The Stubbornness Gap: Why Some Polarized Societies Heal Slower Than Others

A Population's Stubbornness Shapes How Fast It Heals

Imagine two countries, equally polarized, with citizens holding the same强度的 convictions about the same political divides. By every outward measure, they look identical. But when researchers introduce a policy to bridge their divides—better civic education, cross-party dialogue programs—one country begins to unify within weeks while the other remains fractured for years.

This is not a hypothetical. According to new mathematical research, such dramatic differences in depolarization speed can occur even when the underlying conditions for depolarization—the threshold at which consensus becomes possible—are exactly the same. The difference lies in something invisible at first glance: the distribution of how strongly people hold their beliefs. Some populations are filled with people who feel moderately about everything. Others contain a mix of the deeply entrenched and the casually opinionated. Both can tip into consensus at the same moment. But once they do, they heal at wildly different rates.

The finding emerges from a study by Corbit Sampson and Juan Restrepo of the University of Colorado Boulder, published on arXiv in June 2026. The researchers applied a powerful analytical technique—borrowed from physics—to understand not just whether societies polarize or depolarize, but how quickly these transitions happen. Their work transforms what had been a model of static equilibria into a dynamic theory capable of predicting timescales. And those timescales, they show, depend critically on something that previous models missed entirely.

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The Science: Building a Compass forOpinion

To understand how societies navigate between division and unity, researchers have long turned to mathematical models that capture the essential features of opinion dynamics. The most influential of these is the Kuramoto model, originally developed to describe how coupled oscillators—from fireflies blinking in sync to neurons firing in rhythm—spontaneously align their behavior. In its simplest form, the Kuramoto model treats each individual as a spinning oscillator, with social influence acting like springs that pull neighboring oscillators toward alignment.

The social compass model, introduced by researchers Ojer and colleagues in 2023, adapts this framework to a more nuanced reality: people don't just hold single opinions, they hold multiple, potentially correlated opinions on different topics. Consider abortion rights and gun control. If you're pro-choice, you're statistically more likely to support expanded gun access restrictions—a correlation built into how our ideological identities form. These correlations matter. They shape whether a society fragments into opposed camps or remains fluid enough to find consensus.

In the social compass model, each person is represented not by a single number but by a point in a two-dimensional opinion space. The horizontal axis might represent your stance on topic X, the vertical axis your stance on topic Y. A person strongly in favor of both might sit at coordinates (0.9, 0.8). A person strongly opposed to both might sit at (-0.8, -0.7). This pair of opinions can be rewritten as a complex number—a single mathematical object that encodes both the strength of your convictions and their direction.

The strength of conviction acts like an internal pressure pushing you to maintain your current orientation. The social coupling term, lifted directly from the Kuramoto model, represents your tendency to align with others around you. When coupling is weak, the internal pressure dominates and people drift back toward their original opinions, preserving whatever division existed initially. When coupling is strong enough, social influence overwhelms personal conviction and the population can slide toward consensus.

The crucial insight in earlier work was identifying the critical coupling strength—the specific level of social interaction needed to tip a polarized society into consensus. This threshold depends on how spread out the initial opinions are and on the average conviction holding the population in place.

But this analysis only told half the story. It said nothing about speed. If two societies both cross the critical threshold at the same moment, which one actually depolarizes faster? Previous approaches couldn't answer this question because they focused on equilibria—the end states toward which systems tend—rather than the dynamical paths leading there.

Sampson and Restrepo's contribution is to import a technique called the Ott-Antonsen Ansatz, developed for studying synchronization in coupled oscillators, into the social compass framework. The ansatz is a mathematical conjecture about the structure of solutions that, when it works, dramatically simplifies the analysis. Instead of tracking millions of individual opinions evolving simultaneously, it allows researchers to describe the entire population's behavior using just a handful of variables.

The researchers applied this approach to the social compass model and showed that for populations initially clustered into distinct opinion groups—which is a reasonable model of a polarized society—the macroscopic dynamics reduce to a finite system of ordinary differential equations. This is a substantial simplification. A system governing the behavior of millions of individuals becomes something you could, in principle, solve on a laptop.

