Figure 3

Unfiltered network of significant transitions. Here, we show the network of all significant ($s>0.5$) transitions between variables from the data of Grade 5 Mathematics. The network is drawn with the following features: (i) node diameters are proportionate to the prior probability $P_i$ of their corresponding variables, (ii) arrowheads of the edges point in the direction from the precedent to the consequent, (iii) edges thicknesses are proportionate to the weighted frequency with thick lines denoting high weighted transition frequencies, and (iv) the placement of the nodes are determined by a multilevel force algorithm [32] that was included in the graphing software. This algorithm assigns attractive and repulsive forces among nodes according to their connectedness and allows the graph to dynamically relax into a configuration of low energy. The multilevel treatment ensures that the graph relaxes into the global minimum energy configuration instead of being trapped in a local minimum. In this scheme, the distances between nodes are roughly inversely proportionate to the score between them i.e. a pair of nodes with high score will be placed closed together in the network. For this unfiltered network, the density of edges is so great that it is very difficult to discern any meaningful structure from it.