Flow-Based Cartogram Generator

• Flow-based cartograms are density-equalizing map projections that deform regions so their areas represent target quantitative attributes using diffusion equations.
• They leverage numerical methods like FFT-accelerated diffusion and adaptive densification to preserve topology and achieve area errors below 1%.
• Practical implementations such as 5FCarto offer sub-second performance with high fidelity in shape, area accuracy, and topological integrity.

A flow-based cartogram generator is a computational technique for creating density-equalizing map projections in which each geographic region is deformed so that its area is proportional to a target quantitative statistic. These methods, introduced in the cartography literature as “diffusion-based” or “flow-based” cartograms, preserve adjacency (contiguity) and recognizable geography while striving for minimal error in achieved area and minimum topological artifacts. The central mathematical principle is to treat the quantitative attribute (e.g., population) as a continuous density field, and compute a smooth bijective transformation that redistributes this density to a constant value over the map domain.

1. Mathematical and Algorithmic Foundations

Flow-based cartogram generation interprets the variable of interest (e.g., population) as a spatially varying density, $\rho(x, y)$, and seeks a transformation that produces a uniform target density while conserving the total quantity in each region (Nusrat et al., 2016, Gastner et al., 2018, Miaji et al., 11 Nov 2025). The initial density is discretized on a regular grid with values proportional to regional totals. The transformation is governed by the solution to the diffusion partial differential equation (PDE):

$\frac{\partial \rho}{\partial t} = \nabla^2 \rho$

with no-flux (Neumann) boundary conditions on the map boundary. The process evolves $\rho(x, y, t)$ from its initial state until it is spatially uniform, denoted as $\rho^*$. Each region’s final area is then exactly proportional to its prescribed target mass relative to $\rho^*$. By rescaling the entire map so that $\rho^* = 1$, every region’s area equals the original prescribed value.

The coordinate transformation is tracked via a velocity field

$$\vec{v}(x, y, t) = -\frac{\nabla \rho(x, y, t)}{\rho(x, y, t)}$$

The new position of each point is given by

$$x'(x, y) = x + \int_0^\infty \vec{v}(x(t), y(t), t) \, dt$$

Numerically, points are advected along this field as the map diffuses toward the target density. This velocity-based transport guarantees area rescaling is performed continuously, orientation is preserved, and no mass escapes the domain (Nusrat et al., 2016, Miaji et al., 11 Nov 2025, Gastner et al., 2018).

2. Numerical Implementation and Acceleration

The diffusion process is typically discretized on a uniform $N \times M$ grid. The most robust time-stepping schemes are implicit in time (e.g., Crank–Nicolson), with updates given by solving the linear system

$(I - \Delta t \nabla^2) \rho^{n+1} = \rho^n$

at each timestep. The grid must be fine enough to resolve narrow geographic features. For practical speed, fast Fourier transform (FFT)-based solvers are used for the Poisson problem, yielding per-timestep complexity $O(NM \log (NM))$ (Nusrat et al., 2016).

In the “fast flow-based algorithm” (Gastner et al., 2018), the key acceleration is to expand $\rho(x, y, t)$ and the associated fluxes in a spectral (cosine/sine) basis. Linear “equalization” enables a closed-form, time-independent solution for the mass flux field

$\vec{J}(x, y) = \rho(x, y, t) \vec{v}(x, y, t)$

over $t \in [0,1]$, so the entire advection can be computed in a single integration cycle. These spectral sums are evaluated by discrete cosine transforms (DCTs), leveraging highly parallel FFT libraries. Trajectories of each vertex or raster cell are integrated in parallel, typically with higher-order predictor-corrector methods, until all masses are within the target tolerance (maximum relative area error $<1\%$ is routinely achieved in 5–12 iterations).

Grid sizes (e.g., $512 \times 512$) are chosen according to the spatial complexity of the input; FFT-based methods yield sub-second run times for typical maps (Gastner et al., 2018).

