## 3 EMF-Aware Routing Framework for WMNs
### EMF-Aware Routing Algorithm
#### 3.2.1 Network Model  
The main distinctive characteristic of the proposed model is the need to consider both the instantaneous and accumulated transmitted power as routing metrics. The minimization of the transmission power would yield a reduction of the overall exposure within the whole scenario, but it would not avoid the appearance of exposure peaks, as well as great variability between different network areas. In order to avoid this, we need to also consider the accumulated transmitted power (or a function of it), as a proxy of the EMF exposure induced by each node. Hence, those nodes that have caused a higher exposure should be avoided in the future establishment of additional routes, thus leading to a fairer exposure distribution throughout the whole scenario. Note that the metrics used in this section are abstract, since the network model aims to integrate the two previously mentioned costs. The network is modeled as a graph \(G=(\mathcal{V},\mathcal{E})\), where \(\mathcal{V}\) is the set of vertices and \(\mathcal{E}\) the set of the corresponding edges. We assume that two nodes are connected if the euclidean distance between them is shorter than the coverage area of the corresponding  
Figure 1: Evolution of the exposure and its second derivative upon transmission events. communication technology. Since the two routing metrics that we seek to optimize have different characteristics, they are attributed to either nodes or edges:  
* Node accumulated exposure \(\mathcal{X}_{i}\): this cost represents the accumulated exposure that has been induced by a node \(i\) up to a certain time. In this sense, it does not depend on the amount of traffic traversing a node at a particular time, since we are interested in just considering its transmission history. It is worth noting that the proposed routing framework would allow using different definitions of this cost, provided it remains independent of the traffic currently traversing the node. Since the EMF exposure grows with the transmission power, we will later consider \(\mathcal{X}_{i}\) to be proportional to the power transmitted by the node until a particular moment. Nevertheless, more accurate models could be used, for instance considering advanced antenna patterns, or user density around the node;
* Edge power exposure \(\mathcal{P}_{e}\): this reflects the transmission power that the node requires to send a packet to the next hop through an edge \(e\). It is therefore attributed to each of the edges. \(\mathcal{P}_{e}\) should be proportional to the traffic load over every link, since the more a node is traversed by traffic, the more packets it will need to forward. This way, we can define the cost of an edge \(e\) connecting two nodes \(\{i,j\}\) as \(\mathcal{P}_{e}=P_{i,j}\cdot T_{i,j}\); where \(P_{i,j}\) is the required transmission power, and \(T_{i,j}\) represents the traffic load in that edge. Given the fact that our approach also assigns a cost to each node, we need to apply some network transformations before describing the algorithm, which will be used to establish the optimum routes. The reason for this is that most of the routing algorithms assume just edge costs, and so we introduce virtual edges to bear the cost \(\mathcal{X}_{i}\). For our algorithm, each node \(i\) is split into two virtual vertices, which are then connected with an edge, having a cost \(\mathcal{X}_{i}\): \(i^{\prime}\) and \(i^{\prime\prime}\), ingress and egress, respectively. The incoming edges at the original node \(n\) are now connected to \(i^{\prime}\), while the outgoing edges are now starting at \(i^{\prime\prime}\). As an illustrative example, Figure 2 shows the transformation applied to a simple graph of three nodes. Here, \(T_{ij}\) and \(P_{ij}\) are the traffic load and the required transmission power, between nodes \(i\) and \(j\), respectively. On the other hand, \(\mathcal{X}_{i}\) models the overall exposure caused by node \(i\) up to a certain time.