## 4 Mathematical Model  
In the proposed model, we consider that nodes are randomly deployed in an underwater environment, as shown in Figure 1. All nodes have a predefined transmission range associated with them. Initially, nodes do not possess depth and residual energy information of one-hop neighbors. This information is attained in the broadcasting phase. In this phase, all nodes share depth and residual energy information with one-hop neighbors. Along with this, nodes keep on sharing their residual energy information (at regular time intervals) during the network lifetime. Different rules and concepts that are followed in our algorithm are detailed:  
**Rule 1:** Network is divided into 4 distinct regions or sub-graphs (\(R_{k}\mid k=1-4\)) on the basis of depth (\(D\)), such that, in any subgraph all nodes are one-hop apart and lie within each other's transmission range. Figure 1 signifies rule 1 by showing one-hop apart nodes within each sub-graph. D(\(R_{1}\)) **>**\(D(R_{2}\)) **>**D(\(R_{3}\)) **>**D(\(R_{4}\))  
**Rule 2:** In each \(R_{k}\), vertices are divided among three distinct sets, \(V_{Hi}\), \(V_{Me}\) and \(V_{Lo}\) as shown in Figure 1, where  
* \(V_{Hi}\) are high depth vertices in a sub-graph
* \(V_{Me}\) are medium depth vertices in a sub-graph
* \(V_{Lo}\) are low depth vertices in a sub-graph  
A \(DS\) is formed for each \(R_{k}\) on the basis of a certain criterion. We define \(DS\) with respect to this work as;  
\(DS\): \(DS\) is a subset of vertices \(V\) of any \(R_{k}\) such that every vertex not in the \(DS\) is adjacent to at least one vertex in \(DS\). \(DS\) for cooperative routing is computed on the bases of depth and residual energy information of sensor nodes. Nodes having lowest depth and highest residual energy in each \(R_{k}\) form a \(DS\) such that all the other nodes in that \(R_{k}\) are connected to at least one node in \(DS\). **Rule 3:** Source, relay and destination nodes in any \(R_{k}\) belong to \(V_{Lo}\), \(V_{Me}\) and \(V_{Hi}\) such that;  
\(D_{D}\) **<**\(D_{R}\) **<**\(D_{S}\)  
where \(D_{D}\), \(D_{R}\) and \(D_{S}\) represent depth of destination, relay and source nodes respectively. This relation shows that source, relay and destination nodes are present at highest, medium and lowest depth  
Figure 1: Network model. regions in any \(R_{k}\), respectively. Rule 3 is shown in Figure 2. We assume that MSs are periodically transmitting beacon signals because of which nodes are aware of their mobility pattern and are forwarding data accordingly. In wireless transmission, flooding is a routing process in which each incoming data packet is sent via every outgoing link except the one from which it was received. Flooding results in duplication of data and thus wastes resources of the network. In depth-based routing, depth threshold (\(d_{th}\)) is used to avoid flooding by limiting the number of outgoing links for each incoming data packet. DBR [3], EEDBR [4], iAMCTD [5] and CoDBR [6] are existing depth-based routing protocols in UWSNs. In DBR, EEDBR and CoDBR, \(d_{th}\) is pre-defined and remains fixed throughout the network lifetime. In iAMCTD, \(d_{th}\) is selected and varied according to the network density information which requires exchange of control packets at regular time intervals. This exchange acts as an overhead and wastes network resources. We present a way for varying \(d_{th}\) that minimizes this huge control packet exchange. In our work, nodes just require the residual energy information of one-hop neighbors. **Rule 4:** Value of \(d_{th}\) is selected and varied according to the number of alive neighbors of a node, _i.e._,  
\[\forall\,i\in V:d_{th_{i}}(t)\propto N_{i}(t) \tag{4}\]  
where, \(N_{i}\)(t) represents alive neighbors of \(i_{th}\) node at time instant \(t\)  
\[N_{i}=N(i)=j\mid(i,j)\in A \tag{5}\]  
Increase in \(N\) leads to increased \(d_{th}\) and decrease in \(N\) leads to decreased \(d_{th}\). This relation signifies that the greater the number of alive neighbors, higher is the value of \(d_{th}\) and vice versa.