# Elements of Hyperstructure Theory in UWSN Design and Data Aggregation
## 2 Mathematical Model  
_The mathematical model presented in this section was published as an extended abstract of the conference contribution [PERSON], [PERSON], and [PERSON]_ [1] presented by the authors of this paper at International Conference on Numerical Analysis and Applied Mathematics (ICNAAM 2017)._  
UWSNs consist of elements of different types: First, we have _surface stations_, which pass data to a ship or to a data-collecting station located on the sea shore; second, we have _sensor nodes_ deployed at various tiers in water or at the sea bed. The sensors, which are deployed in water, can function as sensors measuring the requested data or as transporters of information from seabed sensors. In any case, information collected from all sensors must be passed to surface stations. From these it can be collected either by a ship passing by or, alternatively, transmitted to a data-collecting station located on the sea shore. The ship or the data-collecting stations are _central nodes_. Denote \(H\) the set of all elements of an arbitrary UWSN. Suppose that all elements are capable of handling (i.e., receiving or transmitting) data in the same way. Also suppose that they perform the same set of tasks. Thus they are, from the mathematical point of view, interchangeable and equal (of course, with respect to their functionality as sinks and sensor nodes). The aim of the system is to collect information. Therefore, our elements of \(H\) must communicate data. This should be done ideally upwards, towards the surface. As we have mentioned above, there are different ways of passing information. In our model we concentrate on multipath and cluster routing approach (see Figure 1 and Figure 2). For details concerning these see [PERSON] et al. and [PERSON] et al. [4; 29]. Multipath routing protocols (Figure 1), forward the data packets to the sink via other nodes while in cluster based routing protocols (Figure 2), data packets are first aggregated to the respective cluster heads and only then forwarded via other cluster heads to the sink. For our purposes, we denote the \(i\)-th cluster by \(cl_{i}\). Its cluster head will be denoted by \(CH_{i}\). We call non-\(CH\) nodes _ordinary_ and sinks will be treated as cluster heads. Now, suppose that the elements of our system are clustered. In other words, some elements of \(H\) function as cluster heads, i.e., masters, while others are ordinary. The data aggregation process goes as follows: Within their cluster, the ordinary elements pass information to their cluster head while between clusters, i.e., supposedly over longer distances, only cluster heads communicate. At a given point in time, each cluster has the unique cluster head, and each element can belong to exactly one cluster. We denote the \(i\)-th cluster by \(cl_{i}\) and its cluster head by \(CH_{i}\). Figure 1: Multipath approach to UWSN data aggregation. Notice the oriented communication between nodes. Now, for a given pair \(a,b\in H\), regard a binary hyperoperation, where \(a*b\) is, for arbitrary \(a,b\in H\), defined by:  
\[a*b=\left\{ \begin{array}{ll}\{a,b\}\cup[a\cdot b)_{\leq}&\text{for }(a=CH_{i},b=CH_{j}) \text{ or }a,b\in cl_{i}\\ \{a,b\}&\text{for }(a\
eq CH_{i}\text{ or }b\
eq CH_{j})\text{ and }(a\in cl_{i},b\in cl_{j},i\
eq j) \end{array} \right. \tag{1}\]  
By \([a\cdot b)_{\leq}\) we mean a set \(\{x\in H\mid a\cdot b\leq x\}\), where \(a\cdot b\) is a result of a single-valued binary operation such that \(a\cdot b\) is, for arbitrary \(a,b\in H\), defined by:  
\[a\cdot b=\left\{ \begin{array}{ll}CH_{i}&\text{for }a,b\in cl_{i}\\ CH_{k}&\text{for }a=CH_{i},b=CH_{j},i\
eq j\\ s&\text{for }((a\
eq CH_{i}\text{ or }b\
eq CH_{j})\text{ and }(a\in cl_{i},b\in cl_{j},i\
eq j)) \text{ or }a=s\text{ or }b=s\end{array} \right.