This method (Fritzke, 1994b, 1995a) is different from the previously described models since the number of units is changed (mostly increased) during the self-organization process. The growth mechanism from the earlier proposed

- 1.
- Initialize the set to contain two units and

with reference vectors chosen randomly according to .Initialize the connection set , , to the empty set:

- 2.
- Generate at random an input signal according to .
- 3.
- Determine the winner and the second-nearest unit by

and

- 4.
- If a connection between and
does not exist already, create it:

Set the age of the connection between and to zero (``refresh'' the edge):

- 5.
- Add the squared distance between the input signal and the
winner to a local error variable:

- 6.
- Adapt the reference vectors of the winner and its direct
topological neighbors by fractions and ,
respectively, of the total distance to the input signal:

Thereby (see equation 2.5) is the set of direct topological neighbors of . - 7.
- Increment the age of all edges emanating from :

- 8.
- Remove edges with an age larger than . If this results in units having no more emanating edges, remove those units as well.
- 9.
- If the number of input signals generated so far is an integer
multiple of a parameter , insert a new unit as follows:
- Determine the unit
*q*with the maximum accumulated error:

- Determine among the neighbors of
*q*the unit*f*with the maximum accumulated error:

- Add a new unit
*r*to the network and interpolate its reference vector from*q*and*f*.

- Insert edges connecting the new unit
*r*with units*q*and*f*, and remove the original edge between*q*and*f*:

- Decrease the error variables of
*q*and*f*by a fraction :

- Interpolate the error variable of
*r*from*q*and*f*:

- 10.
- Decrease the error variables of all units:

- 11.
- If a stopping criterion (e.g., net size or some performance
measure) is not yet fulfilled continue with step 2.

Figure 5.7 shows some stages of a simulation for a simple ring-shaped data distribution. Figure 5.8 displays the final results after 40000 adaptation steps for three other distribution. The parameters used in both simulations were: , , , , , .

**Figure 5.7:** *Growing neural gas* simulation sequence for a ring-shaped uniform probability distribution. a) Initial state. b-f) Intermediate states. g) Final state. h) Voronoi tessellation corresponding to the final state. The maximal network size was set to 100.

**Figure:** *Growing neural gas* simulation results after 40000 input signals for three different probability distributions (described in the caption of figure 4.4).

Sat Apr 5 18:17:58 MET DST 1997