Figure 3.

Two-dimensional toy example of the null model construction process for a 10-bone network intentionally connected using the Gabriel rule, in order to show the validity of the approach. (a) From an initial network model (left diagram) we measure pairwise bone similarity by their TO, which estimates common growth co-dependences among bones. A non-metric multidimensional scaling of these similarities yields a predicted virtual position for each bone (central diagram). Then, the Gabriel rule is applied to each pair of bones to establish whether they are connected or not; gray circles indicate how to apply the Gabriel rule to evaluate, for example, if bone b will be connected to bone c and/or d. In this case, bone b connects to bone c because there are no other bones within the circle; in contrast, bone b does not connect to bone d because bones c and g are situated within the circle. Thus, the Gabriel rule can be interpreted as the result of unconstrained bone growth (i.e. uniform rate, direction and timing) from the virtual position assigned to each bone. Finally, we obtain a null network model (right diagram), which can be compared with the actual network of the left diagram. Note that the dimensions of the virtual positions do not necessarily have the same scale and orientation as in the actual space, but all connections have been predicted correctly, as expected. Solid blue lines represent bone articulations predicted in the NNM; the dashed red line represents an example of a failure to establish a connection by the Gabriel rule. (b) Sequence illustrating the model of unconstrained growth in the vicinity of bone c; connections between two bones are established when the growing fronts meet. Nodes b, c, d and g in (b) refer to the same nodes in (a).