The average connectivity matrix from 14 subjects after significance thresholding at level is in Figure S1. It has 233 non-zero entries out of possible . The actual connectivity data of all subjects are being included as the binary file “connectivity-matrices.mat” in Supporting Information S2.

Left: Normalized wiring cost and path length of cheaper-than-brain networks, obtained from 10 runs of Algorithm A4. Cheaper wiring comes with reduced path length. The jump in path length is a result of the network splitting into two pieces. Right: Wiring cost, total inter-hemispheric connection weight and average path length of the brain and two contrived networks with cheaper wiring cost. Cheaper wiring is achieved by redirecting inter-hemispheric connections to sub-cortical ones, which results in higher path length. The wiring cost and connection weights are in units of millions.

Top: random networks with the same edge weight distribution as the brain. Middle: random networks with the same weight distribution as well as weighted node degree distribution as the brain. Bottom: random networks with the same weight distribution as well as topology as the brain. Wiring cost of the real brain network is shown by the red vertical bar for comparison.

Figure 1 shows histograms of the wiring cost of random networks, in comparison to the brain wiring cost. Part (a) shows results for random networks with preserved weight histogram, and part (b) for those which preserve both weight histogram as well as degree distribution. The brain's wiring cost is much smaller than almost any random network of either type. However, it is possible to obtain cheaper wiring cost than the brain, using algorithms specifically designed to do so. In Figure 2 we show that wiring cost of such networks, obtained from two such algorithms (see Methods ), can be lower than the brain's. However, cheaper wiring comes at the cost of highly reduced network performance, as measured by average path length, which is much greater for these contrived networks than for the brain. We also observe that this reduction in wiring cost was specifically achieved by redirecting most of the inter-hemispheric connections to sub-cortical connections.

Wiring cost II: Constant connectivity, varying placement

Figure 3 depicts randomly initialized runs of the cost function (2) being minimized by Algorithm A6. As the wiring cost metric is successively reduced, the network embedding is getting closer and closer to the true brain configuration as measured by the brain similarity index, calculated as follows (lower value means more similar to the brain). The similarity index has to be rotation-invariant and insensitive to actual node locations, since our formulation does not require that node locations coincide with the actual brain. Therefore we developed an index based on the “local neighborhood” of each node: We ask how many nearest neighbors seen by each node agree between the computed cortical configuration and the true brain configuration? This allows us to compare the similarity between the two configurations without using a Euclidean distance. We show results for the combined healthy network as well as individual subjects' networks (in Supporting Information S1), after several random initializations. Observe that regardless of initialization, the algorithm converges to the same cost function, and also converges to the same similarity measure. This implies that a strong local minimum, or even the global minimum has been found. The mean, upper and lower quartiles are also indicated, and suggest that the process is consistent and independent of the choice of subjects or initialization.

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larger image TIFF original image Download: Figure 3. Convergence performance of wiring cost minimization algorithm. Part (a) shows cost function of combined healthy network after random starting configurations, (b) shows cost over individual subjects' networks, after 10 random initializations, and (c) shows the mean (bold curve), upper and lower quartiles (dotted lines) of (b). Both the wiring cost (in blue, (2)) and a measure of similarity to the brain anatomic configuration (in red) are shown. The y-axis is in arbitrary units, after normalizing each quantity by the value at the initial random configuration. Note that although the algorithm only minimizes for wiring cost, the similarity measure is also getting optimized. https://doi.org/10.1371/journal.pone.0014832.g003

These results indicate that the individual subjects' network analysis gives similar results to the combined healthy group network. However, we need to show the effect of sample size on the group network computation. The statistically established technique for approaching these problems is via rigorous bootstrap sampling. Basically, we want to answer the question: how reliable is the average network, and how would our results change if a smaller or different sample was used? These questions can be answered by repeated resampling with replacement, i.e. bootstrap technique. We implemented this technique in MATLAB, once for each iteration of Algorithm A6, and report the mean objective function (blue curve) as well as the % confidence interval around it (red curve). This is shown in Figure S4, and demonstrates that the objective function for the average network is in fact a very robust and consistent result, which simply could not have arisen due to chance.

As a final result regarding wiring cost minimization, we performed a pairwise t-test to see whether the starting configuration, which is random, is statistically different from the ending configuration. The test returned a p-value of for the wiring cost and for the similarity measure, indicating that both quantities are highly statistically significant.

Figure 4 shows a point cloud of the centroids of cortical and subcortical regions mapped onto the unit sphere for the real brain (left) and the wiring-optimal configuration (right). The points are color coded by lobe and their size denotes node strength. The view is an approximately coronal projection of the brain. Notice how the points tend to locate themselves in roughly the same configurations as the brain, where points belonging to each lobe tend to stay together. The lobes are anatomically correct in relation to each other, although there appear significant variations within the lobes. As a point of comparison, we implemented the exact solution proposed in [11] for this problem in terms of the small eignevectors of the graph Laplacian (see Related Work in Discussion) - this is also presented in the figure. It can be seen that the results of the eigenvector approach are disappointing, and lead to the nodes clustering in s few collinear clusters.

Figure 5 shows 3D rendered surfaces of the unit sphere for the brain as well as for the optimally-placed nodes after running the minimization algorithm described in Methods. The surface is color coded by the lobe to which each point belongs. This view is approximately coronal, showing the parietal lobes in each hemisphere, with both the parieto-frontal (above) and the parieto-occipital (below) interfaces visible. Note the striking similarity in appearance of the two configurations. In figure 6 the same surfaces are shown in a sagittal view. In figure 7 the same surfaces are shown, but this time color coded by each individual cortical regions. The view is approximately the same as above, showing regions in the parietal lobe and adjoining regions in the frontal and occipital lobes. Again note the similarity in appearance. In Figure S2 another example is shown, starting from a different random configuration, but producing remarkably similar optimal surface. We repeated this analysis multiple times, and observed the same tendency each time.

In order to assess whether changes in connectivity and topology can affect the anatomical placement of regions, Figure 8 shows results for a slightly modified network, where we randomly rewired % of brain connections. The resulting “optimal” surface has no resemblance to the brain sphere map, implying that connectivity determined anatomy. In Figure S3 we show another example of the perturbed connectivity matrix with % rewiring.