Noisy Graph Patterns via Ordered Matrices

TL;DR

This work addresses summarizing high-level graph structure under noise by reordering a graph's adjacency matrix to maximize Moran's $I$, turning meaningful patterns into contiguous rectangles that correspond to cliques, bicliques, and stars. It introduces a three-stage pipeline for detecting noisy patterns: matrix reordering via a TSP-path formulation, polynomial-enumeration of candidate patterns under a locally reweighted Moran's $I$ model, and maximal disjoint pattern selection with a weight-based criterion. A novel Ring Motifs visualization encodes the detected patterns and their noise, aided by a force-based layout that preserves coherence with the matrix ordering. The approach is demonstrated on real datasets, showing noise-tolerant pattern discovery and compact visual summaries that generalize traditional motif visualizations.

Abstract

The high-level structure of a graph is a crucial ingredient for the analysis and visualization of relational data. However, discovering the salient graph patterns that form this structure is notoriously difficult for two reasons. (1) Finding important patterns, such as cliques and bicliques, is computationally hard. (2) Real-world graphs contain noise, and therefore do not always exhibit patterns in their pure form. Defining meaningful noisy patterns and detecting them efficiently is a currently unsolved challenge. In this paper, we propose to use well-ordered matrices as a tool to both define and effectively detect noisy patterns. Specifically, we represent a graph as its adjacency matrix and optimally order it using Moran's $I$. Standard graph patterns (cliques, bicliques, and stars) now translate to rectangular submatrices. Using Moran's $I$, we define a permitted level of noise for such patterns. A combination of exact algorithms and heuristics allows us to efficiently decompose the matrix into noisy patterns. We also introduce a novel motif simplification that visualizes noisy patterns while explicitly encoding the level of noise. We showcase our techniques on several real-world data sets.

Figures (11)

• Figure 1: Elements of our noisy pattern detection pipeline. Defining and enumerating (a) cliques and (b) bicliques. (c) Selecting cliques.
• Figure 2: Ring Motifs force-based layout: (a) Rotational force, (b) Link attraction, (c) Glyph repulsion, and (d) Pattern gravity.
• Figure 3: Matrix 1 of FLT data set. (Left) The matrix is not reordered and patterns are found with $\sigma = 0.2$ and $\tau=0.6$. (Right) The reordered matrix is optimal for Moran's I and patterns are found with $\sigma = 0.3$ and $\tau=1.0$. Patterns on the left miss more edges and are more noisy.
• Figure 4: Matrix 58 of the FLT data set. (Left) Patterns are found with $\sigma = 0.5$ and $\tau=0.85$. (Right) Patterns are found with $\sigma = 0.6$ and $\tau=0.95$. Higher parameter settings lead to less noise in the patterns, more missed edges, and the Ring Motifs become more complex.
• Figure 5: Matrix 35 of FLT data set with $\sigma = 0.5$ and $\tau=0.9$.