Copying the local majority usually brings consensus, but straight borders sometimes freeze. When people ignore views far from their own, the population splits into a roughly predictable number of camps.
Put a population on a grid and give each person one of two opinions at random. Each day someone looks at their four neighbors and adopts whatever most of them hold. There is no other interaction.
This is the flocking essay's question asked of opinions: when does copying your neighbors bring a whole population into agreement? Patches form within a few rounds and widen as the square root of time, far more slowly if people trade seats instead of changing their minds or copy one neighbor instead of the majority. On a grid that wraps around, about two runs in three end with one colour everywhere and the rest lock into straight bands. The last section makes opinions continuous and lets people ignore views far from their own, which splits the population into a predictable number of camps.
The model is zero-temperature majority dynamics. Pick a cell at random. Count its four orthogonal neighbors. If three or four disagree with it, it switches. If exactly two disagree, it switches on a coin flip. One sweep is as many random picks as there are cells.
Updating every cell at once with a nine-cell vote over itself and its eight surrounding cells removes ties. It smooths the noise and then stops with domains a few cells across, because no cell on a surviving border is outvoted. Under the random-order rule a cell at a step in a border faces a 2–2 tie, so steps wander until they meet and annihilate, and curved borders keep retreating.
Figure 1. Majority dynamics on a 128×128 grid with wrap-around edges, from a 50/50 random start. The random-order rule usually reaches consensus after several thousand sweeps but sometimes freezes into straight bands. The synchronous vote freezes within about 20 to 40 steps (in about one run in seven a few cells are left blinking between two states). Playback stops when nothing can change.
A cell on a convex bulge is outnumbered by the patch it pokes into and flips, so bulges retreat and small islands vanish. On a straight horizontal or vertical border no cell faces more than one disagreeing neighbor, so the border cannot move: that is where the frozen bands come from.
Measure the typical domain width $L$ as the number of cells divided by the number of neighboring pairs that disagree. A border moves at a speed set by its curvature, roughly $1/L$, and a domain of width $L$ disappears once its border has travelled a distance $L$, which takes a time of order $L^2$. So $L \propto t^{1/2}$, the Allen–Cahn law, worked out for antiphase domains in ordered alloys.
Figure 2. A 200×200 grid coarsening for 2,000 sweeps, with the measured $L(t)$ on log–log axes. The dashed line has slope 1/2, pinned to the data at $t = 10$; the readout is the least-squares slope over $10 \le t \le 1000$. Over repeated runs it lands between about 0.42 and 0.55 and averages 0.48; early sweeps, when domains are a few cells wide, pull it a little under 1/2.
The rule has no length scale of its own, so a late picture should look like an early one magnified. Figure 3 tests this with the correlation $C(r)$, the average agreement between cells $r$ apart along a row or column, which falls from 1 at $r = 0$ toward 0 for cells in unrelated domains.
Figure 3. One 256×256 run of the majority rule, photographed at five times. Each patch is $6L(t)$ cells across, so early patches are blown up and late ones shrunk; they look alike. Below, $C(r)$ at the same times, coloured to match, spreads out as domains grow. Rescale divides distance by $L(t)$ and the curves fall on one.
In the default run every curve falls to one half at 0.51 or 0.52 of its own $L(t)$ while $L$ grows eightfold, from 4.7 to 37 cells. Other seeds agree to within 0.02 up to 300 sweeps; by 1,000 a domain spans a sixth of the grid and the last curve can sit up to 0.08 lower.
The square root comes from two features of the rule: people change their own minds, so a border moves wherever a cell flips, and curvature decides which way it moves. Remove either and the growth law changes.
Suppose people cannot change their minds but can trade seats with a neighbor who disagrees (Kawasaki dynamics, to a physicist). Each opinion's head count is fixed, so consensus is impossible, and a domain grows only when its opinion arrives from a shrinking domain elsewhere. Pure trades stall, so both rules in Figure 4 get some noise $T$: a change that adds $d$ disagreeing neighbor pairs happens with probability $1/(1 + e^{2d/T})$. At $T = 0$ the flip rule is the majority rule; for $T > 0$ it is the Ising model, here always below its critical temperature of 2.27 (see the lattice simulation series).
Figure 4. The same 128×128 start under flips (left, dots) and seat trades (right, squares) at the same noise. Domain size is the distance at which $C(r)$ falls to one half. The flip run stops once that passes 8 cells or one opinion holds three quarters of the grid. Dashed and dotted lines have slopes 1/2 and 1/3. At $T = 0$ the trades stop making progress within about a hundred sweeps, with domains two cells across.
At the default noise flips grow with a slope near 1/2 (0.39 to 0.52 over ten seeds), and within 14 to 21 sweeps they pass the domain size that trades reach only after 10,000. The swap slope keeps rising, from about 0.13 over the first hundred sweeps to between 0.21 and 0.24 over the last decade. Lifshitz and Slyozov's theory of ripening, in which material diffuses from small domains to large ones, gives 1/3; Huse showed that extra transport along the borders delays the approach, which is why lattice simulations had not reached it.
