Aesthetic Discrimination of Graph Layouts - Moritz...

1 September 2018 - Aesthetic Discrimination of Graph Layouts M. Klammler · T. Mchedlidze · A. Pak

26th International Symposium on Graph Drawing and Network Visualization, Barcelona (2018)

Aesthetic Discrimination of Graph LayoutsMoritz Klammler · Tamara Mchedlidze · Alexey Pak

KIT – The Research University in the Helmholtz Association www.kit.edu

Abstract


This paper addresses the following basic question: given two layouts of the same graph, which oneis more aesthetically pleasing? We propose a neural network-based discriminator model trainedon a labeled dataset that decides which of two layouts has a higher aesthetic quality. The featurevectors used as inputs to the model are based on known graph drawing quality metrics, classicalstatistics, information-theoretical quantities, and two-point statistics inspired by methods of con-densed matter physics. The large corpus of layout pairs used for training and testing is constructedusing force-directed drawing algorithms and the layouts that naturally stem from the process ofgraph generation. It is further extended using data augmentation techniques. Our model demon-strates a mean prediction accuracy of 96.48%, outperforming discriminators based on stress andon the linear combination of popular quality metrics by a small but statistically significant margin.

Klammler, M. et al. Aesthetic Discrimination of Graph Layouts., 2018

Klammler, M. Aesthetic value of graph layouts: Investigation of statistical syndromes for automatic quantifica-tion., Master’s thesis, Karlsruhe Institute of Technology, 2018

Klammler, M. et al. Source Code for Aesthetic Discrimination of Graph Layouts., 2018

Problem Statement


Given two vertex layouts Γa and Γb for the same simple graph G = (V , E).Is Γa or Γb more aesthetically pleasing?

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Problem Statement



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Γ : V → R2

v 7→ (xv , yv )

Problem Statement



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→ undirected→ no loops→ no multiple edges

Problem Statement



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Contents


Problem Statement

Related Work

Methodology

Evaluation

Conclusion and Future Work

Related Work


Simple Metricsnumber of edge crossingsminimum crossing angle (cross resolution)minimum angle between incident edges (angular resolution)standard deviation of edge lengths…

COMB(Γi) = ∑M wM zM(Γi) with zM = (M(Γi)− µM) / σM

STRESS(Γ) = ∑n−1i=1 ∑n

j=i+1 kij(distΓ(vi , vj)− L · distG(vi , vj)

)2

Huang, W. et al. J Vis Lang Comput 2013, 24, 262–272

Kamada, T.; Kawai, S. Inf Process Lett 1989, 31, 7–15

Related WorkCombined Metric (COMB)


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COMB

Related Work


Simple Metricsnumber of edge crossingsminimum crossing angle (cross resolution)minimum angle between incident edges (angular resolution)standard deviation of edge lengths…

COMB(Γi) = ∑M wM zM(Γi) with zM = (M(Γi)− µM) / σM

STRESS(Γ) = ∑n−1i=1 ∑n

j=i+1 kij(distΓ(vi , vj)− L · distG(vi , vj)

)2


Kamada, T.; Kawai, S. Inf Process Lett 1989, 31, 7–15

Stress (STRESS)


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STRESS

Contents


Problem Statement

Related Work

Methodology

Evaluation


Methodology Overview


Γa Γb

......

Please take this into account for the Bounding Box!



Γa Γb

......


DiscriminatorModel



Γa Γb

......


DiscriminatorModel

f (Γa)

f (Γb)

Feature Extraction



......


DiscriminatorModel

f (Γa)

f (Γb)

Feature ExtractionLabeled Pairs



......


DiscriminatorModel

f (Γa)

f (Γb)


training

testing



......


DiscriminatorModel

f (Γa)

f (Γb)


Data Acquisition & Augmentation


1. Collect a large number of graphsimported graphs — from public collectionsgenerated graphs — using probabilistic algorithms

2. Compute various layouts for these graphsnative layouts — fall out of the graph generation processproper layouts — using common state-of-the-art algorithmsgarbage layouts — using more or less random placements of vertices

3. Use data augmentation to expand this corpuslayout worsening — degrade proper layout by given ratelayout interpolation — compute layout “between” proper and garbage layout

