Geometric separability of mesoscale patterns in embedding representation and visualization of multidimensional data and complex networks
Fig 2
Geometric separability based on projection or first-neighbor strategies.
(a) Example of how the centroid projection line (CPS) or the linear discriminant line (LDPS) would be drawn considering the linear geometric separability of the two community-based groups of network nodes in Fig 3C. The two black dots at the center of the plot are the centroids of the respective groups of nodes. (b) The nodes are projected on the projection separability line and the AUC-PR is computed to evaluate the extent of linear separability between the two groups. The AUC-PR can be substituted by any other bi-class classification measure for unbalanced data. (c) Travelling salesman tour (with the dashed line) and path (without the dashed line) across the points that are the nodes of the nPSO network (Fig 3) embedded in the two-dimensional space (Fig 3C). The travelling salesman (TS) path approximates the projection separability curve that accounts for the intrinsic nonlinear geometry of the data points. (d) The nodes are aligned on the rectified TS path and the AUC-PR is computed to evaluate the extent of separability between the two groups. (e) and (f) are respectively equivalent to (c) and (d) when the labels of the two communities are uniformly at random reshuffled to generate one instance of the null model. (g) The geometric separability index adopts a strategy defined as the proportion of data points whose classification labels are the same as those of their first-nearest neighbor.