ginestet weighted network analysis - warwick

Weighted Network Analysis for Groups:Separating Differences in Cost from Differences in Topology

Cedric E. Ginestet

Department of Neuroimaging, King’s College London

Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 1 / 35

Connectivity Data

Subject-specific Correlation Matrices

For the i th subject in the j th condition: Rij .

AAL Cortical RegionsA

AL

Cor

tica

lR

egio

ns


Connectivity Data

Experimental Paradigm

J conditions (columns), and n subjects (rows).

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R11 R12 R1J

Rn1 RnJ


Part I

N-back Task on Working Memory


N-back Paradigm

Figure: N-back task. There are here four levels of difficulties from 0-back to 3-back.


Experimental Paradigm

Ginestet et al., Neuroimage, 2011.

i. 43 (incl. 21 females) healthy controls.

ii. Mean age of 68.23 years (sd = 13.17).

iii. 12 randomised blocks lasting each 31 seconds.

iv. 186 T2∗-weighted EPI volumes on 1.5T scanner.

v. TE=40ms, TR=2s, flip angle 90.

Subject-specific Weighted Networks

i. Anatomical Automatic Labeling (AAL) Parcellation.

ii. Regional Mean time series.

iii. Maximal Overlap Discrete Wavelet Transform (MODWT).

iv. Scale 4 Wavelet Coefficient: (0.01-0.03Hz interval).


Wavelet Decomposition + Concatenation

0 10 20 30 40

−4

02

4

Concatenated Volumes

W3

0 10 20 30 40

−1.

00.

00.

51.

0


W3

0 10 20 30 40

−4

02

4


W4

0 10 20 30 40

−1.

00.

00.

51.

0


W4

Figure: Time series of wavelet coefficients and running correlations for the first AALregion. In panels 1 and 3, the time series of the scale 3 and 4 wavelet coefficients arerepresented with region 1 in red. In panels 2 and 4, the running correlations betweenregion 1 and the remaining 89 AAL regions are given for the corresponding waveletcoefficients, with mean correlation in blue. These results are here shown for the fourthexperimental condition (i.e. 3-back) and the first subject in the sample.


Wavelet Decomposition + Concatenation

Concatenation Only

Differences in Correlations (0−back to 1−back)

Den

sity

−1.0 −0.5 0.0 0.5 1.0

0.0

0.5

1.0

1.5

2.0

Concatenation Only


Den

sity

−1.0 −0.5 0.0 0.5 1.0

0.0

0.5

1.0

1.5

2.0

Concatenation Only


Den

sity

−1.0 −0.5 0.0 0.5 1.0

0.0

0.5

1.0

1.5

2.0

Wavelet−Concatenated


Den

sity

−1.0 −0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8



Den

sity

−1.0 −0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8


Differences in Correlations (0−back to 3−back)D

ensi

ty

−1.0 −0.5 0.0 0.5 1.0

0.0

0.4

0.8

1.2


Differences in Cost/Density

Main Effect of N-back Experimental Factor?

0-back 1-back 2-back 3-back

0.0 0.2 0.4 0.6 0.8 1.0

Figure: Heatmaps corresponding to subject-specific correlation matrices for the fourN-back conditions. (Ginestet et al., Neuroimage, 2011).


Part II

Statistical Parametric Networks (SPNs)


Statistical Parametric Networks (SPNs)

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Mass-univariate Approaches to Network Inference

Previous Approaches

i. Achard et al. (Jal of Neuroscience, 2006).

ii. He et al. (PLoS one, 2009).

iii. Kramer et al. (Phys. Rev. E., 2009).

Method

i. Z -test on Fisher-transformed correlation coefficients.

ii. Parametric/Non-parametric significance testing.

iii. Control for multiple comparison (False Discovery Rate).


