« Brain Connectivity Graph Classification »

Romain Chion

tutored by: S. Achard, M. Desvignes, F. Forbes, D. Vandeville

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SYNOPSIS

INTRODUCTION TO GRAPHS

USUAL METHODS

LOCAL MEASURES

RESULTS

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INTRODUCTION

METHODS

3

CONTEXT

• How to compare graphs to each other?

• Is it possible to model brain connectivity graphs (BCG)?

• To which extent can we characterize BCGs?

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INTRODUCTION

METHODS

4

GENERATIVE MODELSS

Illustration « Small World », Collective dynamics of‘small-world’ networks, D. J. Watts & S. H. Strogatz

Illustration « Preferential Attachment », Choice-driven phasetransition in complex networks, P. L. Krapivsky and S. Redner

• Erdos-Renyi

• Forest Fire

• Kronecker

• Preferential Attachment

• Random k-regular

• Random Power Law

• Random Typing

• Small-World

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INTRODUCTION

METHODS

5

GRAPH COMPARISON

• Transformation from a graph to anotherex : Edition distance

STRUCTURAL MEASURES

• Nodes tendency to form clusters, degree distribution, path between nodesex : Clustering, Characteristic Path Length

LOCAL MEASURES(for each node)

• Average for all local measures, core and community formationex : Assortativity, Centrality, Modularity, Diameter

OVERALL MEASURES

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METHODS

LOCAL MEASURES

6

Graphlets coutnting

Learning Set Graph Instance

Amount of Graphlets

Classifier

Graph model

classifier inputclassifier learning

STATE OF THE ART : JANSSEN et al. 2012

Amount of Graphlets

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METHODS

LOCAL MEASURES

7

STATE OF THE ART : MOTALLEBI et al. 2013

Complex NetworksClassification

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METHODS

LOCAL MEASURES

8

BCGs MODELISATION

BCGs classification in 4 generative models(Erdos-Renyi, Preferential Attachment, Random k-regular, Small-World)

Classe Prédiction E-R P A R k-R S-W

Control Small-World 0.2502 0.2501 0.2492 0.2505

Patient Small-World 0.2502 0.2501 0.2492 0.2505

Characterization with global measures and SVM classifier

Confidence interval ~25%

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METHODS

LOCAL MEASURES

9

BCGs IDENTIFICATION

true Control true Patient class precisionpred. Control 13 11 54.17%pred. Patient 7 6 46.15%

class recall 65.00% 35.29% 50.16%

Identification results with global measures and SVM classifier

Classification accuracy 50.16%, random at 50%

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METHODS

LOCAL MEASURES

RESEARCH QUESTION

« Global measures are not representative of local properties of graphs »

Local clustering coefficient histograms for 3 generative models

10

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LOCAL MEASURES

RESULTS

HISTOGRAMME NORMALISE• Clustering Coefficient

• Characteristic Path Length

• Degrees Distribution

• Efficiency Learning Set Graph Instance

Average Normalized Histograms

Histograms Distances

Graph model

Local measures histograms

NormalizedHistograms

distance minimum or classifier

11

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LOCAL MEASURES

RESULTS

HISTOGRAMS DISTANCE

12

• Bin to bin (dis)similarity measures :

Battacharyya :

Chi²

Hellinger :

• Shape preservation dissimilarity measures:

Earth Mover Distance : Optimisation of minimal work someone has to

provide to move earth from a pile to an other one.

Match : Cumulated histograms bin to bin measures

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RESULTS

GENERATED DATA

13

Performances graphlets : 78% global measures : 88% to 97.3% 6 measures and more local measures : 86% or 100% only 1 measure

Accuracy

SW 100%

RPL 100%

RkR 100%

PA 100%

KG 100%

FF 100%

ER 100%

100%

Accuracy

SW 100%

RTG 96%

RPL 98%

PA 99%

KG 96%

FF 98%

ER 93%97.2%

Classification results

local measures global measures

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RESULTS

CONNECTIVITY GRAPHS

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GLOBAL

A.N.N.

