How Cities Choose Their Centers — And Why Small Changes Can Create Entirely Different Urban Futures

On a simulated city grid, two computer runs start with identical geography, transport networks, and economic rules. The only difference: one begins with slightly more activity near a hilltop. By the end, they’ve settled into completely different urban worlds — one with three bustling hubs, another with four. Not because of external shocks or policy shifts, but because the system itself allows multiple stable futures.

This isn’t speculation. It’s what Alexander Hellervik demonstrates in his 2026 paper on preferential centrality with a density penalty, where he shows that at 172 out of 420 tested combinations of feedback strength and transport cost, the same underlying landscape produces distinct equilibria depending only on initial conditions (Hellervik, 2026). In 75 of those cases, the number of urban centers differs. A city could evolve into a polycentric network or consolidate around fewer dominant nodes — not due to geography alone, but due to path dependence baked into how people and activity reinforce each other.

That finding flips a core assumption in urban modeling: that given enough time and data, cities converge to one predictable outcome. Instead, Hellervik’s work suggests we live in a world of contingent urbanism — where history, chance, and starting points matter as much as infrastructure and economics.

The Science

Preferential centrality builds on classic spatial interaction models like the gravity model, where flows between places depend on their size and distance (Wilson, 1970; Harris & Wilson, 1978). But unlike traditional models where attraction is fixed by land use or population, preferential centrality makes attraction endogenous: locations become more attractive simply because they already have activity. This creates positive feedback — a “rich-get-richer” dynamic where early advantages compound.

The original model, introduced by Hellervik et al. (2019), formalizes this loop:

• Each zone $j$ has a capacity $R_j$ — think buildable area or road segment length.
• Activity $a_j$ generates interaction flows according to a production-constrained gravity model:$$S_{ij}=a_i\frac{W_j{K}_{ij}}{{\sum }_k{W}_k{K}_{ik}},K_{ij}=m_{ij}(c_{ij}+d_0{)}^{-\beta }$$where $S_{ij}$ is flow from $i$ to $j$, $c_{ij}$ is generalized travel cost, $d_0$ is access cost, and $m_{ij}$ masks self-interactions and disconnected pairs.
• Attraction $W_j$ combines intrinsic capacity and existing activity:$$W_j=\gamma a_j+R_j$$with $\gamma \ge 0$ controlling the strength of feedback.
• Equilibrium occurs when inflow equals activity:$$a_j=\sum _i{S}_{ij}$$and total activity is conserved at ${\sum }_j{R}_j$.

At $\gamma =0$, this reduces to eigenvector-like centrality; at $\gamma >0$, it captures self-reinforcing agglomeration.

Hellervik’s new contribution is adding a density penalty — a corrective force that dampens concentration. The intuition is simple: as activity crowds into a place, something pushes back — congestion, rising rents, zoning limits, or social friction. Rather than modeling these mechanisms explicitly, the penalty responds to local density ${\tilde{a}}_j=a_j/R_j$ through a function $g({\tilde{a}}_j)$, subtracting influence where density exceeds the average.

Crucially, the correction preserves total activity and respects subdivision consistency: splitting a zone into two identical halves shouldn’t change aggregate dynamics. To satisfy both constraints, the adjustment takes the form:

$$C_j(a)=I_j(a)-\kappa R_j\left(g({\tilde{a}}_j)-{\bar{g}}_R\right)$$

where $I_j(a)$ is inflow, $\kappa \ge 0$ is penalty strength, and ${\bar{g}}_R$ is the capacity-weighted mean of $g({\tilde{a}}_j)$. This ensures adjustments sum to zero and scale correctly under subdivision.

The response function used throughout is $g(\tilde{a})={\tilde{a}}^2$, making the marginal penalty grow faster than linear — mimicking accelerating costs in dense environments.

To test and share these ideas, Hellervik released prefcent, an open-source Python implementation built on NumPy and SciPy. It includes convergence checks, feasibility monitoring, and full reproducibility logging — down to bit-level repeatability when possible.

What They Found

The computational experiments reveal several key insights:

First, the density penalty works as intended: it reduces extreme concentration. In a strong-feedback scenario on a ring-shaped network with correlated capacity (Figure 1), the unpenalized model concentrates 98% of activity within 10 km of the peak after thousands of iterations. With $\kappa =0.0128$, the same system reaches equilibrium in fewer than 1,700 steps — and spreads activity across four detectable peaks instead of collapsing into one.

Second, penalty strength reshapes urban structure. On the same ring at lower feedback ($\gamma =0.5$), increasing $\kappa$ from 0 to 0.0255 reduces peak density from 20 to 2.2 and increases the number of detected centers from one to four (

). The largest penalty cuts maximum density by nearly 90%, redistributing activity while preserving alignment with high-capacity zones.

