Structure-Aware Simplification for Hypergraph Visualization

TL;DR

This paper addresses visual clutter in large hypergraphs by introducing a structure-aware simplification framework rooted in a bipartite graph representation. It decomposes hypergraphs into topological blocks, bridges, and branches, and defines an entanglement-based metric to identify where overlaps are unavoidable. Two topology-preserving and topology-altering atomic operations are developed, guided by a tight cycle basis, to reduce clutter while preserving meaningful structures like cycles. The approach yields more reliable multi-scale visualizations and demonstrates improved efficiency and structural preservation on real-world datasets, with an implementation available for broader use.

Abstract

Hypergraphs provide a natural way to represent polyadic relationships in network data. For large hypergraphs, it is often difficult to visually detect structures within the data. Recently, a scalable polygon-based visualization approach was developed allowing hypergraphs with thousands of hyperedges to be simplified and examined at different levels of detail. However, this approach is not guaranteed to eliminate all of the visual clutter caused by unavoidable overlaps. Furthermore, meaningful structures can be lost at simplified scales, making their interpretation unreliable. In this paper, we define hypergraph structures using the bipartite graph representation, allowing us to decompose the hypergraph into a union of structures including topological blocks, bridges, and branches, and to identify exactly where unavoidable overlaps must occur. We also introduce a set of topology preserving and topology altering atomic operations, enabling the preservation of important structures while reducing unavoidable overlaps to improve visual clarity and interpretability in simplified scales. We demonstrate our approach in several real-world applications.

Figures (13)

• Figure 1: The bipartite representations (b) of the primal hypergraph (a) and dual hypergraph (b) are identical. We use a black dot to represent a node in the primal hypergraph and a gray diamond to represent a node in the dual hypergraph.
• Figure 2: The three simple cycles $C_1,C_2,C_3$ of a bipartite graph (center) are highlighted by the green, orange, and purple dotted lines. The same cycles are highlighted in the matching primal hypergraph (left) and dual hypergraph (right). Cycles $C_1$ and $C_2$ can be combined to form $C_3$.
• Figure 3: We use our block decomposition (a) to generate a topological decomposition (b) of the bipartite graph representation for the hypergraph in \ref{['fig:bipartite']}. In (a), the blue bubbles indicate single edge blocks and the purple bubbles multi-edge blocks. This leads to a number of extracted structures in (c) including topological blocks (purple bubbles), bridges (orange bubbles), and branches (green bubbles).
• Figure 4: The forbidden sub-hypergraphs of polygon hypergraph drawings: (a1) 3-adjacent hyperedge bundle of 2 hyperedges, (a2) 2-adjacent hyperedge bundle of 3 hyperedges, (b1) strangled vertex cycle variant, (b2) strangled hyperedge cycle variant, and (c1, c2) strangled vertex and hyperedge star variant. Notice that (a2) is the dual of (a1), (b2) is the dual of (b1), and (c2) is the dual of (c1). The cycle adjacency graph for each primal-dual pair is drawn in blue over the corresponding bipartite graph representation in (a3), (b3), and (c3).
• Figure 5: An example of a topological block (a) containing multiple forbidden clusters. In (c), the cycle adjacency graph is superimposed over the bipartite representation (b), with the minimal basis cycles drawn as blue nodes, and the long basis cycles drawn as green nodes. Removing the green cycles leaves two 2-connected components of blue nodes, corresponding to the forbidden clusters highlighted in blue in (d).

Theorems & Definitions (1)

• proof