C PC 11 9 55%P 5 12 71%

69% 57% 63%

MAX global measures 63% V.S. 83% MAX histograms

Confusion matrix of Control / Patient identification

HISTOGRAM

CLUSTERING

AND CHI²

C PC 18 2 90%P 4 13 76%

82% 87% 83%

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RESULTS

BCGs MODELISATION

15

7 Clustering DegreeER 0,418 0,133FF 0,207 0,074KG 0,112 0,211RPL 0,156 0,088PA 0,437 0,242

RkR 0,459 0,183SW 0,103 0,238

EMD distance between BCGs and models for two histograms

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RESULTS

SCALABILITY

16

100 200 300 400 500 600 700 800 900 100011001200130014001500160017001800190020000,01 11% 10% 10% 14% 14% 12% 7% 6% 14% 11% 23% 29% 29% 30% 29% 34% 34% 36% 36% 45%0,02 12% 18% 18% 16% 20% 22% 30% 39% 41% 42% 43% 42% 42% 44% 42% 43% 43% 42% 42% 43%0,03 10% 19% 20% 27% 28% 41% 41% 45% 43% 43% 43% 42% 41% 40% 44% 43% 44% 43% 43% 43%0,04 17% 26% 32% 40% 43% 41% 44% 40% 43% 43% 43% 43% 43% 43% 45% 43% 43% 43% 43% 42%0,05 16% 25% 41% 42% 41% 43% 42% 43% 38% 40% 43% 42% 42% 43% 42% 43% 42% 43% 43% 43%0,06 33% 41% 43% 44% 43% 42% 42% 46% 41% 43% 43% 43% 42% 43% 43% 43% 49% 43% 44% 43%0,07 36% 57% 54% 65% 62% 70% 67% 72% 71% 72% 69% 71% 68% 72% 85% 85% 83% 86% 84% 86%0,08 44% 69% 72% 72% 72% 75% 69% 86% 84% 86% 86% 86% 86% 86% 86% 86% 84% 86% 86% 86%0,09 41% 81% 85% 93% 96% 93% 90% 97% 94% 90% 86% 86% 84% 85% 71% 71% 70% 72% 71% 71%0,1 49% 88% 86% 100% 96% 100% 99% 84% 81% 85% 86% 86% 86% 86% 86% 86% 86% 86% 86% 86%

0,11 52% 99% 93% 90% 89% 91% 92% 78% 74% 72% 71% 71% 69% 71% 71% 71% 71% 71% 71% 71%0,12 62% 83% 85% 72% 71% 68% 72% 68% 74% 73% 72% 71% 71% 71% 72% 71% 71% 72% 71% 71%0,13 62% 64% 70% 64% 68% 68% 71% 68% 67% 67% 70% 66% 69% 57% 65% 61% 56% 43% 48% 43%0,14 59% 57% 48% 43% 43% 44% 44% 44% 43% 43% 43% 43% 43% 43% 43% 43% 43% 43% 43% 43%0,15 54% 49% 45% 49% 42% 42% 45% 42% 42% 43% 43% 43% 43% 43% 43% 43% 43% 43% 43% 43%0,16 45% 44% 43% 43% 44% 45% 42% 44% 43% 43% 43% 43% 43% 43% 43% 43% 43% 42% 43% 43%0,17 42% 41% 41% 40% 42% 42% 42% 42% 43% 44% 43% 43% 42% 41% 41% 42% 42% 41% 39% 39%0,18 45% 43% 44% 43% 43% 45% 42% 42% 43% 42% 43% 41% 41% 37% 40% 37% 34% 32% 31% 31%0,19 44% 45% 43% 39% 43% 42% 41% 42% 42% 40% 36% 33% 30% 32% 29% 29% 29% 29% 29% 29%0,2 43% 43% 41% 45% 41% 41% 40% 35% 30% 35% 31% 29% 29% 29% 29% 29% 29% 29% 29% 29%

Increasing number of nodes

Incr

ea

sing

den

sity

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RESULTS

LEARNING STABILITY

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CROSS-VALIDATION d100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 200067% 70% 67% 69% 70% 71% 73% 75% 74% 77% 77% 78% 77% 78% 79% 80% 80% 78% 79% 81%

