ginestet weighted network analysis - warwick
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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
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 2 / 35
Connectivity Data
Experimental Paradigm
J conditions (columns), and n subjects (rows).
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...
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R11 R12 R1J
Rn1 RnJ
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 3 / 35
Part I
N-back Task on Working Memory
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 4 / 35
N-back Paradigm
Figure: N-back task. There are here four levels of difficulties from 0-back to 3-back.
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 5 / 35
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).
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 6 / 35
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).
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 6 / 35
Wavelet Decomposition + Concatenation
0 10 20 30 40
−4
02
4
Concatenated Volumes
W3
0 10 20 30 40
−1.
00.
00.
51.
0
Concatenated Volumes
W3
0 10 20 30 40
−4
02
4
Concatenated Volumes
W4
0 10 20 30 40
−1.
00.
00.
51.
0
Concatenated Volumes
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.
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 7 / 35
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
Differences in Correlations (0−back to 2−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
Differences in Correlations (0−back to 3−back)
Den
sity
−1.0 −0.5 0.0 0.5 1.0
0.0
0.5
1.0
1.5
2.0
Wavelet−Concatenated
Differences in Correlations (0−back to 1−back)
Den
sity
−1.0 −0.5 0.0 0.5 1.0
0.0
0.2
0.4
0.6
0.8
Wavelet−Concatenated
Differences in Correlations (0−back to 2−back)
Den
sity
−1.0 −0.5 0.0 0.5 1.0
0.0
0.2
0.4
0.6
0.8
Wavelet−Concatenated
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
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 8 / 35
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).
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 9 / 35
Part II
Statistical Parametric Networks (SPNs)
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 10 / 35
Statistical Parametric Networks (SPNs)
. . .
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. . .
...
...
...
R11 R12 R1J
Rn1 RnJ
. . . . . .
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 11 / 35
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).
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 12 / 35
Cost/density Decreases with Cognitive Load
Sagittal SPNj
0-back 1-back 2-back 3-back
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).
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 13 / 35
Task-related Physiological Variability
Sagittal SPNj
0-back 1-back 2-back 3-back
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).
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 14 / 35
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).
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 15 / 35
Differential SPNs
. . .
. . .
. . .
. . .
. . .
. . .
...
...
...
R11 R12 R1J
Rn1 RnJ
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 .
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 16 / 35
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).
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 17 / 35
Part III
Differences in Topology vs. Differences in Density
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 18 / 35
Differences in Topology vs. Differences in Density
Regular Random
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 19 / 35
Differences in Topology vs. Differences in Density
Regular Random
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 19 / 35
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.
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 20 / 35
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.
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 21 / 35
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 )|.
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 22 / 35
‘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).
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 23 / 35
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)
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 24 / 35
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).
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 25 / 35
Part IV
Weighted Metrics for Weighted Networks?
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 26 / 35
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)
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 27 / 35
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.
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 28 / 35
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).
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 29 / 35
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).
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 30 / 35
Part V
Some Conclusions.
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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.
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 32 / 35
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.
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 32 / 35
Activity vs. Connectivity
Sepulcre et al. (PLoS CB, 2010).
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 33 / 35
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
Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 34 / 35
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