## 1 Introduction
Other parameters, shown in Equation (1), are used to weigh each component's contribution, and the contribution brought by uplink/downlink communications. The reader may refer to [6] for a more detailed description of the LEXNET \(EI\). \[EI^{SAR}\left[\frac{W}{Kg}\right]=\frac{1}{T}\sum_{k}^{N_{EC}}fr_{k}\cdot\left[ \sum_{u}^{N_{u}}(d_{k,u}^{UL}\cdot\overline{P}_{tx_{k}})+d_{k}^{DL}\overline{ S}_{rx_{k}}\right] \tag{1}\]  
The LEXNET EI is related to the Specific Absorption Rate (SAR) (the same measurement unit as well: \(\Psi\)/kg), which is a parameter defined for mobile devices that is strongly correlated with the accumulated power involved in wireless transmissions. Since the goal is to propose a routing protocol that can be used over real networks, the routing metrics that will be exploited are to serve as proxies for the overall EI. The assumption is that, if the proposed scheme is able to optimize some of these proxy parameters, among which the most important is the transmit power, from which the accumulated power is directly derived, it would also lead to an eventual reduction of the overall EI. To the best of our knowledge, when it comes to routing in multi-hop decentralized networks, even if the EMF accumulated exposure (such as the LEXNET EI), or other related parameter, has been used to optimize the physical and media access layer of deployed networks, it has not yet been taken into consideration as a routing metric, this being one of the key original contributions of this paper. This is a novel metric, defined within [6], where the EMF EI is defined as the summation of the entire SAR over a particular area, and over all people within such an area. Even if the LEXNET EI, which we also use as metric in this paper, highly depends of the transmit power of each node, it is not identical to this, especially due to its accumulated nature. In our work we indeed use the instant transmit power as a proxy for the more complicated EI metric, whenever this was appropriate, but in the end our routing decision at each node's level is taken with respect to the entire accumulated exposure of each node, that is, the sum of all the transmit powers that the node had to use at various previous moments in time. This way, besides minimizing the induced nodes' exposure, the algorithm and protocol that we have designed and implemented also leverage a load balancing of the decentralized distributed system, namely it does not over-stress a popular route and thus does not create EMF hotspots. We achieve this avoidance of EMF hotspots by gradually increasing costs for any route, when the exposure of the involved nodes becomes too high. This yields a changing network topology, since some graph edges from the initial multi-hop network may be un-usable after having been used too intensively, thus favoring less popular routes. In our work, we were more interested in the network and node behavior as theoretical models, rather on the particular details of a technique to route the packets between the nodes. This is why we have chosen, as the underlying routing protocol, the well-known Ad hoc On-Demand Distance Vector Routing (AODV) [7], modified to solve the minimum cost flow problem that arises from minimizing the exposure, as we will describe in the following sections. In particular, we adopt the Cycle Canceling Algorithm (CCA) algorithm to solve the corresponding optimization problem. The integration of CCA into the AODV operation leads to the so-called EMF Aware AODV (EA-AODV). On top of the routing protocol and algorithm, we have used techniques similar to peer-to-peer relay networks, in order to minimize each node's exposure, as defined in the LEXNET EI. Before designing the routing protocol, we formulate an optimization problem and propose an algorithm based on graph theory to solve it. This will establish the performance baseline for the routing protocol.