Cost/density Decreases with Cognitive Load

Sagittal SPNj


Figure: Mean Statistical Parametric Networks (SPNj), based on wavelet coefficients inthe 0.01–0.03Hz frequency band. The locations of the nodes correspond to thestereotaxic centroids of the cortical regions (Ginestet et al., Neuroimage, 2011).


Task-related Physiological Variability

Sagittal SPNj


i. Could N-back connectivity differences be solely explained by task-correlatedphysiological variability, such as breathing?

ii. As breathing accelerates with task difficulty, its frequency 0.03Hz.

iii. See Birn et al. (HBM, 2008), and Birn et al. (Neuroimage, 2009).


Connectivity Strength Predicts Task Performance

0−back 1−back 2−back 3−back

400

600

800

1000

1200

1400

1600

1800

(a) Penalized RT

pRT(m

s)

0−back 1−back 2−back 3−back0.

20.

40.

60.

8

(b) Weighted Cost

K(G

)

Figure: Boxplots of (a) penalized reaction time and (b) weighted cost. Regression ofpRT on subject-specific weighted cost (KW (Gij) for the i th subject under the j th

condition) after controlling for the N-back factor was found to be significant (p < .001)(Ginestet et al., Neuroimage, 2011).


Differential SPNs

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F-testfor all e ∈ E(G), v ∈ V (G):

rei = Xei β

e + Zei be

i + εei ;

yvi = Xvi β

v + Zvi bv

i + εvi .


Differential SPNs

L R

Figure: Differential SPN. Sagittal section of the negative differential SPN, whichrepresents the significantly ‘lost’ edges, due to the N-back experimental factor. Thepresence of an edge is determined by the thresholding of p-values at .01, uncorrected(Ginestet et al., Neuroimage, 2011).


Part III

Differences in Topology vs. Differences in Density


Differences in Topology vs. Differences in Density

Regular Random


Classical Measures of Topology

Efficiencies (Latora et al., 2001)

For any unweighted graph G = (V, E), connected or disconnected,

E (G ) :=1

NV (NV − 1)

NV∑i=1

NV∑j 6=i

d−1ij , (1)

where dij is the length of the shortest path between vertices i and j in G .

Global and Local Efficiencies

E Glo(G ) := E (G ), and E Loc(G ) :=1

NV

NV∑i=1

E (Gi ), (2)

where Gi is the subgraph of G that includes all the neighbors of the i th node.


Efficiencies are Monotonic Increasing with Density

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

(a) Global Efficiency

Cost

E(G

lo)

0−back1−back2−back3−back

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

(b) Lobal Efficiency

Cost

E(L

oc)

0−back1−back2−back3−back

Figure: Efficiencies under the four conditions of the N-back task, with density-equivalentrandom (red) and regular (blue) networks, for each condition.


Integrating over Densities

Cost-integrated Topological Metrics

Given a weighted graph G = (V, E ,W) and a topological metric T (·),

Tp(G ) :=∑k∈ΩK

T (γ(G , k))p(k), (3)

where γ(G , k) thresholds G and returns an unweighted graph with density/cost k.

Treating Cost/Density as a Random Variable

Here, the number of edges in G , denoted k , is given distribution p(k), defined over

Ωk :=

1, . . . ,

(NV

2

), (4)

with NV := |V(G )|.


‘Prior Distribution’ over Graph Densities

0 1000 2000 3000 4000

0e+

002e

−04

4e−

046e

−04

Beta−Binomial Distribution

Ne

p(K

=k)

n=Ne

a=b=1a=b=2a=b=3a=b=4a=b=5

Figure: Symmetric versions of the Beta-binomial distribution for different choices ofparameters, with NE = 4005 (Ginestet et al., PLoS one, 2011).


Integrating over Cost/Density

Proposition (Ginestet et al., PLoS one, 2011)

Let a weighted undirected graph G = (V, E ,W). For any monotonic function h(·)acting elementwise on a real-valued matrix, W, corresponding to the weight setW, and any topological metric T , the cost-integrated version of that metric,denoted Tp, satisfies

Tp(W) = Tp(h(W)). (5)

Proof.