3. Topology-Preserving Line Densification

While the flow-based mapping, $\Pi$, is mathematically a globally bijective and orientation-preserving projection, projecting the edges of a coarse polygonal map can nonetheless introduce invalid topologies such as gaps or overlaps (polyline crossings) post-projection. To guarantee strict topology preservation, the 5FCarto framework (Miaji et al., 11 Nov 2025) performs adaptive line densification:

1. Quadtree Adaptive Subdivision: The input domain is adaptively rasterized so that local density variations are captured at fine scale, with the leaf nodes subdivided until a parameterized threshold $L_{\text{target}}$ is reached.
2. Delaunay Triangulation: The corners of quadtree leaves are triangulated in the original and projected domains, yielding a mesh $T$ that enables local affine mapping of boundary segments.
3. Edge Subdivision: Each polygon boundary edge is divided at its intersections with the triangulation edges, ensuring every segment lies entirely within a local affine patch, preventing inversion and crossing.
4. Simplification: Redundant or nearly colinear vertices are pruned to maintain manageable complexity, while the use of a mesh with minimum angle bounds guarantees geometric regularity and numerical stability.

The mathematical guarantee is as follows (Miaji et al., 11 Nov 2025): since $\Pi$ is smooth with positive Jacobian, and the mesh triangles are mapped affinely with shared edge coherence, no segment self-intersects (P1), no overlaps occur between polygons (P2), and junction winding order is preserved (P3).

4. Empirical Accuracy, Fidelity, and Performance

Quantitative evaluations on diverse, real-world maps that span density ratios $10^1$ to $8 \times 10^5$ show that flow-based cartograms with topology-preserving densification (e.g., 5FCarto) achieve:

• Area accuracy: Maximum relative error $E_{\text{max}} < 10^{-7}$ on all maps in (Miaji et al., 11 Nov 2025); diffusion-based methods without densification or force-based variants (F4Carto) show substantially higher errors, especially under high density disparities.
• Topology: No self- or overlap-intersections on all tested maps with densification; hundreds of crossings without densification under high disparity.
• Shape fidelity: Polygon-level metrics (Fréchet, Hausdorff, symmetric-difference distances) are nearly identical for density-equalizing projections with or without densification, and better than force-based alternatives.
• Speed: 5FCarto produces correct output in under a second on challenging maps; FFT-based acceleration offsets the cost of densification, yielding overall faster performance compared to force-based and standard diffusion methods.

5. Variants: Circular-Arc and Network Flow Cartograms

Flow-based principles extend to circular-arc cartograms (Kämper et al., 2011), which model country boundaries as polygons whose edges are bent to circular arcs. Here, a dual-graph max-flow problem encodes the redistribution of area: each region is a node, and the edges have capacities based on maximal safe area transfer (computed via the straight skeleton geometry). The capacity for each edge ensures that bending does not create invalid overlaps, and the max-flow solution provides the arc adjustments. The actual geometric construction then involves numerical inversion of the circular segment area formula to determine arc radii. The heuristic offers area redistribution up to geometric feasibility limits alongside maintainable computational complexity: $O(N \log^2 N)$ for skeletons and $O(n \log^3 n)$ for max-flow (Kämper et al., 2011).

6. Design Choices, Limitations, and Recommendations

Major design parameters include grid resolution (to capture geometric detail), densification threshold (e.g., $L_{\text{target}} \approx n^2/256$ in (Miaji et al., 11 Nov 2025)), and the area error tolerance for iterative refinement. Practical guidelines:

• Fast Fourier acceleration is critical for performance, especially on large grids.
• Quadtree adaptivity in densification balances accuracy vs. computational effort.
• Topology preservation requires automated subdivision and projection of boundary points.
• For spherical geometries, extension of the diffusion solver to spherical harmonics is recommended.
• Residual floating-point errors or domains with extreme curvature may necessitate extended precision or post-processing.

5FCarto is available under GPL v3, integrates directly with GIS/web-mapping pipelines, and preserves topology and geometric accuracy without the need for post-hoc repairs (Miaji et al., 11 Nov 2025).

7. Context and Significance

Flow-based cartogram generators form the leading class of algorithms for producing contiguous, area-accurate, and topologically correct cartograms. Their efficiency and robustness stem from the coupling of PDE-based area scaling with geometric mesh refinement for guaranteed topology. These methods are widely adopted in geovisualization, demographic mapping, and spatial data normalization. Ongoing research addresses further acceleration, improved handling of extreme density contrasts, and extensions to alternative geometries and output formats (Nusrat et al., 2016, Miaji et al., 11 Nov 2025, Gastner et al., 2018, Kämper et al., 2011).