Data Acquisition & AugmentationImported Graphs


ROME



ROMENORTH



ROMENORTH

RANDDAG



ROMENORTH

RANDDAG

BCSPWR



ROMENORTH

RANDDAG

BCSPWR

GRENOBLE



ROMENORTH

RANDDAG

BCSPWR

GRENOBLE

PSADMIT



ROMENORTH

RANDDAG

BCSPWR

GRENOBLE

PSADMIT

SMTAPE



ROMENORTH

RANDDAG

BCSPWR

GRENOBLE

PSADMIT

SMTAPEIMPORT

Data Acquisition & AugmentationGenerated Graphs


LINDENMAYER



LINDENMAYER

MOSAIC1



LINDENMAYER

MOSAIC1

MOSAIC2



LINDENMAYER

MOSAIC1

MOSAIC2GRID



LINDENMAYER

MOSAIC1

MOSAIC2GRID

TORUS1



LINDENMAYER

MOSAIC1

MOSAIC2GRID

TORUS1

TORUS2



LINDENMAYER

MOSAIC1

MOSAIC2GRID

TORUS1

TORUS2

QUASI3D



LINDENMAYER

MOSAIC1

MOSAIC2GRID

TORUS1

TORUS2

QUASI3DQUASI4D



LINDENMAYER

MOSAIC1

MOSAIC2GRID

TORUS1

TORUS2

QUASI3DQUASI4D

QUASI5D



LINDENMAYER

MOSAIC1

MOSAIC2GRID

TORUS1

TORUS2

QUASI3DQUASI4D

QUASI5D

QUASI6D



LINDENMAYER

MOSAIC1

MOSAIC2GRID

TORUS1

TORUS2

QUASI3DQUASI4D

QUASI5D

QUASI6D BOTTLE



LINDENMAYER

MOSAIC1

MOSAIC2GRID

TORUS1

TORUS2

QUASI3DQUASI4D

QUASI5D

QUASI6D BOTTLE TREE

Data Acquisition & AugmentationLayouts


prop

er

NATIVE FMMM STRESS

garbage

RANDOM_UNIFORM RANDOM_NORMAL PHANTOM

Data Acquisition & AugmentationLayout Worsening


Data Acquisition & AugmentationLayout Worsening


PERTURB FLIP_NODES FLIP_EDGES MOVLSQ

r=

10%

r=

20%

r=

50%

Data Acquisition & AugmentationLayout Interpolation


r = 0% r = 20% r = 40%

r = 60% r = 80% r = 100%



......


DiscriminatorModel

f (Γa)

f (Γb)


Feature Extraction Overview


PRINCOMP1

PRINCOMP2

ANGULAR

EDGE_LENGTH

TENSION

RDF_GLOBAL

RDF_LOCAL(d)



PRINCOMP1

PRINCOMP2

ANGULAR

EDGE_LENGTH

TENSION

RDF_GLOBAL

RDF_LOCAL(d)

{x1, x2, . . . . . .}

{x1, x2, . . . . . .}

{x1, x2, . . .}

{x1, x2, . . .}

{x1, x2, . . .}

{x1, x2, . . . . . . . . .}

{x1, x2, . . . . . . . . .}(d)

Statistical Syndromes of Graph Layouts


PRINVEC1 and PRINVEC2 — first and second principal axis of vertexcoordinatesPRINCOMP1 and PRINCOMP2 — projections of vertex coordinates ontoprincipal axesANGULAR — angles between incident edgesEDGE_LENGTH — edge lengthsTENSION – ratios of Euclidean and graph-theoretical distances computedfor all vertex pairsRDF_GLOBAL — pairwise distances between vertex coordinatesRDF_LOCAL(d) — pairwise distances between vertex coordinates where thegraph-theoretical distance between them is bounded by d ∈ N

Statistical Syndromes of Graph Layouts


PRINVEC1 and PRINVEC2 — first and second principal axis of vertexcoordinatesPRINCOMP1 and PRINCOMP2 — projections of vertex coordinates ontoprincipal axesANGULAR — angles between incident edgesEDGE_LENGTH — edge lengthsTENSION – ratios of Euclidean and graph-theoretical distances computedfor all vertex pairsRDF_GLOBAL — pairwise distances between vertex coordinatesRDF_LOCAL(d) — pairwise distances between vertex coordinates where thegraph-theoretical distance between them is bounded by d ∈ N

RDF_LOCAL( 1 ) = EDGE_LENGTH

RDF_LOCAL(∞) = RDF_GLOBAL

Angles Between Incident Edges (ANGULAR)


0 0.5 1 1.5 2φ/π

0 0.5 1 1.5 2φ/π

0 0.5 1 1.5 2φ/π

Radial Distribution Function (RDF_GLOBAL)


0 5 10 15r/100

0 5 10 15r/100

0 5 10 15 20 25 30 35r/100



PRINCOMP1

PRINCOMP2

ANGULAR

EDGE_LENGTH

TENSION

RDF_GLOBAL

RDF_LOCAL(d)

{x1, x2, . . . . . .}

{x1, x2, . . . . . .}

{x1, x2, . . .}

{x1, x2, . . .}

{x1, x2, . . .}

{x1, x2, . . . . . . . . .}

{x1, x2, . . . . . . . . .}(d)

(y1, y2, y3, y4, y5, y6)

(y1, y2, y3, y4, y5, y6)

(y1, y2, y3, y4)

(y1, y2, y3, y4)

(y1, y2, y3)

(y1, y2, y3, y4)

(y1, y2, y3)(d)



PRINCOMP1

PRINCOMP2

ANGULAR

EDGE_LENGTH

TENSION

RDF_GLOBAL

RDF_LOCAL(d)