Third, transport costs shape center count — but not monotonically. When line-haul costs are scaled by factors $s=10,3,1,0.3$ (holding access cost constant), both ring and torus landscapes show fewer detected centers at lower $s$ — i.e., better transportation. On the torus, center count drops from three to zero; on the ring, from six to one (

). Yet the tallest peak emerges not at fastest transport, but at intermediate speeds ($s=1$). This suggests there’s a “sweet spot” for polycentricity — too slow, and movement is too costly; too fast, and everything collapses toward dominance.

But the most profound result comes from multi-start simulations across parameter grids. Using a $64\times 64$ toroidal grid with lognormal capacity fields, Hellervik ran the model from 16 different initial states at each of 420 points in $(\gamma ,s)$-space. At 172 of these points, multiple distinct equilibria emerged. In 75, the number of urban centers differed.

One striking example: at $\gamma =3$, $s=3$, $\sigma =0.5$, twelve of sixteen starts led to a two-center configuration, while four produced a three-center state (

). Both are stable — perturbing either returns it to itself. There is no single “true” equilibrium. There are possible ones.

And the software proves its rigor: when reproducing outputs from a prior study on Porto Alegre’s 65,357-segment road network, prefcent matched results with median relative differences of order $10^{-14}$ — essentially machine precision.

Number of Distinct Converged States by Feedback Strength

Label	Value
γ = 0.5	1
γ = 1.0	2
γ = 1.5	3
γ = 2.0	5
γ = 2.5	8
γ = 3.0	12
γ = 3.5	14
γ = 4.0	14

Data derived from multi-start sweeps in Figure 6 and Appendix A of Hellervik (2026).

Why This Changes Things

Urban economists have long debated whether cities are shaped primarily by fundamentals — geography, transport, resources — or by historical accidents and tipping points. This model offers a synthesis: fundamentals set the stage, but feedback and nonlinear responses allow multiple plausible endings.

Consider the implications for real-world planning. If a city’s final configuration depends sensitively on early decisions — say, where to place the first transit line or industrial zone — then small interventions today could lock in large structural differences tomorrow. Conversely, attempts to reverse entrenched patterns may fail not because they’re poorly designed, but because the system has settled into a deep basin of attraction.

The density penalty also reframes how we think about congestion. Traditionally, congestion is seen as a negative externality to be priced or engineered away. Here, it plays a constructive role: by penalizing overconcentration, it enables a more distributed urban form. Without it, the model tends toward winner-take-all outcomes. With it, diversity of centers becomes sustainable.

This aligns with empirical observations. Many modern metropolitan areas — Los Angeles, Paris, Tokyo — have evolved into polycentric systems despite strong agglomeration forces. Standard models struggle to explain this without invoking exogenous zoning or policy. Here, endogenous density penalties offer a parsimonious mechanism: even without top-down control, natural frictions can prevent monopolization of urban space.

Moreover, the existence of multiple equilibria challenges the validity of point predictions in urban forecasting. Most models assume uniqueness — that given inputs, there is one answer. But if reality allows several stable configurations, then prediction must give way to possibility mapping. Planners shouldn’t ask “Where will development go?” but rather “Which of several potential futures are accessible from here?”

This shift mirrors advances in climate science, where ensembles of simulations reveal ranges of plausible outcomes rather than single trajectories. Similarly, urban modeling may need to embrace multi-equilibrium thinking — running many starts, varying initial seeds, and cataloging attractors.

Detected Centers vs. Transport Cost Scaling

Label	Value
s = 10	6
s = 3	6
s = 1	4
s = 0.3	1

From Figure 4 in Hellervik (2026); γ = 0.5, κ = 0.0128.

What's Next

Several open questions remain. First, why do multiple equilibria occur? The paper documents their existence but leaves analysis to future work. Are they due to bifurcations in the dynamical system? Hysteresis? Basin fragmentation?

Second, can we characterize the basins of attraction — the set of initial states leading to each equilibrium? If so, planners might assess resilience: how easily can a city shift from monocentric to polycentric forms? Or intervene to expand the basin of a desired outcome?

Third, how sensitive are results to the choice of $g(\tilde{a})$? The quadratic form ${\tilde{a}}^2$ was chosen for simplicity, but real density costs may follow different scaling laws — logarithmic, sigmoidal, or threshold-based. Testing alternative functions could reveal which features are robust.

Fourth, the current model assumes static capacities and fixed travel costs. A next step is coupling it with dynamic elements: letting congestion affect travel times, or allowing capacity to expand in response to demand. Such extensions would bring the model closer to integrated land-use transport frameworks — but with richer behavioral foundations.

Finally, empirical validation is needed. While the Porto Alegre reproduction confirms computational fidelity, does the model actually reflect real urban evolution? Applying it to historical datasets — say, U.S. county development from 1850–1950 — could test whether observed paths align with predicted basins.

For now, the biggest contribution may be philosophical. By showing that identical landscapes can yield divergent cities, Hellervik reminds us that urban form is not inevitable. It is contingent. And that opens space for agency: if multiple futures are possible, then choices matter.

As computing power grows and open tools like prefcent lower barriers to entry, we may see a new generation of participatory urban modeling — where communities explore not just what will happen, but what could happen, and decide which version they want to build toward.