# An Algorithm to Generate a Weighted Network Voronoi Diagram Based on Improved PCNN
## 4 Network Voronoi Diagram Construction Based on the Improved PCNN
### Transmission of the Auto-Wave
#### 4.2.2 Transmission of the Auto-Wave  
In the improved PCNN, the auto-wave which is generated simultaneously by initial neurons will expand along the shortest path between neurons with the above-calculated speed, until all the neurons are fired and the transmitting auto-wave stops. The concrete steps are described as follows:  
1. Initialization of networks' parameters The initial iteration number is set as \(n=0\). The projected points in point cluster and all the nodes in road network are regarded as neurons, and the projected points are chosen as initial neurons (\(b_{m}\)) whose initial state is set to be fired (\(Y_{bm}\) [0] = 1). The initial neurons are regarded as the mother neurons of other neurons (\(par(p)=b_{m}\)). \(V_{E}\) is a constant satisfying condition: \(V_{E}>(n-1)\)\(W_{max}\), in which \(N\) is the total number of the neurons, and \(W_{max}\) represents the maximum value of the connection weight between neurons. The initial dynamic threshold of iteration is set as \(E_{P}\) [0] = \(V_{E}\), \(Y_{P}\) [0] = 0, \(S\) [0] = 0, \(\triangle S\) [0] = 0. 2. Operation of the improved PCNN Step1: Dynamic threshold of iteration (\(E_{P}\)[\(n\)]) is calculated by the Formula (8). Step2: The transmission speed of the auto-wave (\(L_{Bn}\)) is updated according to Formula (16), and the current wave velocity is set as \(\triangle S[n]=L_{Bn}\). Step3: The current wavelength is calculated by Formula (9) as: \(S[n]=S[n-1]+\triangle S[n]\). Step4: The current dynamic threshold (\(E_{P}\)[\(n\)]) is compared with the current wavelength (\(S[n]\)). If \(S[n]\geq E_{P}\)[\(n\)], then \(Y_{P}\)[\(n\)] = 1, which means neuron \(P\) is fired successfully in the \(n\)th iteration. Otherwise, \(Y_{P}\)[\(n\)] = 0 and the current neuron \(P\) is not fired yet. Step5: The connection weight \(W_{ap}\)[\(n+1\)] is updated by Formula (11). Step6: Turn to step 1 to repeat the above steps, until all the neurons are fired and the transmitting auto-wave stops. The initial state of the points and the road network is shown as Figure 7a. If the weights of the point and the road segments are ignored, which means all the points have the same importance and all the road segments are two-way and have the same importance, the end state of auto-wave transmission is shown as Figure 7b. When the weights of the points and the road networks are all considered and the auto-wave transmits with different speed on different road segments, the end state is shown in Figure 7c. In Figure 7c, the auto-wave generated from the projection points expands along road segments at the speed calculated by Formula (9). It will transmit along the shortest path between neurons until all the neurons are fired. In Figure 7b,c, the black dotted lines represent the road segments which were not passed by the auto-wave. The appearance of this type of road segment is because the auto-wave travels along the shortest path between neurons in the  
Figure 6: The speed of the auto-wave under constraints. improved PCNN. As soon as the neurons ahead are fired (i.e., the shortest path toward the current neurons is found already), the current auto-wave that travels toward them will stop. For example, in Figure 7c, the auto-wave generated from point \(P_{3}{}^{\prime}\) (marked in purple) stops at point \(r_{2}\), because at that very moment the neuron ahead of the auto-wave (\(r_{2}\)) is fired by another auto-wave (marked in red). In the end, \(r_{1}\)\(r_{2}\) is the road segment that is never reached by any auto-wave. This kind of road segment is assigned to two initial neurons if the two nodes of the road segment are fired by auto-waves generated from two initial neurons (\(r_{1}\)\(r_{2}\) for example). Otherwise, the road segments are assigned to the only initial neurons (\(r_{3}\)\(r_{4}\) for example). According to this method, these untouched road segments are assigned to the corresponding neurons, and the result is as shown in Figure 8. It can be seen that the road segments (marked in different color) and the corresponding space are assigned to different points.