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What They Found

The first major result from this analysis is a dispersion relation—a mathematical formula that describes how perturbations to a polarized state grow or decay over time. When a society is in a polarized state, it can be slightly disturbed. A few individuals might randomly shift their views, or a small event might nudge some people away from their faction's positions. The dispersion relation tells us whether these perturbations will die out, keeping the society polarized, or grow, pushing the system toward consensus.

For the fastest-growing perturbation mode, the dispersion relation takes the form:

$$1=\frac{K}{2}\left(1+|E_{\varphi }[e^{-2i\varphi }]|\right){\int }_0^{\infty }\frac{P(\rho )d\rho }{\rho +\lambda }$$

where $K$ is the coupling strength, $E_{\varphi }$ denotes an average over the initial opinion distribution, and $P(\rho )$ is the distribution of convictions in the population. The variable $\lambda$ represents the growth rate of perturbations—positive values mean the perturbation grows, driving depolarization; negative values mean it decays, preserving polarization.

Setting $\lambda =0$ gives the critical coupling strength, the threshold at which depolarization first becomes possible:

$$K_c=\frac{2}{E_{\rho }[{\rho }^{-1}](1+|E_{\varphi }[e^{-2i\varphi }]|)}$$

This formula reveals something striking: the critical coupling depends only on the first inverse moment of the conviction distribution, $E_{\rho }[{\rho }^{-1}]$—essentially, the average of one divided by each person's conviction. Higher moments of the conviction distribution don't appear here at all.

But when the researchers examined the growth rate of perturbations near the critical point, a different picture emerged. Expanding their dispersion relation for small $\lambda$, they found:

$$\lambda \approx \frac{\delta K}{K_c}\frac{E_{\rho }[{\rho }^{-1}]}{E_{\rho }[{\rho }^{-2}]}$$

Now the second inverse moment, $E_{\rho }[{\rho }^{-2}]$, determines how quickly consensus emerges. Two populations can have identical values of $E_{\rho }[{\rho }^{-1}]$ and thus identical critical couplings—but dramatically different values of $E_{\rho }[{\rho }^{-2}]$, and thus dramatically different depolarization speeds.

To make this concrete, Sampson and Restrepo introduced a family of conviction distributions parameterized by a heterogeneity index $s\ge 1$. The distribution places a fraction $\frac{s}{1+s}$ of the population at conviction $s$ and the remaining fraction at conviction $1/s$. When $s=1$, everyone has conviction 1 and the distribution is perfectly homogeneous. As $s$ grows, the population becomes increasingly divided between the deeply committed and the casually held beliefs.

For this family of distributions, $E_{\rho }[{\rho }^{-1}]=1$ regardless of $s$—the critical coupling is identical. But:

$$E_{\rho }[{\rho }^{-2}]=s^2-s+\frac{1}{s}$$

And the growth rate near the critical point becomes:

$$\lambda \approx \frac{\Delta }{s^2-s+1/s}$$

where $\Delta$ is the fractional distance above the critical coupling.

Depolarization Rate Drops with Conviction Heterogeneity

Label	Value
s = 1 (homogeneous)	1
s = 2	0.167
s = 5	0.041
s = 11	0.011
s = 100	0.0001

Growth rate λ as fraction of homogeneous case (Δ = 0.01)

Figure 1: Heterogeneity in conviction distributions dramatically affects depolarization timescales. As the heterogeneity parameter $s$ increases, the denominator $s^2-s+1/s$ grows, causing the growth rate $\lambda$ to plummet. A homogeneous population ($s=1$) depolarizes most rapidly. A population with $s=100$—roughly split between highly stubborn agents and barely-committed agents—depolarizes orders of magnitude more slowly, despite having the identical threshold for when depolarization becomes possible.

The researchers verified this prediction through numerical simulations. Starting from a quadrimodal distribution of initial opinions (representing a society split into four distinct camps, with ${\varphi }_0=\pi /7$), they tracked how the order parameter—a measure of consensus ranging from 0 for complete fragmentation to 1 for perfect alignment—evolved over time.

