## 2 Conventional SOM, Its Selected Generalizations, and Related Work
### Incremental Grid Growing (IGG) Approach  
IGG method is able to increase the number of neurons in the map (however, only in the areas of the map perimeter and not \"inside\" the map) as well as to remove or add some topological connections in the map [9]. Figure 2 (Initialization-, WTM-algorithm-, and IGG-blocks) presents further extension of the conventional-SOM learning algorithm for IGG approach. This extension consists in modification of the Initialization block (in a similar way as for GSOMs and GGNs, i.e., by setting the initial dimensions of the map to \(m_{1}\times m_{2}=2\times 2\)) and adding the three-module IGG block, which is activated after each learning epoch. The first module of the IGG block controls the process of adding new neurons into the area of the map perimeter. The second module governs the removal of existing connections between neighboring neurons located far apart in the data space. The third module manages the adding of new connections between neighboring neurons located close to each other in the data space.  
The main idea behind the IGG approach is to build and organize the map in an incremental way. Such an approach initially organizes a small number of neurons in the structure. Next, after each learning epoch, the Euclidean distance between unconnected neighboring neurons in the map is examined. If the distance is below the \"connect\"-control parameter, a connection between the neurons is introduced. In an analogous way, using the \"disconnect\"-control parameter, existing connections between neighboring neurons located far apart in the data space are removed. It is also possible to add some neurons at the perimeter areas that inadequately represent the data.  
Figures 12 and 13 illustrate the application of IGG approach to the synthetic data sets of Figures 5a and 6a, respectively. The evolution of the number of IGG's substructures as the learning progresses is shown in Figures 12j and 13j for particular data sets. The number of substructures is equal to the number of detected clusters. Similarly as for GSOMs and GGNs, there are two stages in the map formation. In the first stage, the map concentrates on \"unfolding\" its structure by an intensive increase of the number of neurons (see Figures 12k and 13k) without disconnecting its structure into substructures (see Figures 12j and 13j). In turn, in the second stage, the map with a relatively stable number of neurons concentrates on its essential task, i.e., detecting clusters in data by disconnecting its structure into substructures representing particular clusters. It is evident that IGG much better copes with simple data of Figure 5a, finally dividing its structure into six substructures (including one very small) representing five clusters. However, IGG completely fails in the case of more complex data of Figure 6a. It is clear that IGG lacks the ability to reduce the number of neurons in the network and to reconnect some substructures representing different part of a given cluster. It results in an \"explosion\" of separate, small substructures in the final part of the learning process (see Figure 13j) and completely false image of the cluster distribution in the considered data set.