Matrix Representations of Graphs Benjamin Martin, Christopher Mueller, Joseph Cottam, and Andrew...
-
Upload
kelley-mcbride -
Category
Documents
-
view
222 -
download
0
Transcript of Matrix Representations of Graphs Benjamin Martin, Christopher Mueller, Joseph Cottam, and Andrew...
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)