Matrix Representations of Graphs

Benjamin Martin, Christopher Mueller, Joseph Cottam, and Andrew Lumsdaine

Open Systems Lab/Indiana University

Introduction

Introduction to Visual Similarity Matrices Interpreting Visual Similarity Matrices

Christopher Mueller, Benjamin Martin, and Andrew Lumsdaine. Interpreting Large Visual Similarity Matrices. In Asia-Pacific Symposium on Visualization, February 2007

A Comparison of Ordering Algorithms Christopher Mueller, Benjamin Martin, and Andrew Lumsdaine.

A Comparison of Vertex Ordering Algorithms for Large Graph Visualization. In Asia-Pacific Symposium on Visualization, February 2007

BFS Case Study

Visual Similarity Matrices

Essentially, draw the adjacency matrix Axes are labeled with the vertex names Matrix dots represent graph edges

Visual Similarity MatricesAdvantages: Capable of representing much larger graphs No risk of occlusion or edge crossings Good for large and dense graphs, for most tasks

See M. Ghoniem, J.-D. Fekete, and P. Castagliola. “On the readability of graphs using node-link and matrix-based representations: a controlled experiment and statistical analysis.” In Information Visualization, volume 4, pages 114–135, 2005.

Visual Similarity Matrices

NCI AIDs Screen DataExpert Ordering Random Ordering

But VSMs must be ordered

So how do we do that?

Questions

How do we order a VSM? How do we decide between different

methods? How do we interpret the results?

Interpreting VSMs

Some features of VSMs are dependent on the matrix ordering, some are not

Interpreting VSMs

Straight horizontal or vertical lines show “star” connectivity patterns in a graph.

Noncontiguous lines carry the same information as contiguous lines

Diagonal lines carry information if they are more than one pixel wide.

New edges

May have to look at information elsewhere in the matrix…

Interpreting VSMs

Wedges appear on the diagonal and represent cliques

Off-diagonal blocks are bi-partite sub-graphs.

Blank rows or columns for a vertex remove the vertex from the component.

Blocks and wedges can be joined across rows and columns:

- A is a clique (mirrored on the diagonal)- B, C, D are bi-partite sub-graphs- A + B, A + C are cliques- A + B + C + D is a clique- A and D are independent w/o additional supporting structures

Interpreting VSMs

Some ordering algorithms may produce characteristic patterns in the VSM

Envelope footprints are indicative of breadth-first search algorithms.

Horizon footprints suggest depth-first algorithms.

Galaxy footprints are caused by algorithms that don’t follow direct paths through the graph.

Ordering

Algorithms Breadth first search Depth first search Degree ordering Reverse Cuthill-McKee (RCM) King’s algorithm Sloan’s algorithm Separator tree Spectral Ordering

Data

Synthetic Graphs (100, 500, 1000 vertices) Erdős-Renyi Small World Power-law K-partite “K-linear-partite”

Real COGSimilar - 1770 vertices, 290k edges COGDissimilar - 2030 vertices, 158k edges NCIca - 436 vertices, 18.6k edges NCIall - 42,750 vertices, 3.29M edges

Methods: VSM Preparation

…

(1) (2) (3) (4)

1) Create or load each graph2) Create two initial vertex orderings in memory:

original and random3) Apply each algorithm to each initial ordering4) Generate hi-res (1000x1000) and lo-res (100x100)

version of each VSM

Methods: Evaluation

Coarse/fine structure Coarse structure Minimal structure No structure

Stability: Compare original and random pairs with new ordering

Interpretability: Evaluate quality of structure in each image

Stable: similar structure Structure: dissimilar structure Ordered: only original has

structure No structure

Results: Stability

Synthetic graphs tended to be stable for all algorithms

Real graphs “Stable” for degree, partitioning, and spectral

algorithms “Structured” for search-based algorithms

Stability is dependent on the graph and algorithm

Results: Interpretability

Graphs with regular or dense connection patterns exhibited coarse and fine structure

Structural artifacts from each algorithm were evident across all graph types

Results: BFS

“Envelope” footprint Retains some internal structure from original

ordering Imparts structure on ER graphs

Results: DFS

“Horizon” footprints Strong diagonal Some internal structure added to visualization

Results: Degree

Visually reveals degree distribution Poorest overall results

Results: RCM and King

Both impart additional structure within the envelope created by the BFS

Results: Separator Tree

Characterized by a “fat” diagonal Nearly reproduces ordering for small world

graph

Results: Sloan

Best overall Similar results, regardless of initial ordering, for

all graph types

Results: Spectral

“Galaxy” footprint Performed well on structured graphs and fully

resolved KL5 graphs

Ordering Conclusions

If structure is present in the data, all algorithms provide clues to it

Amount of connectedness has the largest positive impact on ordering quality

Randomness in data has the largest negative impact on ordering quality

Algorithms that looked at global and local properties performed best

Implementation Issues

Interactivity is essential for exploring large graphs.

Alternate orderings allow for different views and interpretations.

Linked views, especially for vertex properties, help explore structural features.

Edges (dots) can be colored by weight or category.

Anti-aliasing is essential for large, sparse graphs:

The graph on the right is easily interpreted as a dense graph with little structure. The image on the left provides a more accurate

rendering of the graph.

Some Lessons

VSMs are compelling tools for exploring large graphs

Care must be taken to ensure proper interpretations are made

Interactive tools that provide multiple views of the graph are useful for exploring large VSMs

BFS Case Study

BFS shows remarkable consistency over several graph types, especially SWGs

Can we better characterize its behavior?

Specifically, can we quantify some of the things we’re seeing?

BFS Case Study

Visual Parameters

Bandwidth and average envelope width Envelope jump Envelope terminal Number of envelope gaps Average gap width

Graph Measurements

Diameter Characteristic path length Global efficiency Clustering coefficient Model parameters n, k, p

Graph Measurements

BFS Case Study

We begin by considering fixed n and k,

and looking at diameter and average envelope width.

Average Envelope Width and Diameter

Global Efficiency

Global Efficiency

Global Efficiency

Generalizing

What about other n and k?

Generalizing

Generalizing

Generalizing

Generalizing

Generalizing

Summary

Average envelope width is an effective and simple predictor of global efficiency for Watts-Strogatz graphs

Which indirectly gives us the diameter And hopefully this works with other graph

types

Possible directions

Look at more diverse data Look at other orderings, especially

spectral

Acknowledgements

Funding: Lily Endowment National Science Foundation grant EIA-0202048.

Software and Algorithms: Doug Gregor (Boost Graph Library) Jeremiah Willcock (Separator Tree)

Data: Sun Kim (PLATCOM) David Wild (NCI)