Fig 1.
Illustration of pairwise vs partial correlation networks.
Thicker edges represent stronger absolute correlations. Left: true network of partial correlations (blue), with 8 connections, no triangles. Middle: associated pairwise correlation network, with erroneous direct connections (red) that form 84 triangles. Right: pruned network of 8 strongest pairwise correlations, with two isolated nodes (yellow) and two erroneous connections (red) that form 2 triangles (2-3-8 and 3-7-8). Comparing the true partial correlation network on the left with the pruned pairwise correlation network on the right, which consists of the same number of edges as the underlying network, three differences stand out. Firstly, indirect connections may appear as direct connections (i.e., nodes 2–8 and nodes 3–7). This results in an excessive number of triangles, affecting network measures such as small-worldness. Secondly, while the true network is connected (i.e., there exists a path between each pair of nodes), pruned pairwise correlation networks tend to consist of isolated (groups of) nodes (i.e., nodes 1 and 9). Thirdly, the number of connections of a node may differ from the true number of connections (e.g., node 3 has four instead of three edges). In larger networks, hub nodes may emerge erroneously.
Fig 2.
Exemplary network with path from node 1 to 5, showing partial covariances γij.
Fig 3.
Recovery of degree distributions based on 500 observations.
Densities of the true (black) and recovered node degrees of shrinkage (blue), ridge (orange), and lasso (green) estimated partial correlations, and of pairwise correlations (red). NB: x-axis cut off.
Table 1.
Characteristics of simulated networks.
Fig 4.
The small-worldness index for the four networks and the four estimation methods pairwise correlations (red), lasso (green), ridge (orange), and shrinkage (blue), compared to the true value −− (black).
The thickness of the line represents the number of selected edges. Pairwise correlation networks always overestimate the small-worldness.
Fig 5.
Clustering coefficient (upper) and average pathlength (lower) for the four networks and estimation methods.
Fig 6.
The number of components (upper) and the size of the largest component (lower) obtained for the four networks and estimation methods.
Fig 7.
The mean betweenness centrality of the four networks and estimation methods.
Fig 8.
Recovery of node degrees based on 10000 observations.
Scatter plots of true (x-axis) vs recovered (y-axis) node degrees of shrinkage (blue), ridge (orange), and lasso (green) estimated partial correlations, and of pairwise correlations (red).
Fig 9.
Overview of absolute differences between true and recovered network characteristics of shrinkage (blue), ridge (orange), and lasso (green) estimated partial correlations, and of pairwise correlations (red), in the condition where the correct number of edges is selected.
For node characteristics (i.e., degree, strength, and betweenness), sums of absolute differences of linearly transformed variables x* are shown (; i.e., 0 was mapped on the minimum of the true variable, and 1 was mapped on the maximum of the true value). SWI = Small-worldness index, CC = Clustering coefficient, APL = Average path length, #Comp = Number of components, n = Number of observations. NB: x-axis on logarithmic scale; if absolute difference is zero, the method’s symbol is not shown.
Fig 10.
The false positive rate for the four networks and estimation methods.
The dotted line ⋯ shows the level of the false positive rate above which the absolute number of false positive edges even exceeds the absolute number of edges in the true network.
Fig 11.
The true positive rate for the four networks and estimation methods.
Fig 12.
Overview of TPR and of 1 − f(FPR) for shrinkage (blue), ridge (orange), and lasso (green) estimated partial correlations, and for pairwise correlations (red), averaged over all three selection criteria (i.e., correct number of edges, 20% less edges, and 20% more edges).
f(FPR) = exp(−102*FPR); n = Number of observations.
Fig 13.
True positive rate as a function of node degree (given 10000 observations) of shrinkage (blue), ridge (orange), and lasso (green) estimated partial correlations, and of pairwise correlations (red).
For each network, nodes were divided into 6 bins according to degree: 5 equally-sized bins, and a 6th bin containing the 50 nodes with the highest degree (i.e., the hubs in the hub networks). TPR is shown for each pairing of degree bins (e.g., sixteenth pair refers to edges between the nodes with lowest degrees and the nodes with highest degrees; rightmost pair
refers to edges between the nodes with highest degrees).
Fig 14.
False positive rate as a function of node degree (given 10000 observations) of shrinkage (blue), ridge (orange), and lasso (green) estimated partial correlations, and of pairwise correlations (red).
For each network, nodes were divided into 6 bins according to degree: 5 equally-sized bins, and a 6th bin containing the 50 nodes with the highest degree (i.e., the hubs in the hub networks). FPR is shown for each pairing of degree bins (e.g., eleventh pair (1,6) refers to edges between the nodes with lowest degrees and the nodes with highest degrees; rightmost pair (6,6) refers to edges between the nodes with highest degrees).
Fig 15.
Recovery results of the four estimation methods for additional random network (with the high density of 3%).
True −− network metrics indicated where appropriate. The dotted line ⋯ shows the level of the false positive rate, above which the absolute number of false positive edges even exceeds the absolute number of edges in the true network.
Fig 16.
Networks of 68 ROIs based on 3% strongest partial correlations (blue) and pairwise correlations (red) of all 5 participants.
Left hemisphere is on left side. ROIs with larger nodes have higher betweenness centralities. Networks are superimposed on transverse MNI152 T1 template for illustration purposes (Copyright (C) 1993–2004 Louis Collins, McConnell Brain Imaging Centre, Montreal Neurological Institute, McGill University). Figure prepared with the R-package qgraph.
Fig 17.
Overlap between networks at different numbers of ROIss (parcellations).
Dashed black lines −− show the proportion of edges that were present both in the pairwise and in the partial correlation network of a given parcellation. Separate lines for each participant (numbered 1–5). Blue (or red) lines show the comparison of the base-line 68-ROI parcellation with higher-resolution parcellations for pairwise correlation (red) networks (or partial correlation (blue) networks). Plain blue (or red) lines − show the proportion of areas of low-resolution parcellation that were internally connected by at least one edge in the higher-resolution parcellations, given that the area was split (within-area connectivity). Dotted blue (or red) lines … show the proportion of areas that were inter-connected in the low-resolution parcellation, that were also inter-connected by at least one edge in the higher-resolution parcellations (between-area connectivity).
Fig 18.
Overlap between networks at different numbers of volumes (i.e., time-series lengths).
Shown is the proportion of identical edges present in two respective networks. Black lines −− show overlap between the pairwise correlation network and the partial correlation network of a participant, based on a given number of volumes (i.e., time-series length). Separate lines for each participant (numbered 1 − 5). Red (or blue) lines indicate overlap between the pairwise correlation (red) (or partial correlation (blue)) network based on the full time-series of 240 volumes and the pairwise correlation (red) (or partial correlation (blue)) network based on smaller numbers of volumes (i.e., shorter time-series length.
Fig 19.
Global network metrics of interest of pairwise correlation (red) and partial correlation (blue) networks for different numbers of ROIs (parcellations).
Numbered lines for participants 1 to 5.
Fig 20.
Local transitivity of left (L) and right (R) hemisphere ROIs in pairwise correlation (red) and partial correlation (blue) networks with 68 ROIs, averaged over participants.
Fig 21.
Betweenness centrality of left (L) and right (R) hemisphere ROIs in pairwise correlation (red) and partial correlation (blue) networks with 68 ROIs, averaged over participants.