# Elements of Hyperstructure Theory in UWSN Design and Data Aggregation
## 3 Use of the Theory of Quasi-Automata
In a completely analogous manner we get for the second sum, i.e., for the second component of the vectors:  
\[\left.\sum\limits_{i=1}^{n}v_{i}b_{i1}0+\sum\limits_{i=2}^{n}v_{i }b_{i2}0+\ldots+\sum\limits_{i=n-1,}^{n}v_{i}b_{in-1,}a_{n-1,n-1}+\sum \limits_{i=n}^{n}v_{i}b_{in}a_{nn},\ldots\right.\]5 \[\left.\sum\limits_{i=1}^{n}v_{i}b_{i1}0+\sum\limits_{i=2}^{n}v_{i }b_{i2}0+\ldots+\sum\limits_{i=n-1,}^{n}v_{i}b_{in-1,}a_{n-1,n-1}+\sum \limits_{i=n}^{n}v_{i}b_{in}a_{nn},\ldots\right.\]6 \[\left.\sum\limits_{i=1}^{n}v_{i}b_{i1}0+\sum\limits_{i=2}^{n}v_{i }b_{i2}0+\ldots+\sum\limits_{i=n-1,}^{n}v_{i}b_{in-1,}a_{n-1,n-1}+\sum \limits_{i=n}^{n}v_{i}b_{in}a_{nn},\ldots\right.\]7 \[\left.\sum\limits_{i=1}^{n}v_{i}b_{i1}0+\sum\limits_{i=2}^{n}v_{i }b_{i2}0+\ldots+\sum\limits_{i=n-1,}^{n}v_{i}b_{in-1,}a_{n-1,n-1}+\sum \limits_{i=n}^{n}v_{i}b_{in}a_{nn},\ldots\right.\]8 \[\left.\sum\limits_{i=1}^{n}v_{i}b_{i1}0+\sum\limits_{i=2}^{n}v_{i }b_{i2}0+\ldots+\sum\limits_{i=n-1,}^{n}v_{i}b_{in-1,}a_{n-1,n-1}+\sum \limits_{i=n}^{n}v_{i}b_{in}a_{nn},\ldots\right.\]9  
which will give us condition that \(b_{ii}\geq b_{i2}\) for all \(i\in\{3,4,\ldots n\}\), etc. until the \((n-1)^{th}\) component will result in condition \(b_{ii}\geq b_{i,n-1}\) for \(i=n\); and the first and the last condition \(b_{ii}\geq b_{ii}\) for all \(i\in\{1,n\}\) hold trivially. Thus, altogether we get that \(b_{ii}\geq b_{ij}\) for all \(i\in\{1,2,\ldots,n\}\) and \(j\in\{1,\ldots,i\}\). In other words, each diagonal element must be greater than all other elements in the given row, which is exactly what we suppose \(\hat{I}_{M}\) to be. **Remark 3**. : _If we want to find out whether we are able to reach any state from any other state with the help of our transition functions (i.e., test strong conectedness), we must take into account that the fact that we regard diagonal matrices means that for known vectors \(\vec{u}\), \(\vec{v}\) and an unknown matrix \(\mathbf{A}\) the equation \(\vec{u}\mathbf{A}=\vec{v}\) results in a linear system in the row echelon form and the task of finding the unknown matrix \(\mathbf{A}\) is equivalent to finding its solution. Of course, we have to consider the fact that \(\vec{v}\) might have components being zero, which will affect the solvability of the linear system._  
**Example 2**. : _The hyperoperation on the input set, i.e., the fact that we work with hypergroups, gives us a better tool for controlling the process of data aggregation. Suppose that we have the same vector as in Example 1, i.e., \(\vec{v}=(0.3;0.1;0;0.5)\). Furthermore, assume the same context as in Example 1. Let the matrix be:_  
\[\mathbf{A}= \begin{bmatrix}1&0&0&0\\ 0.9&1&0&0\\ 0.2&1&1&0\\ 0.01&0.3&1&0\end{bmatrix} \in\hat{I}_{M}.\]  
_Suppose that in our quasi-multiautomaton this matrix is applied on vector \(\vec{v}\), i.e._  
\[\delta(\mathbf{A},\vec{v})=(0.3;0.1;0.5)\cdot \begin{bmatrix}1&0&0&0\\ 0.9&1&0&0\\ 0.2&1&1&0\\ 0.01&0.3&1&0\end{bmatrix} .\]  
_Now, focus on the multiplication of \(\vec{v}\) and the first column of \(\mathbf{A}\). We get that:_  
\[(0.3\cdot 1+0.1\cdot 0.9+0\cdot 0.2+0.5\cdot 0.01;a;b;c)=(0.395;a;b;c),\]  
_which means that the capacity of tier 1 is filled by \(39.5\%\) in such a way that the original \(30\%\) was accompanied by further \(9.5\%\) from tier 2 (which had originally been filled up by \(10\%\), i.e., we have a \(1\%\) transmission loss), \(0\%\) from tier 3, and \(0.5\%\) from tier 4. In this case it is obvious that the transmission from tier 4 is extremely difficult with a substantial data loss. Theoretically, we could model the process of data aggregation with matrices with all-one columns. However, this would not be a real-life case. When looking for an optimal transition input (i.e., an optimal matrix), we could regard matrices such as those in Equation (8) with increasing entries. The definition of the hyperoperation (8) means that from a certain moment on, the memory capacity could be filled to maximum and data will start being lost, i.e., not stored in cluster heads on upper tiers anymore.