NCP: Neighborhood-Preserving Non-Uniform Circle Packing for Visualization

Circle packing is widely used in visualization due to its aesthetic appeal and simplicity, particularly in tasks where the spatial arrangement and relationships between data are of interest, such as understanding proximity relationships (e.g., images with categories) or analyzing quantitative data (e.g., housing prices). Many applications require preserving neighborhood relationships while encoding a quantitative attribute using radii for data analysis. To meet these two requirements simultaneously, we present a neighborhood-preserving non-uniform circle packing method, NCP. This method preserves neighborhood relationships between the data represented by non-uniform circles to comprehensively analyze similar data and an attribute of interest. We formulate neighborhood-preserving non-uniform circle packing as a planar graph embedding problem based on the circle packing theorem. This formulation leads to a non-convex optimization problem, which can be solved by the continuation method. We conduct a quantitative evaluation and present two use cases to demonstrate that our NCP method can effectively generate non-uniform circle packing results.

💡 Research Summary

The paper introduces NCP, a novel method for generating non‑uniform circle packings that simultaneously preserve neighborhood relationships among data items and encode a quantitative attribute in the circles’ radii. The authors observe that many visualization tasks require both spatial proximity (to reveal structural or similarity information) and a size‑based encoding of a numeric variable, yet existing circle‑packing approaches either assume uniform radii or focus solely on compactness, ignoring neighborhood preservation.

To address this gap, the authors reformulate the problem as a planar graph embedding task based on the classic circle‑packing theorem, which guarantees a one‑to‑one correspondence between a set of non‑uniform circles and a maximal planar graph. In this graph, vertices represent circles and edges encode desired neighbor relations. By embedding the graph in the plane, the positions of the circles are obtained, while the radii are directly tied to the quantitative attribute through a scaling factor.

Three global objectives and two hard constraints guide the optimization: (G1) compactness – maximize spatial efficiency; (G2) convexity – keep each cluster’s shape convex for perceptual clarity; (G3) neighborhood preservation – keep similar items close; (C1) size – radii must be proportional to the attribute; (C2) non‑overlap – circles must not intersect. The multi‑objective problem is turned into a single weighted‑sum objective, with empirically chosen weights (α = 0.2, β = 1.0).

Because the resulting optimization is highly non‑convex with many local minima, the authors employ the continuation method. This technique solves a sequence of increasingly difficult sub‑problems, using the solution of each as the initialization for the next, thereby guiding the search toward better regions of the solution space. The pipeline consists of three stages:

1. Neighborhood‑preserving planar graph initialization – data are projected to 2‑D, a Delaunay triangulation is built, and a maximal planar graph is extracted, capturing as many neighbor links as possible.
2. Power‑diagram‑based layout refinement – a weighted Voronoi (power diagram) partitions the plane; circles are moved toward the centers of their cells and their radii are increased until they tightly fill the cells. This step improves compactness while maintaining the graph’s adjacency structure.
3. Force‑directed refinement – circles are treated as physical particles subject to attractive, repulsive, and spring‑like forces. The forces are tuned to increase convexity of cluster boundaries and to eliminate any residual overlaps.

The authors evaluate NCP against several baselines: a front‑chain method (SimiFC), a pure power‑diagram approach (FD), and hybrid variants. Metrics include neighborhood preservation (NMI, ARI), convexity (convex‑hull ratio), compactness (area utilization), and radius‑scaling error. NCP consistently achieves higher neighborhood preservation scores and better convexity while matching or slightly exceeding the compactness of the baselines.

Two real‑world use cases illustrate practical benefits. In a clothing‑image dataset, circles sized by prediction‑uncertainty highlight mislabeled or ambiguous samples that lie on cluster borders, enabling analysts to spot noisy labels quickly. In a housing‑price visualization, circles sized by price reveal spatial price patterns while preserving similarity‑based neighborhoods, aiding policy makers in regional analysis.

The paper’s contributions are threefold: (1) a formal mapping of non‑uniform circle packing to maximal planar graph embedding; (2) a continuation‑method‑driven optimization that integrates power‑diagram and force‑directed techniques to balance compactness, convexity, and neighborhood preservation; (3) an open‑source implementation released on GitHub. Limitations include dependence on the initial 2‑D projection and computational cost for very large datasets; future work may explore dimensionality‑reduction strategies and GPU acceleration.

Overall, NCP provides a robust, theoretically grounded, and practically useful framework for visualizing data where both relational proximity and quantitative magnitude must be conveyed simultaneously.