MHeatmap: Two-Walk Laplacian Reordering and Mosaic Layout for Rectangular Matrix Visualization

Weighted rectangular matrices record relations between two types of objects, such as occupations and industries, documents and terms, origins and destinations, or annotated classes and computed clusters. Their heatmaps become difficult to read when arbitrary row and column orders scatter related objects, while equal size cells hide large differences in row and column totals. We present MHeatmap (Mosaic Heatmap), which addresses these problems in two stages. Two-Walk spectral seriation first represents the matrix as a weighted bipartite graph. Ordinary bipartite ordering uses only observed row to column links. Two-Walk also uses length-two walks, so rows that share columns and columns that share rows receive explicit similarity links. Sorting a Fiedler vector of the resulting graph Laplacian gives coordinated row and column orders. The Mosaic layout then partitions a unit square so that row height is proportional to the row total and column width is proportional to the column total, while color represents the observed entry. We prove how the Two-Walk energy combines direct alignment with shared-neighbor smoothness, derive an exact matrix vector product that permits sparse computation, and formalize the tiling and area semantics of Mosaic. Across 300 synthetic matrices and seven real data sets, Two-Walk obtains the lowest normalized 2-SUM in all 15 synthetic regimes and the highest Band@10% in 14. The largest gains occur when shared neighborhoods express the latent relational structure.