Structural Differences Between Two Graphs through Hierarchies Daniel - - PowerPoint PPT Presentation

Structural Differences Between Two Graphs through Hierarchies

Daniel Archambault1

1GRAVITE, INRIA Bordeaux Sud-Ouest, France Daniel Archambault Graph Difference 1/ 24

Introduction

The Wine Difference Problem

Wikimedia Project, Creative Commons Licence, André Karwath

Carefully studying this important problem...

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Introduction

The Graph Difference Problem Structural evolution of dynamic graphs

how does a dynamic graph evolve over time

Typical technique in graph drawing community

show graph evolution through animation

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Introduction

Video

• Y. Frishman and A. Tal. Online Dynamic Graph Drawing. EuroVis 2007

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Introduction

Difference Map and Small Multiples Approach ∆t = 10 and 11 ∆t = 11 and 12 ∆t = 12 and 13 Is this an improvement?

Need an experiment to support this argument

Contributes new algorithms to generate coarsened difference map

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Previous and Related Work

Animation as Dynamic Graph Evolution

Krists Boitmanis, Ulrik Brandes, and Christian Pich. Visualizing Internet Evolution on the Autonomous Systems Level.

• Symp. Graph Drawing (GD ’07)

By far, most common method for dynamic graph visualization Changes faded in and out, node movement interpolated

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Previous and Related Work

Small Multiples and Dynamic Attributes

• E. H. Chi and S. Card. Sensemaking of Evolving Web Sites using Visualization Spreadsheets (InfoVis ’99)

Not frequently used to depict structural evolution of graphs Experiments suggest may be better for dynamic data

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Previous and Related Work

Hierarchy-Based Graph Visualization

(a) DA-TU, 2000 (b) ASK-GraphView, 2006 (c) GrouseFlocks, 2008

Abstract away parts of the graph where details not required

used to abstract away structural difference

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Previous and Related Work

Path-Preserving Hierarchy

(a) (b)

Defined in GrouseFlocks work A path in the hierarchy means at least one path in the graph Path-preserving hierarchies respect this property

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Previous and Related Work

Path-Preserving Hierarchy

(a) Edge Conservation (b) Connectivity Conservation

Metaedge if and only if a pair of descendants connected Metanodes contain connected subgraphs If preserved, paths in cuts are also in underlying graph

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Algorithm

Algorithm Overview

1

Construct difference map

2

Difference hierarchy construction

3

Degree One Coarsening

4

Betweenness Centrality Coarsening

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Algorithm

Difference Map Construction Each node in graph is guaranteed a unique labelling Can be done through a single scan of the node and edge list of each graph. O(|N| + |E|) for the nodes in both graphs

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Algorithm

Difference Hierarchy Construction (1)

A B C D E F G H I J K L M N

(a) Input

A B C D E F G H I J E E E G K L M N F

(b) Edge Decomposition

Decompose the graph into connected components by edge difference

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Algorithm

Difference Hierarchy Construction (2)

A B C D E F G H I J K L M N

(a) Node Decomposition

A D E F G {B, C} {H, I, J} {K, L, M} N

(b) Graph Hierarchy

Decompose by node difference Create a hierarchy based on node and edge difference

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Algorithm

Degree One Coarsening

A B C D E

(a) Before Coarsening

A {B, C} {D, E}

(b) After Coarsening

Group degree one nodes attached same root together Does not need to be connected because paths begin/end here

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Algorithm

Betweenness Centrality Coarsening Coarsens away nodes with:

little change in betweenness centrality large metanodes common to both graphs

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Results

Results: Threads ∆t = 10 and 11 ∆t = 10 and 11 ∆t = 11 and 12 ∆t = 12 and 13 ∆t = 13 and 14 ∆t = 14 and 15 Newsgroup discussion between 62-70 nodes and 72-87 edges

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Results

Results: Opte (1)

(a) Difference Map (b) Hierarchy

Internet scan of about 40,000 nodes and 47,000 edges

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Results

Results: Opte (2)

(a) Hierarchy (b) Degree One Coarsening

Internet scan of about 40,000 nodes and 47,000 edges Degree one coarsening applied

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Results

Results: Opte (3)

(a) Degree One Coarsening (b) Betweenness Centrality Coarsening

Internet scan of about 40,000 nodes and 47,000 edges Betweenness centrality coarsening applied

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Results

Results: RouteViews Internet scan of about 24,000 nodes and 58,000 edges Work still to be done Possible coarsening techniques to simplify hierarchy

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Future Work and Conclusion

Future Work Do difference maps help in understanding structural evolution?

currently designing an experiment to provide evidence

Can people properly interpret hierarchies in a difference context? Coarsening techniques to scale to larger graphs

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Future Work and Conclusion

Conclusion Visualizing structural difference between two graphs Contributions

use hierarchy to coarsen away areas of similarity/difference path-preserving coarsening technique betweenness centrality coarsening technique

Scales to graphs of tens of thousands of nodes

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Future Work and Conclusion

Acknowledgements Thank you Guy Melançon and David Auber for numerous discussions Thank you LaBRI for comments on the work Partially funded by the INRIA GRAVITÉ project Questions?

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