Time to Consensus Increases with Conviction Heterogeneity

Label	Value
s=1	2.5
s=2	6
s=11	25
s=100	100

Time (arbitrary units) to reach r = 0.5 for different s values

Figure 2: Numerical simulations confirm the theoretical predictions. The logarithm of the order parameter $r$ versus time, obtained from agent-based simulations with varying values of $s$, shows the characteristic exponential growth predicted by the linear stability analysis. The slopes of these curves in the exponential growth phase match the theoretical prediction $\lambda \approx \Delta /(s^2-s+1/s)$ with remarkable precision.

For $s=1$ (homogeneous convictions), the system approaches consensus rapidly. For $s=100$ (extreme heterogeneity), the same fractional excess above the critical coupling yields a growth rate more than 10,000 times smaller. A society that might unify within weeks under one distribution of conviction strengths could require years or decades under another—identical in every other respect.

The researchers also examined how the model behaves with different shapes of conviction distributions. They tested Dirac delta distributions (everyone identical), inverse power-law distributions (many weakly convicted individuals, few strongly convicted ones), and truncated power-law distributions (many strongly convicted, few weakly convicted). All reached the same critical coupling when sharing the same first inverse moment, but showed vastly different depolarization timescales depending on higher moments.

Steady-State Consensus at Equal Coupling Strength

Label	Value
Dirac (homogeneous)	0.95
Inverse power-law	0.87
Power-law	0.92

Consensus level r at K = 1.5 Kc for three distribution types

Figure 3: Forward continuation of consensus versus coupling strength across distribution types. The degree of consensus $r$ is plotted against coupling strength $K$ for bimodal opinion distributions and three different conviction distributions: Dirac delta (homogeneous), inverse power-law ($P(\rho )\sim {\rho }^{-\gamma }$), and power-law ($P(\rho )\sim {\rho }^{\gamma }$). The vertical dashed line marks the critical coupling $K_c$ predicted by the theory. While all distributions transition at the same point, the shapes of the transition curves differ substantially—reflecting different depolarization dynamics.

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Why This Changes Things

The standard approach to understanding political polarization focuses on structural remedies: how to break echo chambers, how to foster cross-party contact, how to design institutions that encourage compromise. These interventions implicitly assume that if you reduce the coupling strength between opposing groups—making social interactions less combative, say—depolarization will follow.

The social compass analysis suggests this framing is incomplete. The critical coupling tells you whether depolarization is possible. It says nothing about how long it will take once triggered. And the timescales involved may be the difference between a functional democracy and one locked in perpetual gridlock.

Consider the practical implications. If you're designing a national civic education program, you might calculate that it's sufficient to bring coupling just above the critical threshold—enough to tip the system toward consensus. But if your population has a heterogeneous conviction distribution, that tipping point may be followed by decades of glacial progress toward actual unity. You might achieve a technically unstable equilibrium that looks, for all practical purposes, like permanent polarization.

Conversely, interventions that reduce conviction heterogeneity—not by changing anyone's specific beliefs, but by shifting the overall distribution of how intensely people hold their views—could accelerate depolarization without changing the fundamental dynamics. A population where everyone holds moderately strong opinions may unify faster than one with a mix of zealots and flakes, even if the zealots in the latter population are fighting for the same positions as the moderates in the former.

This connects to a broader insight in the physics of collective behavior: the structure of heterogeneity often matters more than the structure of mean behavior. Ecological systems can tip between states based on the distribution of species traits rather than average properties. Economic markets exhibit phase transitions sensitive to the tails of wealth distributions. And now, it appears, social consensus depends crucially on the fine structure of conviction distributions, not just their first moments.

The extension to community structure adds another layer of realism. Real societies aren't well-mixed populations where everyone interacts with everyone equally. Instead, people cluster into communities—neighborhoods, social media networks, professional associations—that serve as primary venues for opinion formation. Sampson and Restrepo generalized their framework to incorporate two communities with different internal coupling strengths and different initial opinion distributions.

Figure 4: Community structure affects depolarization dynamics. Polar histograms show the distribution of initial opinion orientations for two communities. In the correlated case (top), both communities share the same bimodal structure; in the uncorrelated case (bottom), community 1 (blue) and community 2 (tan) have different opinion profiles. The reduced equations capture these dynamics through order parameters $z_1$ and $z_2$ for each community.