CROSS-VALIDATION nd = 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,1 0,11 0,12 0,13 0,14 0,15 0,16 0,17 0,18 0,19 0,2

PREC 76% 82% 95% 97% 97% 97% 99% 99% 99% 99% 97% 99% 98% 99% 99% 99% 99% 99% 99% 99%MIN

PREC. 32% 31% 80% 88% 89% 91% 96% 96% 96% 96% 91% 93% 92% 94% 97% 96% 96% 98% 98% 98%MIN

CLASS. ER ER FF FF KG KG KG SW SW SW KG SW SW SW SW SW SW SW SW SW

Increasing density

Increasing number of nodes

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RESULTATS

RANDOMIZATION STABILITY

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0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90%ER 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%FF 100% 100% 97% 97% 100% 97% 100% 100% 100% 100% 100% 97% 100% 100% 100% 100% 100% 100% 100%KG 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%PA 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%

RkR 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%RPL 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%SW 100% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

Increasing randomness

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RESULTS

REMOVING CLASSES

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Erdos-Renyi

FFRPL

Forest Fire

RPLSW

Kronecker Graph

FF

77%SW 23%RPL

Preferential Attachment

FFRPL

Random k-Regular

FFRPL

Random Power Law

FF

92% SW 8% PA

Small-World

FFRPL

Graphes de Connectivités

FFRPLPASW…

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RESULTS

PCA : RESULTS

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PC 1 0.415 0.750 0.750PC 2 0.170 0.126 0.876PC 3 0.132 0.076 0.952PC 4 0.101 0.044 0.996PC 5 0.028 0.004 0.999PC 6 0.011 0.000 1.000PC 7 0.003 0.000 1.000

NUMBER OF PRINCIPAL COMPONENT

CUMULATIVE VARIANCE

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RESULTS

PCA : STABILITY

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14% 14% 14% 2% 0% 20% 11% 5% 14% 15% 14% 15% 15% 15% 15% 17% 17% 20% 23%

1% 14% 25% 15% 20% 28% 25% 24% 19% 36% 36% 37% 37% 40% 40% 55% 43% 42% 43%

6% 26% 31% 30% 35% 43% 46% 63% 62% 64% 63% 58% 60% 61% 67% 61% 66% 63% 61%

27% 34% 42% 45% 53% 55% 60% 66% 67% 67% 66% 68% 66% 64% 63% 51% 55% 57% 55%

31% 43% 48% 57% 59% 60% 63% 70% 66% 69% 71% 70% 70% 71% 71% 58% 70% 78% 70%

32% 51% 56% 69% 70% 66% 68% 72% 71% 74% 72% 86% 86% 85% 84% 71% 85% 86% 83%

34% 62% 68% 71% 70% 71% 83% 86% 87% 86% 86% 85% 84% 85% 86% 79% 84% 86% 86%

36% 67% 67% 79% 85% 86% 86% 86% 84% 86% 86% 86% 86% 86% 86% 86% 85% 86% 86%

40% 76% 94% 99% 100% 100% 98% 99% 99% 100% 100% 100% 99% 100% 94% 96% 96% 83% 82%

46% 96% 99% 100% 98% 100% 99% 98% 98% 98% 100% 100% 100% 100% 88% 88% 87% 88% 86%

