. Social media moderation aims to reduce virality by limiting exposure. Financial regulators expect panic to subside as memories fade.

But this work shows that forgetting can enable persistence. By removing stiflers, it clears space for new spreaders to emerge. It prevents the formation of permanent immune barriers. In one dimension, that’s the difference between inevitable extinction and potential survival.

This reframes how we think about memory in social systems. Maybe the reason some ideas persist for centuries isn’t because they’re unforgettable — but because they’re forgettable. They fade, then reignite. They retreat, then return. Like forest fires that depend on regrowth, they need the system to reset.

2. Spatial Structure Changes Everything

In a well-mixed population — the assumption behind most epidemic and rumor models — the mean-field approximation predicts that survival depends only on $\lambda > 1$, regardless of $\alpha$. But on a lattice, the critical threshold depends strongly on $\alpha$.

This is a reminder that network geometry matters. In real life, we don’t interact with everyone equally. We have neighbors, friends, colleagues — local structures that shape how information flows. Ignoring that structure leads to wrong predictions.

The one-dimensional lattice is the simplest possible spatial model. Yet it already reveals dynamics invisible in mean-field theories. If we want to understand real-world spread — whether of misinformation, innovations, or diseases — we must account for space.

3. The Paradox of Skepticism

The models show that higher stifling rates $\alpha$ raise the bar for survival. That makes intuitive sense: if people lose interest faster, the rumor needs to spread faster to keep up.

But here’s the twist: without some level of forgetting, no amount of transmission can sustain the rumor. So while high $\alpha$ makes survival harder, zero $\alpha$ makes it impossible.

This suggests a Goldilocks zone for information ecosystems: too much skepticism kills new ideas; too little allows permanent polarization. But a moderate level of forgetting and skepticism might allow healthy turnover — the death of bad ideas and the rebirth of good ones.

What’s Next

The paper opens several new directions.

First, the results are for $\mathbb{Z}^1$. What happens in higher dimensions? On $\mathbb{Z}^2$, could rumors survive even without forgetting? Preliminary work by Coletti et al. suggests yes — but the critical thresholds remain unknown. Extending these methods to 2D and beyond could reveal how dimensionality shapes information resilience.

Second, the models assume symmetric, regular lattices. Real networks are irregular — scale-free, small-world, modular. Do phase transitions persist on random graphs? How do hubs or communities affect the critical threshold? These questions are wide open.

Third, the current models treat all spreaders identically. But in reality, some people are more influential, more persuasive, or more connected. Introducing heterogeneity — in transmission rates, memory spans, or network positions — could yield richer dynamics.

Finally, there’s the question of validation. Can we observe these phase transitions in real data? Social media platforms have vast logs of information spread. Could we detect a critical threshold in meme virality, protest mobilization, or stock market panic? If so, we might be able to forecast tipping points — not just in rumors, but in collective behavior itself.

The most profound implication may be philosophical. We often assume that truth spreads because it’s true, or that lies die because they’re false. But this work suggests that structure and memory may matter more than content. A false idea with the right transmission rate on the right network, with the right forgetting dynamics, could survive forever — not because it’s convincing, but because the system allows it.

That’s not a reason for despair. It’s a call to design better systems. If forgetting enables survival, perhaps we can engineer healthy forgetting — through education, media literacy, or algorithmic design. If spatial structure determines fate, perhaps we can reshape networks to favor truth over noise.

In the end, the rumor doesn’t die because people lose interest. It dies because the system traps it. And it lives — not despite forgetting — but because of it.