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
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Connectivity Data
Subject-specific Correlation Matrices
For the i th subject in the j th condition: Rij .
AAL Cortical Regions
AAL Cortical Regions
Cedric E. Ginestet (KCL)
Weighted Network Analysis
24th January 2012
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Connectivity Data
Experimental Paradigm
J conditions (columns), and n subjects (rows).
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Part I
N-back Task on Working Memory
Cedric E. Ginestet (KCL)
Weighted Network Analysis
24th January 2012
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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
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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◦ .
Cedric E. Ginestet (KCL)
Weighted Network Analysis
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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
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0
−4
W3
2
4
Wavelet Decomposition + Concatenation
0
10
20
30
40
30
40
0.0 0.5 1.0
−1.0
W3
Concatenated Volumes
0
10
20
Concatenated Volumes
Cedric E. Ginestet (KCL)
Weighted Network Analysis
24th January 2012
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Wavelet Decomposition + Concatenation
0.0
0.5
1.0
1.5
Density
0.5
0.0
0.5
0.0
−0.5
1.0
2.0
1.5
1.0
Density
1.5
1.0
−1.0
−0.5
0.0
0.5
1.0
−1.0
−0.5
0.0
0.5
1.0
Differences in Correlations (0−back to 2−back)
Differences in Correlations (0−back to 3−back)
Wavelet−Concatenated
Wavelet−Concatenated
Wavelet−Concatenated
−1.0
−0.5
0.0
0.5
1.0
Differences in Correlations (0−back to 1−back)
Cedric E. Ginestet (KCL)
0.8
Density
0.4
0.0
0.0
0.2
0.4
Density
0.6
0.8
1.2
Differences in Correlations (0−back to 1−back)
0.0 0.2 0.4 0.6 0.8
Density
0.5
0.0
−1.0
Density
Concatenation Only
2.0
Concatenation Only
2.0
Concatenation Only
−1.0
−0.5
0.0
0.5
1.0
Differences in Correlations (0−back to 2−back)
Weighted Network Analysis
−1.0
−0.5
0.0
0.5
1.0
Differences in Correlations (0−back to 3−back)
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Differences in Cost/Density
Main Effect of N-back Experimental Factor?
0-back
1-back
0.0
0.2
2-back
0.4
0.6
0.8
3-back
1.0
Figure: Heatmaps corresponding to subject-specific correlation matrices for the four
N-back conditions. (Ginestet et al., Neuroimage, 2011).
Cedric E. Ginestet (KCL)
Weighted Network Analysis
24th January 2012
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Part II
Statistical Parametric Networks (SPNs)
Cedric E. Ginestet (KCL)
Weighted Network Analysis
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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).
Cedric E. Ginestet (KCL)
Weighted Network Analysis
24th January 2012
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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 in
the 0.01–0.03Hz frequency band. The locations of the nodes correspond to the
stereotaxic centroids of the cortical regions (Ginestet et al., Neuroimage, 2011).
Cedric E. Ginestet (KCL)
Weighted Network Analysis
24th January 2012
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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-correlated
physiological 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
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Connectivity Strength Predicts Task Performance
1800
(a) Penalized RT
(b) Weighted Cost
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1200
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0.2
600
800
pRT(ms)
1400
0.8
1600
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0−back 1−back 2−back 3−back
0−back 1−back 2−back 3−back
Figure: Boxplots of (a) penalized reaction time and (b) weighted cost. Regression of
pRT 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
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Differential SPNs
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F-test
for all e ∈ E (G ), v ∈ V (G ):
e
ri = Xei β e + Zei bei + ei ;
yvi = Xvi β v + Zvi bvi + vi .
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Cedric E. Ginestet (KCL)
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Differential SPNs
L
R
Figure: Differential SPN. Sagittal section of the negative differential SPN, which
represents the significantly ‘lost’ edges, due to the N-back experimental factor. The
presence 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
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Part III
Differences in Topology vs. Differences in Density
Cedric E. Ginestet (KCL)
Weighted Network Analysis
24th January 2012
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Differences in Topology vs. Differences in Density
Regular
Random
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Differences in Topology vs. Differences in Density
Regular
Random
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Classical Measures of Topology
Efficiencies (Latora et al., 2001)
For any unweighted graph G = (V, E), connected or disconnected,
N
N
V X
V
X
1
dij−1 ,
E (G ) :=
NV (NV − 1)
(1)
i=1 j6=i
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
NV
1 X
(G ) :=
E (Gi ),
NV
(2)
i=1
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
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Efficiencies are Monotonic Increasing with Density
0.8
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0.2
0.0
0−back
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3−back
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3−back
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1.0
(b) Lobal Efficiency
1.0
(a) Global Efficiency
0.0
Cost
0.2
0.4
0.6
0.8
1.0
Cost
Figure: Efficiencies under the four conditions of the N-back task, with density-equivalent
random (red) and regular (blue) networks, for each condition.