52% 100% 96% 100% 99% 100% 100% 95% 94% 92% 93% 96% 92% 91% 86% 86% 86% 86% 86%

57% 98% 100% 99% 100% 98% 87% 73% 74% 77% 75% 72% 72% 73% 72% 71% 71% 72% 71%

58% 80% 85% 68% 73% 67% 70% 57% 55% 59% 57% 57% 57% 58% 58% 57% 57% 57% 57%

61% 64% 69% 64% 66% 61% 63% 59% 58% 58% 57% 57% 58% 57% 57% 57% 57% 58% 57%

65% 57% 67% 62% 59% 60% 58% 56% 58% 57% 57% 58% 57% 57% 58% 58% 58% 58% 57%

68% 59% 61% 53% 56% 57% 57% 58% 58% 57% 57% 57% 57% 57% 57% 57% 58% 57% 57%

66% 56% 56% 42% 45% 52% 57% 57% 57% 58% 57% 58% 57% 57% 57% 57% 57% 57% 57%

62% 57% 61% 43% 43% 47% 54% 58% 58% 55% 57% 57% 57% 57% 58% 58% 58% 57% 57%

60% 58% 57% 43% 43% 43% 44% 49% 54% 47% 46% 56% 57% 57% 57% 57% 57% 56% 57%

57% 59% 52% 46% 43% 42% 43% 42% 41% 43% 44% 43% 43% 43% 43% 43% 43% 43% 43%

11% 10% 10% 14% 14% 12% 7% 6% 14% 11% 23% 29% 29% 30% 29% 34% 34% 36% 36%

12% 18% 18% 16% 20% 22% 30% 39% 41% 42% 43% 42% 42% 44% 42% 43% 43% 42% 42%

10% 19% 20% 27% 28% 41% 41% 45% 43% 43% 43% 42% 41% 40% 44% 43% 44% 43% 43%

17% 26% 32% 40% 43% 41% 44% 40% 43% 43% 43% 43% 43% 43% 45% 43% 43% 43% 43%

16% 25% 41% 42% 41% 43% 42% 43% 38% 40% 43% 42% 42% 43% 42% 43% 42% 43% 43%

33% 41% 43% 44% 43% 42% 42% 46% 41% 43% 43% 43% 42% 43% 43% 43% 49% 43% 44%

36% 57% 54% 65% 62% 70% 67% 72% 71% 72% 69% 71% 68% 72% 85% 85% 83% 86% 84%

44% 69% 72% 72% 72% 75% 69% 86% 84% 86% 86% 86% 86% 86% 86% 86% 84% 86% 86%

41% 81% 85% 93% 96% 93% 90% 97% 94% 90% 86% 86% 84% 85% 71% 71% 70% 72% 71%

49% 88% 86% 100% 96% 100% 99% 84% 81% 85% 86% 86% 86% 86% 86% 86% 86% 86% 86%

52% 99% 93% 90% 89% 91% 92% 78% 74% 72% 71% 71% 69% 71% 71% 71% 71% 71% 71%

62% 83% 85% 72% 71% 68% 72% 68% 74% 73% 72% 71% 71% 71% 72% 71% 71% 72% 71%

62% 64% 70% 64% 68% 68% 71% 68% 67% 67% 70% 66% 69% 57% 65% 61% 56% 43% 48%

59% 57% 48% 43% 43% 44% 44% 44% 43% 43% 43% 43% 43% 43% 43% 43% 43% 43% 43%

54% 49% 45% 49% 42% 42% 45% 42% 42% 43% 43% 43% 43% 43% 43% 43% 43% 43% 43%

45% 44% 43% 43% 44% 45% 42% 44% 43% 43% 43% 43% 43% 43% 43% 43% 43% 42% 43%

42% 41% 41% 40% 42% 42% 42% 42% 43% 44% 43% 43% 42% 41% 41% 42% 42% 41% 39%

45% 43% 44% 43% 43% 45% 42% 42% 43% 42% 43% 41% 41% 37% 40% 37% 34% 32% 31%

44% 45% 43% 39% 43% 42% 41% 42% 42% 40% 36% 33% 30% 32% 29% 29% 29% 29% 29%

43% 43% 41% 45% 41% 41% 40% 35% 30% 35% 31% 29% 29% 29% 29% 29% 29% 29% 29%

CROSS-VALIDATION d : 5 to 16% increase average from 75 to 84%

CROSS-VALIDATION n : UP to 5% increase average from 96 to 97%

BEFORE AFTER

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RESULTS

PCA : INTERPRETATION

22

CO

MP

ON

EN

T 2

COMPONENT 1

Biplot: visual representation

RANDOM POWER LAW

COMPONENT 1

SMALL WORLD

FOREST FIRE

PREF ATTACHMENT

ERDOS RENYI

K REGULAR

FORMER ATTRIBUTESVECTORS

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CONCLUSION

Excellent performances on generated data

Histograms of local measures are useful

Local clustering is particularly important

Still dependent on existing and number of models

Results on connectivity data are still lacking

Combined model are to be considered

Basis of histograms to be constructed