{x1, x2, . . . . . .}

{x1, x2, . . . . . .}

{x1, x2, . . .}

{x1, x2, . . .}

{x1, x2, . . .}

{x1, x2, . . . . . . . . .}

{x1, x2, . . . . . . . . .}(d)

(y1, y2, y3, y4, y5, y6)

(y1, y2, y3, y4, y5, y6)

(y1, y2, y3, y4)

(y1, y2, y3, y4)

(y1, y2, y3)

(y1, y2, y3, y4)

(y1, y2, y3)(d)

FeatureVector

Feature Extraction


We need to condense syndromes into a feature vector of fixed size.

arithmetic meanroot mean squared (RMS)

entropy of distributionProblem: depends on data aggregation (histogram bin / filter width)→ Compute entropy for several histograms with different bin widths→ Perform linear regression→ Use regression parameters instead of entropy

Feature Extraction





0 0.5 1 1.5 2φ/π

0 0.5 1 1.5 2φ/π

Feature Extraction





Feature Extraction




entropy of distributionProblem: depends on data aggregation (histogram bin / filter width)→ Compute entropy for several histograms with different bin widths→ Perform linear regression→ Use regression parameters instead of entropy−1

0

1

2

3

4

5

6

7

8

3 4 5 6 7 8 9

E/

bit

log2(β)

f (x) = −1.97+ 0.60 xf (x) = −2.18+ 0.74 xf (x) = −2.25+ 0.91 xf (x) = −1.94+ 0.98 xf (x) = −1.44+ 0.99 x



......


DiscriminatorModel

f (Γa)

f (Γb)


Siamese Neural Network


drop

out

dens

e

drop

out

dens

e57 57 15 15 11

~v ~σ

conc

aten

ation

dens

e

dens

e

11

2 2

13 1~σa −~σb

~vG

t

Bromley, J. et al. Adv Neural Inf Process Syst 1994, ed. by Jiang, X.; Wang, P. S. P., 737–744

Contents


Problem Statement

Related Work

Methodology

Evaluation


Comparison With Other Metrics


Metric Success Rate Advantage

DISC_MODEL ( 96.48 ± 0.85 )% ( 0.00 ± 0.00 )%STRESS ( 93.49 ± 0.86 )% ( 2.99 ± 1.01 )%COMB ( 92.76 ± 1.03 )% ( 3.71 ± 1.22 )%

10-fold Cross validation via random sub-samplingSTRESS was compared for best scaleCOMB weights were fitted to training data set

Welch, E.; Kobourov, S. Comput Graph Forum 2017, 36, 341–351


Comparison With Other Metrics


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COMBSTRESS

Significance of Individual Syndromes


Property Sole Exclusion Sole Inclusion

PRINCOMP1 ( 96.37 ± 0.84 )% ( 55.51 ± 6.50 )%PRINCOMP2 ( 96.20 ± 0.76 )% ( 61.08 ± 5.24 )%EDGE_LENGTH ( 96.33 ± 0.59 )% ( 71.65 ± 3.38 )%ANGULAR ( 96.40 ± 0.34 )% ( 77.79 ± 6.06 )%RDF_GLOBAL ( 95.92 ± 0.94 )% ( 86.37 ± 3.43 )%TENSION ( 96.83 ± 0.31 )% ( 89.78 ± 0.95 )%RDF_LOCAL ( 90.04 ± 2.04 )% ( 94.78 ± 1.60 )%

Baseline Using All Properties ( 96.48 ± 0.85 )%

Note that RDF_LOCAL refers to a whole set of syndromes.

Contents


Problem Statement

Related Work

Methodology

Evaluation




Binary discrimination instead of absolute aesthetic measureAvoid a priori assumptions about influence on aestheticsUse of statistical syndromes inspired by Statistical Physics andCrystallographyData driven approach (machine learning)Accuracy usually outperforms other metricshttps://github.com/5gon12eder/msc-graphstudy

Identification of necessary and sufficient syndromesOptimization of the neural networkValidation against human-labeled data

https://github.com/5gon12eder/msc-graphstudy

Bibliography


Bromley, J.; Guyon, I.; LeCun, Y.; Säckinger, E.; Shah, R. Adv Neural Inf Process Syst1994, ed. by Jiang, X.; Wang, P. S. P., 737–744.

Huang, W.; Eades, P.; Hong, S.-H.; Lin, C.-C. J Vis Lang Comput 2013, 24, 262–272.

Kamada, T.; Kawai, S. Inf Process Lett 1989, 31, 7–15.

Klammler, M. Aesthetic value of graph layouts: Investigation of statisticalsyndromes for automatic quantification., Master’s thesis, Karlsruhe Institute ofTechnology, 2018.

Klammler, M. et al. Source Code for Aesthetic Discrimination of Graph Layouts.,2018.

Klammler, M.; Mchedlidze, T.; Pak, A. Aesthetic Discrimination of Graph Layouts.,2018.

Welch, E.; Kobourov, S. Comput Graph Forum 2017, 36, 341–351.

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