Since h(·) is applied elementwise to W, we have

Rij(h(W)) =1

2

NV∑u=1

NV∑v 6=u

Ih(wij) ≥ h(wuv ) = Rij(W). (6)


Topological Differences does not Predict Performance

0−back 1−back 2−back 3−back

0.55

0.60

0.65

0.70

(a) Global EfficiencyE

(Glo

)

0−back 1−back 2−back 3−back0.

700.

750.

80

(b) Local Efficiency

E(L

oc)

Figure: Boxplots of subject-specific cost-integrated global and local efficiencies in panels(a) and (b), respectively, where Gij denotes the functional network for the i th subject inthe j th condition (Ginestet et al., Neuroimage, 2011).


Part IV

Weighted Metrics for Weighted Networks?


Weighted Topological Metrics

Weighted Global Efficiency

As introduced by Latora et al. (2001),

EW (G ) :=1

NV (NV − 1)

NV∑i=1

NV∑j 6=i

1

dWij

. (7)

where G is a weighted graph, G = (V, E ,W).

Weighted Shortest Path

The weighted shortest path dWij is defined as

dWij := min

Pij∈Pij (G)

∑wuv∈W(Pij )

w−1uv , (8)


Integrating over Cutoff

Proposition (Ginestet et al., PLoS one, 2011)

For any weighted graph G = (V, E ,W), whose weight set is denoted by W(G ), ifwe have

minwij∈W(G)

wij ≥1

2max

wij∈W(G)wij , (9)

thenEW (G ) = KW (G ). (10)

Proof.

Assume that dWij 6= w−1

ij for at least one edge (i , j), and then show that thiscontradicts the hypothesis.


Modularity & Edge Density

A B

Random Rewirings

Num

ber

of M

odul

es

0 200 400 600 800

0

2

4

6

Number of Edges

Num

ber

of M

odul

es

100 600 1100 1600 2100 2600 3100

0

2

4

6

8

10Networks

RandomRegular

Figure: Topological randomness and number of edges predict number of modules.(A) Relationship between the number of random rewirings of a regular lattice and thenumber of modules in such a network. (B) Relationship between the number of edges ina network and its number of modules (Bassett et al., PNAS 2011).


Modularity & Edge Density

C

NE = 100 NE = 600 NE = 1100 NE = 1600 NE = 2100

D

NE = 100 NE = 600 NE = 1100 NE = 1600 NE = 2100

Figure: Topological randomness and number of edges predict number of modules.Modular structures of regular (C) and random (D) networks for different number ofedges, NE (Bassett et al., PNAS 2011).


Part V

Some Conclusions.


Summary

Main Messages

1 Thresholding: Discrete mathematics on Continuous (Real-valued) data.2 What matters when comparing weighted networks:

i. Weighted cost/density (e.g. mean correlation).ii. Cost-integrated topological metrics.iii. Problem does not vanish with weighted metrics.

3 Cost-integration approximated using Monte Carlo sampling scheme.

4 R package for cost-integration: NetworkAnalysis on CRAN.

Future Work

1 Replicate these findings in other MRI cognitive tasks.

2 Weighted network analysis in neuropharmacological studies.


Activity vs. Connectivity

Sepulcre et al. (PLoS CB, 2010).


Collaborators & Funding Agencies

Collaborators

1 Andy Simmons, Mick Brammer, Andre Marquand, Vincent Giampietro, OrlaDoyle, Jonny O’Muircheartaigh, Owen G. O’Daly (King’s College London)

2 Arnaud Fournel (Lyon, France)

3 Ed Bullmore (Cambridge, UK)

4 Tom Nichols (Warwick, UK)

5 Randy Buckner (Harvard, MA)

6 Dani Bassett (UCLA, CA)

Funding Agencies


ginestet weighted network analysis - warwick

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