Matrix Representations of Graphs
Benjamin Martin, Christopher Mueller,
Joseph Cottam, and Andrew Lumsdaine
Open Systems Lab/Indiana University
Introduction
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Introduction to Visual Similarity Matrices
Interpreting Visual Similarity Matrices
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A Comparison of Ordering Algorithms
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Christopher Mueller, Benjamin Martin, and Andrew Lumsdaine.
Interpreting Large Visual Similarity Matrices. In Asia-Pacific
Symposium on Visualization, February 2007
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 Matrices
Advantages:
 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
But VSMs must be ordered
Expert Ordering
NCI AIDs Screen Data
So how do we do that?
Random Ordering
Questions
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How do we order a VSM?
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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.
May have to look at information
elsewhere in the matrix…
New edges
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
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Algorithms
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Breadth first search
Depth first search
Degree ordering
Reverse Cuthill-McKee (RCM)
King’s algorithm
Sloan’s algorithm
Separator tree
Spectral Ordering
Data
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Synthetic Graphs (100, 500, 1000 vertices)
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Erdős-Renyi
Small World
Power-law
K-partite
“K-linear-partite”
Real
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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 graph
2) Create two initial vertex orderings in memory:
original and random
3) Apply each algorithm to each initial ordering
4) Generate hi-res (1000x1000) and lo-res (100x100)
version of each VSM
Methods: Evaluation
Stability: Compare original and random pairs with new ordering
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Stable: similar structure
Structure: dissimilar structure
Ordered: only original has
structure
No structure
Interpretability: Evaluate quality of structure in each image
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Coarse/fine structure
Coarse structure
Minimal structure
No structure
Results: Stability
Stability is dependent on the graph and algorithm
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Synthetic graphs tended to be stable for all
algorithms
Real graphs
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“Stable” for degree, partitioning, and spectral
algorithms
“Structured” for search-based algorithms
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
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“Envelope” footprint
Retains some internal structure from original
ordering
Imparts structure on ER graphs
Results: DFS
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“Horizon” footprints
Strong diagonal
Some internal structure added to
visualization
Results: Degree
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Visually reveals degree distribution
Poorest overall results
Results: RCM and King
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Both impart additional structure within
the envelope created by the BFS
Results: Separator Tree
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Characterized by a “fat” diagonal
Nearly reproduces ordering for small world
graph
Results: Sloan
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Best overall
Similar results, regardless of initial ordering, for
all graph types
Results: Spectral
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“Galaxy” footprint
Performed well on structured graphs and fully
resolved KL5 graphs
Ordering Conclusions
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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
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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
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Bandwidth and average envelope width
Envelope jump
Envelope terminal
Number of envelope gaps
Average gap width
Graph Measurements
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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
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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
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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)