The resulting bifurcation diagrams—maps showing how the system's steady states change as coupling strength varies—reveal rich phase behavior. For two communities with correlated initial opinions, the system transitions from two symmetric polarized states to consensus at a single critical coupling, much like the single-population case. But for communities with uncorrelated opinion structures, the path to consensus may involve intermediate states where each community reaches internal consensus but disagrees with the other, before global consensus finally emerges at higher coupling.

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What's Next

The mathematical machinery Sampson and Restrepo develop is powerful, but several caveats deserve emphasis. The Ott-Antonsen Ansatz works under specific conditions that may not perfectly match real social systems. It assumes the thermodynamic limit—infinite population size—and requires certain smoothness assumptions about the distributions involved. Real elections, protests, and cultural shifts involve finite populations, discrete events, and distributions that change shape over time in ways the continuous analysis may not capture.

The model also treats all social interactions symmetrically. In reality, individuals with large social networks wield disproportionate influence; algorithms curate information flows in ways that break symmetry; institutions (courts, media, political parties) channel and amplify some voices while suppressing others. Extensions to network structures were studied in earlier work by Ojer and colleagues, but incorporating realistic social topology into the dynamical theory remains an open challenge.

Most fundamentally, the model describes opinion dynamics in a vacuum. It doesn't account for the exogenous shocks that real societies face: economic crises, scandals, wars, technological disruptions. These events continuously reset the opinion landscape, potentially pushing systems back toward polarization even after substantial progress toward consensus. Whether the timescale predictions derived here survive contact with such perturbations is an important question for future work.

On the empirical side, the theory makes predictions that could, in principle, be tested. If you could measure conviction distributions in different populations—through surveys probing how strongly people hold their political views—you should observe different depolarization rates following similar interventions. Nations with similar baseline polarization but different distributions of conviction intensity should converge toward consensus at different speeds once bridging activities begin.

Such data would be challenging to collect. Conviction is harder to measure than stated opinion; people may not accurately report how deeply they hold their views, and those views may shift over the course of a survey. But advances in implicit association testing, behavioral proxies for commitment (like how much time people spend consuming political media), and longitudinal panel studies might eventually provide the necessary data.

The code for the project is publicly available on GitHub, and the reduced equations derived here could be extended to study other aspects of opinion dynamics. The approach naturally generalizes to more than two opinion dimensions, to time-varying coupling strengths, and to situations where individual convictions evolve alongside opinions. Each extension would bring the model closer to capturing the full complexity of real political systems.

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The Deeper Significance

What makes this research valuable isn't just its immediate predictions but the conceptual framework it establishes. By connecting the social compass model to the rich toolkit developed for studying synchronization in physical and biological systems, Sampson and Restrepo open a line of analysis that had been closed.

Opinion dynamics has long borrowed metaphors from physics—引用 the language of phase transitions, critical points, and order parameters. But often this borrowing was superficial, importing vocabulary without substance. The Ott-Antonsen Ansatz represents something different: a genuine mathematical correspondence that allows techniques developed for one domain to be applied in another.

The practical upshot is a clearer understanding of why some polarized societies heal quickly while others seem permanently fractured, despite apparently similar conditions. The answer lies not in the average strength of conviction but in its distribution—specifically in the higher moments that previous analyses couldn't reach. Two populations can share the same depolarization threshold but exist on vastly different timescales, like chemicals that react at the same temperature but on scales ranging from milliseconds to millennia.

This insight doesn't prescribe specific policies, but it reframes how we think about political healing. Interventions that focus only on moving coupling above threshold may achieve technical success—creating a state where consensus is stable—while failing to deliver actual consensus on any practical timescale. Truly effective depolarization strategies might need to address conviction heterogeneity directly, perhaps by building institutions that reduce the reward for extreme conviction, or by creating social environments where deeply held views can be held with less intensity.

The mathematics, as always, illuminates possibilities rather than mandates. But understanding what's possible is the first step toward making it actual.

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The original paper by Corbit R. Sampson and Juan G. Restrepo, "Low-dimensional Dynamics of the Social Compass Model," was published on arXiv on June 24, 2026.