The other change keeps flips but drops the majority: in the voter model a random person copies one randomly chosen neighbor. A cell on a flat border switches as readily as one on a bulge, so curvature exerts no pull (there is no surface tension) and borders wander.
Figure 5. Majority rule (dots) and the voter model (squares) from the same 256×256 start. $\rho$ is the fraction of neighboring pairs that disagree, $1/(2L)$ in Section 2's terms. Top: on log–log axes majority rule is a straight line and the voter model barely bends. Bottom: plotted as $1/\rho$ against log time the voter model runs straight, beside a dashed line of slope $2/\pi$ started at its value at $t = 10$; the readout fits $t \ge 10$.
Frachebourg and Krapivsky solved the voter model exactly: at large times the fraction of disagreeing pairs falls as $\pi/(2\ln t + \ln 256)$, so $1/\rho$ gains $2/\pi \approx 0.64$ for every factor of $e$ in time. Over ten seeds the fitted gain is 0.55 to 0.69, mean 0.62, while majority rule's $\rho$ falls as $t^{-0.45}$ to $t^{-0.51}$. After 1,000 sweeps 14 to 16 percent of voter pairs still disagree, against about 1 percent under majority rule. The domains stay ragged but still coarsen, and since any cell with a disagreeing neighbor can switch, the voter model never freezes into stripes.
Back to majority rule. Spirin, Krapivsky and Redner found that on a periodic square grid a finite fraction of runs freeze into straight stripes, and Barros, Krapivsky and Redner tied that fraction to crossing probabilities in critical percolation, giving about 0.34. Figure 6 measures it, then tilts the start toward black.
Figure 6. Outcomes of random-order majority dynamics for starting black fractions from 50% to 56%, 40 runs per bar (20 on the 64×64 grid), each run to consensus or a frozen stripe state. Dark is black consensus, light is white consensus, blue is frozen stripes; a grey cap marks any run still moving at the sweep limit. At 50% about a third of runs end in stripes.
At 53% black the 32-cell grid still gives white a few wins, while the 64-cell grid almost never does. The dynamics erase detail finer than the domain width, so the winner is whichever colour owns the large connected regions, and a small excess spread over a bigger area is a more reliable majority at that scale. Stripes thin out with bias because they need both colours to span the grid.
Majority rule has two opinions, and every neighbor counts however much they disagree. The bounded-confidence model of Deffuant, Neau, Amblard and Weisbuch drops both assumptions: opinions are numbers between 0 and 1, anyone can meet anyone, and two people who meet both move to the midpoint of their opinions if those differ by less than a tolerance $\varepsilon$. Clusters further apart than $\varepsilon$ never interact again, so the population freezes into camps.
Figure 7. 300 people with opinions drawn uniformly from [0, 1]; each faint trace is one person, time runs left to right in meetings per person. Changing $\varepsilon$ restarts the run. The cluster count ignores groups under 5% of the people, such as the small clusters that often strand near the ends of the range.
A forming cluster captures everyone within about $\varepsilon$ on either side, so it owns a stretch of opinion about $2\varepsilon$ wide. That allows at most about $1/(2\varepsilon)$ major camps, the rough bound Deffuant and colleagues gave. In Hegselmann and Krause's variant everyone moves at once to the average of all opinions within $\varepsilon$ of their own, a meeting rather than a conversation. Figure 8 runs both across the range of $\varepsilon$.
Figure 8. Final camps of 300 people at 91 values of $\varepsilon$, six runs per rule at each, every run to convergence. Top: each dot is a camp at its final opinion, sized by head count; small grey dots are camps under 5% of the people. Bottom: mean number of major camps against $1/(2\varepsilon)$. Click a top panel, or focus it and use the arrow keys, to read off one $\varepsilon$.
Under pairwise meetings neighboring major camps settle 2.17$\varepsilon$ apart in the default sweep (2.12 to 2.21 over five seeds), not $2\varepsilon$, so for $\varepsilon$ up to 0.2 the count runs 11 to 13 percent under the bound. Ben-Naim, Krapivsky and Redner's rate equation for an infinitely large population gives 2.155$\varepsilon$, with tiny minor camps stranded between the majors and at the ends; here more than half the runs leave at least one. The meeting rule strands a minor camp in about one run in twenty, spaces its majors about 2.5$\varepsilon$ apart, and reaches consensus sooner: its mean count falls to 1.5 camps by $\varepsilon \approx 0.21$, against 0.26 to 0.28 for pairwise meetings (Castellano, Fortunato and Loreto's review puts the meeting rule's threshold near 0.2 for large populations). Fortunato showed that under pairwise meetings every last person joins only once $\varepsilon$ exceeds 1/2.
Consensus fails in the two models for different reasons. Majority dynamics stops only when every border is straight, a geometric accident that grows rarer when one side starts with more territory. Bounded confidence stops because clusters more than $\varepsilon$ apart never interact, so their number is set by the tolerance. In majority rule every neighboring pair of cells pulls toward agreement; the next essay keeps the lattice and makes some of those pairs pull toward disagreement.