Weighted Network Analysis for Groups:

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: R ij. AAL Cortical Regions AAL Cortical Regions Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 2 / 35

Connectivity Data Experimental Paradigm J conditions (columns), and n subjects (rows). R 11 R 12 R 1J............... R n1 R nj...... 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

Wavelet Decomposition + Concatenation W3 4 0 2 4 0 10 20 30 40 Concatenated Volumes W3 1.0 0.0 0.5 1.0 0 10 20 30 40 Concatenated Volumes Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 7 / 35

Wavelet Decomposition + Concatenation Concatenation Only Concatenation Only Concatenation Only Density 0.0 0.5 1.0 1.5 2.0 Density 0.0 0.5 1.0 1.5 2.0 Density 0.0 0.5 1.0 1.5 2.0 1.0 0.5 0.0 0.5 1.0 Differences in Correlations (0 back to 1 back) 1.0 0.5 0.0 0.5 1.0 Differences in Correlations (0 back to 2 back) 1.0 0.5 0.0 0.5 1.0 Differences in Correlations (0 back to 3 back) Wavelet Concatenated Wavelet Concatenated Wavelet Concatenated Density 0.0 0.2 0.4 0.6 0.8 Density 0.0 0.2 0.4 0.6 0.8 Density 0.0 0.4 0.8 1.2 1.0 0.5 0.0 0.5 1.0 Differences in Correlations (0 back to 1 back) 1.0 0.5 0.0 0.5 1.0 Differences in Correlations (0 back to 2 back) 1.0 0.5 0.0 0.5 1.0 Differences in Correlations (0 back to 3 back) 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 four N-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) R 11 R 12 R 1J............... R n1 R nj............ 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 SPN j 0-back 1-back 2-back 3-back Figure: Mean Statistical Parametric Networks (SPN j ), 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 13 / 35

Task-related Physiological Variability Sagittal SPN j 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 14 / 35

Connectivity Strength Predicts Task Performance (a) Penalized RT (b) Weighted Cost prt(ms) 400 600 800 1000 1200 1400 1600 1800 K(G) 0.2 0.4 0.6 0.8 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 (K W (G ij ) 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 R 11 R 12 R 1J............... F-test for all e E(G), v V (G): r e i = X e i βe + Z e i be i + ɛ e i ; y v i = X v i βv + Z v i bv i + ɛ v i. R n1 R nj...... 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, 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 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

Classical Measures of Topology Efficiencies (Latora et al., 2001) For any unweighted graph G = (V, E), connected or disconnected, E(G) := 1 N V (N V 1) N V N V i=1 j i d 1 ij, (1) where d ij 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 N V N V i=1 E(G i ), (2) where G i 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 (a) Global Efficiency (b) Lobal Efficiency E (Glo) 0.0 0.2 0.4 0.6 0.8 1.0 0 back 1 back 2 back 3 back E (Loc) 0.0 0.2 0.4 0.6 0.8 1.0 0 back 1 back 2 back 3 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 Cost 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 21 / 35

Integrating over Densities Cost-integrated Topological Metrics Given a weighted graph G = (V, E, W) and a topological metric T ( ), T p (G) := k Ω K 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 { ( )} NV Ω k := 1,...,, (4) 2 with N V := V(G). Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 22 / 35

Prior Distribution over Graph Densities Beta Binomial Distribution p(k=k) 0e+00 2e 04 4e 04 6e 04 0 1000 2000 3000 4000 n=ne a=b=1 a=b=2 a=b=3 a=b=4 a=b=5 Ne Figure: Symmetric versions of the Beta-binomial distribution for different choices of parameters, with N E = 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 set W, and any topological metric T, the cost-integrated version of that metric, denoted T p, satisfies T p (W) = T p (h(w)). (5) Proof. Since h( ) is applied elementwise to W, we have R ij (h(w)) = 1 2 N V N V I{h(w ij ) h(w uv )} = R ij (W). (6) u=1 v u Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 24 / 35

Topological Differences does not Predict Performance (a) Global Efficiency (b) Local Efficiency E (Glo) 0.55 0.60 0.65 0.70 E (Loc) 0.70 0.75 0.80 0 back 1 back 2 back 3 back 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 G ij 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 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), E W (G) := 1 N V (N V 1) where G is a weighted graph, G = (V, E, W). N V N V i=1 j i 1 dij W. (7) Weighted Shortest Path The weighted shortest path d W ij is defined as d W ij := min P ij P ij (G) w uv W(P ij ) w 1 uv, (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), if we have min w ij 1 max w ij W(G) 2 w ij, (9) w ij W(G) then E W (G) = K W (G). (10) Proof. Assume that dij W w 1 ij 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 28 / 35

Modularity & Edge Density A Number of Modules 6 4 2 B Number of Modules 10 8 6 4 2 Networks Random Regular 0 0 0 200 400 600 800 Random Rewirings 100 600 1100 1600 2100 2600 3100 Number of Edges 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). Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 29 / 35

Modularity & Edge Density C N E = 100 N E = 600 N E = 1100 N E = 1600 N E = 2100 D N E = 100 N E = 600 N E = 1100 N E = 1600 N E = 2100 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, N E (Bassett et al., PNAS 2011). Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 30 / 35

Part V Some Conclusions. Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January 2012 31 / 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. 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, Orla Doyle, 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