Cedric E. Ginestet (KCL)
Weighted Network Analysis
24th January 2012
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Integrating over Densities
Cost-integrated Topological Metrics
Given a weighted graph G = (V, E, W) and a topological metric T (·),
X
Tp (G ) :=
T (γ(G , k))p(k),
(3)
k∈ΩK
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
NV
Ωk := 1, . . . ,
,
(4)
2
with NV := |V(G )|.
Cedric E. Ginestet (KCL)
Weighted Network Analysis
24th January 2012
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‘Prior Distribution’ over Graph Densities
6e−04
Beta−Binomial Distribution
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Ne
Figure: Symmetric versions of the Beta-binomial distribution for different choices of
parameters, with NE = 4005 (Ginestet et al., PLoS one, 2011).
Cedric E. Ginestet (KCL)
Weighted Network Analysis
24th January 2012
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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 set
W, 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
N
Rij (h(W)) =
N
V X
V
1X
I{h(wij ) ≥ h(wuv )} = Rij (W).
2 u=1
(6)
v 6=u
Cedric E. Ginestet (KCL)
Weighted Network Analysis
24th January 2012
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Topological Differences does not Predict Performance
(b) Local Efficiency
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0.75
0.60
E(Glo)
E(Loc)
0.65
0.80
0.70
(a) Global Efficiency
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0.70
0.55
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0−back 1−back 2−back 3−back
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0−back 1−back 2−back 3−back
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 in
the j th condition (Ginestet et al., Neuroimage, 2011).
Cedric E. Ginestet (KCL)
Weighted Network Analysis
24th January 2012
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Part IV
Weighted Metrics for Weighted Networks?
Cedric E. Ginestet (KCL)
Weighted Network Analysis
24th January 2012
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Weighted Topological Metrics
Weighted Global Efficiency
As introduced by Latora et al. (2001),
N
EW (G ) :=
N
V X
V
X
1
1
.
NV (NV − 1)
dijW
(7)
i=1 j6=i
where G is a weighted graph, G = (V, E, W).
Weighted Shortest Path
The weighted shortest path dijW is defined as
dijW :=
Cedric E. Ginestet (KCL)
min
Pij ∈Pij (G )
X
−1
wuv
,
(8)
wuv ∈W(Pij )
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24th January 2012
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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 ), if
we have
1
min wij ≥
max wij ,
(9)
2 wij ∈W(G )
wij ∈W(G )
then
EW (G ) = KW (G ).
(10)
Proof.
Assume that dijW 6= wij−1 for at least one edge (i, j), and then show that this
contradicts the hypothesis.
Cedric E. Ginestet (KCL)
Weighted Network Analysis
24th January 2012
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Modularity & Edge Density
A
B
10
Networks
Number of Modules
Number of Modules
6
4
2
0
Random
Regular
8
6
4
2
0
0
200
400
600
Random Rewirings
800
100
600
1100 1600 2100
Number of Edges
2600
3100
Figure: Topological randomness and number of edges predict number of modules.
(A) Relationship between the number of random rewirings of a regular lattice and the
number of modules in such a network. (B) Relationship between the number of edges in
a network and its number of modules (Bassett et al., PNAS 2011).
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Modularity & Edge Density
C
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NE = 100
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Figure: Topological randomness and number of edges predict number of modules.
Modular structures of regular (C) and random (D) networks for different number of
edges, NE (Bassett et al., PNAS 2011).
Cedric E. Ginestet (KCL)
Weighted Network Analysis
24th January 2012
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Part V
Some Conclusions.
Cedric E. Ginestet (KCL)
Weighted Network Analysis
24th January 2012
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Summary
Main Messages
1
2
Thresholding: Discrete mathematics on Continuous (Real-valued) data.
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.
Cedric E. Ginestet (KCL)
Weighted Network Analysis
24th January 2012
32 / 35
Summary
Main Messages
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Thresholding: Discrete mathematics on Continuous (Real-valued) data.
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.
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Cost-integration approximated using Monte Carlo sampling scheme.
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R package for cost-integration: NetworkAnalysis on CRAN.
Future Work
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Replicate these findings in other MRI cognitive tasks.
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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
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Andy Simmons, Mick Brammer, Andre Marquand, Vincent Giampietro, Orla
Doyle, Jonny O’Muircheartaigh, Owen G. O’Daly (King’s College London)
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Arnaud Fournel (Lyon, France)
3
Ed Bullmore (Cambridge, UK)
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Tom Nichols (Warwick, UK)
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Randy Buckner (Harvard, MA)
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Dani Bassett (UCLA, CA)
Funding Agencies
Cedric E. Ginestet (KCL)
Weighted Network Analysis
24th January 2012
34 / 35