Random parcellations have some advantages over template-based parcellations, such as enabling multiscale network analysis that is not constrained by a template or comparing cohorts with different brain sizes^1,2,3. An important criterion for assessing the “goodness” of a random parcellation is the parcel size variability. Variations of parcel sizes introduce an additional dimension of variability in characterizing the network, hence, roughly equally-sized parcels are desired. Previous work has achieved an inter-quartile range to median ratio of around 0.5 using an expectation–maximization algorithm, however the number of parcels deviated from the expected number⁴. In this work we propose an algorithm that generates an exact prespecified number of parcels, while also achieving reduced parcel-size variability. We tested our approach on the cortical surface of one subject and constructed corresponding connectomes for multiple instantiations of random parcellations.

Parcellations are obtained by growing voxel neighborhoods around a set of randomly selected voxel-seeds, as described previously⁵. Because the cortical surface is very irregular, using Euclidean distance as a measure to ensure that seeds are evenly placed throughout the cortical surface results in large parcel-size variability. Here, we introduce a new distance metric D(i,j), which is proportional to the length of the shortest path between voxels i and j, where such path is restricted to traversing voxels within the gray matter surface. We define path-lengths between voxels by representing each gay matter voxel as a node that is connected to its spatially contiguous neighbors. Connection weights between adjacent voxels are defined as follows: $$$w_{ij}=1$$$ if voxels i and j share a face; $$$w_{ij}=\sqrt{2}$$$ if i and j share a side; $$$w_{ij}=\sqrt{3}$$$ if i and j share a vertex. Thus, D(i,j) is defined as$$D(i,j)=\frac{2w_{ij}}{L(i)+L(j)}$$

where L(i) is the sum of shortest-path lengths between voxel i and its M nearest neighbors, and M is defined as the expected number of voxels within a parcel, given a specified number of parcels N.

The parcellation algorithm is implemented in Matlab. Here we show results for N = 125, 250, 500, and 1000 parcels; we ran 200 repetitions for each value of N, except for N = 250 nodes, where 600 repetitions were run. Structural networks were constructed for each parcellation using the method described in 6. Network metrics computed were degree, clustering coefficient, betweenness centrality, and global efficiency.

An example of a parcellation and the corresponding parcel-size distributions for N = 250 nodes are shown in Fig. 1. The ratio of standard deviation to the mean parcel size is 8.4%. Across all 600 trials, 95% of the parcel-sizes are between 291 voxels and 413 voxels, and 99% of the parcel-sizes are between 257 voxels and 434 voxels. If we define the normalized maximum variation (NMV) as the difference between the largest and smallest parcels in a parcellation, divided by the smallest parcel size, the NMV is 38.8% with a mean value of 79.3% across 600 repetitions, with 87.7% resulting in NMV of less than 100%. Table 1 summarizes some features of the distributions obtained for different values of N.

The distributions of some network metrics from 600 trials of the random parcellations with N = 250 are shown in Fig. 2. A Lilliefors test showed that the distributions are not significantly different from a normal distribution. The standard deviation to mean ratio is 1.7% for mean degree, 3.6% for mean clustering coefficient, 4.2% for mean betweenness centrality, and 2.94% for global efficiency.

Small parcel-size variability is critical to ensure the construction of structural networks whose topological properties are reliable across several parcellation trials. We propose a random parcellation algorithm that can produce any given number of parcels with a small parcel-size variability. The inter-quartile range to median ratio is around 0.10, which is significantly smaller than achieved in previous work, e.g. 0.77 (Fornito, parc890), 0.52 (Echtermeyer, parc813). While global network properties vary across different repetitions of the random parcellation, we find that the variability is small and comparable with inter-subject variability. Given the current lack of a standard parcellation scheme, random parcellation schemes such as the one introduced here continue to be a plausible method to construct human connectomes for performing network analysis.