Unfolding Ordered Matrices into BioFabric Motifs

TL;DR

This paper shows how to use well-ordered matrices as a tool to efficiently find good vertex and edge orders for BioFabrics and how to "unfold" the ordered matrix and its patterns into a high-quality BioFabric.

Abstract

BioFabrics were introduced by Longabaugh in 2012 as a way to draw large graphs in a clear and uncluttered manner. The visual quality of BioFabrics crucially depends on the order of vertices and edges, which can be chosen independently. Effective orders can expose salient patterns, which in turn can be summarized by motifs, allowing users to take in complex networks at-a-glance. However, so far there is no efficient layout algorithm which automatically recognizes patterns and delivers both a vertex and an edge ordering that allows these patterns to be expressed as motifs. In this paper we show how to use well-ordered matrices as a tool to efficiently find good vertex and edge orders for BioFabrics. Specifically, we order the adjacency matrix of the input graph using Moran's $I$ and detect (noisy) patterns with our recent algorithm. In this note we show how to "unfold" the ordered matrix and its patterns into a high-quality BioFabric. Our pipelines easily handles graphs with up to 250 vertices.

Figures (8)

• Figure 1: Matrix 58 of the FLT data set with $\sigma = 0.6$ and $\tau = 0.95$.
• Figure 2: The MIS data set, with vertex ordering proposed in BioFabricsMotifs2026 and pipeline parameters $\sigma = 0.6$ and $\tau = 0.96$. The matrix (left) shows identified patterns; our automatically generated motif simplification (middle) finds the same cliques as BioFabricsMotifs2026 (right).
• Figure 3: Automated motifs for matrix 2 in the SCH data set with pipeline parameters $\sigma = 0.5$ and $\tau = 0.95$
• Figure 4: Automated motifs for MIS with $\sigma = 0.3$ and $\tau = 0.9$.
• Figure 5: Automated motifs for ZKC with $\sigma = 0.5$ and $\tau = 0.95$