😊 Happiness & Similar Neighbours

The effect is driven by feedback: when an unhappy agent relocates, it typically increases the same-type density around its new location while decreasing it around its old one. This makes the new neighbourhood more attractive to other same-type agents and the old neighbourhood less attractive, so subsequent moves reinforce the emerging boundary. Because thousands of small relocation events compound over generations, even a modest per-agent tolerance for diversity (say, needing only 3 of 8 neighbours to match) produces regions that are far more segregated than any individual agent's stated preference would suggest.

Both simulations share the same underlying mechanism — agents check their neighbourhood and relocate if unsatisfied — but they differ in framing and parameterisation. The Schelling model on this site is typically presented with a "tolerance percentage" and language emphasizing racial or group segregation dynamics, echoing the historical urban-sociology context in which Schelling's original model was discussed. This simulation instead uses an abstract "Type A / Type B" vocabulary and an explicit integer threshold (k out of 8 neighbours), deliberately avoiding group-identity framing so the focus stays on the mathematics of local similarity preference and its geographic implications, in the spirit of Tobler's First Law rather than segregation studies specifically.

Geographer Waldo Tobler proposed in 1970 that "everything is related to everything else, but near things are more related than distant things." This simulation is a direct, mechanistic illustration of that principle: an agent's happiness depends only on its immediate spatial neighbourhood, not on the whole grid. Because relocation decisions are made purely from local information, the resulting global pattern — large same-type regions separated by sharp boundaries — emerges without any agent, or any central planner, ever considering the grid as a whole.

Generally yes, but the relationship is not perfectly linear. At very low k (0 or 1), almost every layout is already happy and the grid stays close to its random initial state. As k rises toward 4 (half of 8 neighbours), clustering increases sharply because agents become far more selective. Beyond that, very high thresholds (7 or 8) can make convergence difficult or unstable, since agents may struggle to find any empty cell where they satisfy such a strict neighbour requirement, leading to persistent relocation cycles rather than a stable equilibrium.

Beyond its historical use in urban residential-sorting studies, the same local-threshold relocation mechanism has been applied to explain clustering in online social networks (users migrating toward communities with similar views), retail location patterns (similar businesses clustering into districts), species habitat selection in ecology, and opinion polarization in social simulations. The general lesson — that mild local preferences can produce strong global separation without anyone intending it — is a recurring theme across many complex systems, which is why the model remains widely taught outside its original urban-sociology context.

At high density and a high happiness threshold, the supply of empty cells that satisfy an agent's requirement can become scarce, especially near cluster boundaries. An agent might move into a cell that looks satisfactory at the moment it relocates, but as other agents around it also move in the same generation, its new neighbourhood composition changes and it can become unhappy again on a later tick. This can produce long transient periods of continued churn before the grid — if it converges at all — settles into a stable pattern with zero moves per tick.

The info bar's "Converged" indicator reads "yes" whenever a completed generation results in zero relocations — meaning every occupied agent already meets its happiness threshold given the current grid state. Because the grid is finite and moves are driven purely by local rules, most parameter combinations do eventually reach this zero-move state, though the number of generations required grows with higher thresholds, higher density, and more balanced Type A / Type B mixes.