## 3 Preliminaries
### Distribution Function and Conductivity Value  
Temperature transfer formula considers several dimensions of network in one value and we can examine its behavior by simplification. At first, this formula considers distance from the sink. This can be clearly shown by setting thermal conductivity a constant value for all sensor nodes (\(\xi_{1}=\xi_{i}=const\)). Another dimension is the number of neighbors, thus density of the network. It can be shown that the more neighbors with the higher temperature, the higher temperature of the node itself. When conductivity value is very close to zero (\(\xi=const\approx 0\)), the density of the network has very significant impact on the temperature value. Vice versa, if the thermal conductivity value is close to 1 (\(\xi=const\approx 1\)), the distance from the data sink (number of hops) has significant impact on the temperature. Thus when the sensor nodes, which are closer to the data sink, are in good conditions (their conductivity values are close to 1), then the shortest path is preferred as the most optimal one. When those nodes are in bad conditions (their conductivity values are close to 0), more alternative paths can be found (for detailed example see description of Figure 2), thus the probability of data delivery is increased through path diversity.  
This behavior can have significant impact on the network performance when thermal conductivity value is mapped to the real characteristics of node. The thermal conductivity value \(\xi\) can be considered as the node's quality factor, which directly influences the temperature of node. In general, the better are the characteristics of the node, the higher is the conductivity value.  
One example of mapping the thermal conductivity value to the characteristics of node is prioritization. We can assign high value (e.g., \(\xi=0.9\)) to those nodes, which are preferred forwarders (high priority) and low conductivity value (e.g., \(\xi=0.2\)) to the nodes which should be used for data forwarding  
Figure 2: Heat computation algorithm (N-neighbor, \(\phi\)-temperature, P-priority).  
only if necessary (low priority). Those nodes with higher thermal conductivity values are strongly influencing the temperature values of their direct neighbors, thus, are more likely the next hop forwarders for those neighbors.  
Another interesting example is mapping the thermal conductivity value to the transmission power of the node. By mapping the nodes with high transmission power to high conductivity values, those nodes will be likely used as next hop forwarders. Vice versa, the nodes with low transmission power could be mapped to low conductivity values, thus, having more potential next hop forwarders (refer to Figure 2(c)). This is especially important when the node with mobility has low transmission radius (due to low transmission power) and the probability of disconnection with other nodes with mobility is high.  
Mapping of Thermal Conductivity to Residual Energy of Node  
In our proposed temperature transfer-based forwarding method we adapt the thermal conductivity value to a relative residual energy of a sensor node. The only necessary assumption is the knowledge of node's own energy. Then the value of thermal conductivity adapted to the node's energy can be calculated as follows:  
\[\xi_{E_{NOW}}=\frac{E_{NOW}}{E_{INIT}} \tag{2}\]  
where \(E_{NOW}\) is current energy status (_i.e._, remaining capacity of the battery) and \(E_{INIT}\) is initial (maximum) energy of sensor node. Thus, when node posses full battery, the residual energy is close to 100%, thus, the thermal conductivity will be close to 1. Vice versa, when the battery is almost dissipated, the thermal conductivity of this node will be very low (close to 0).