# The Isomorphic Version of [PERSON]'s and [PERSON]'s Nestedness
## 1 Introduction
When matrix \(M\) is interpreted as the bi-adjacency matrix, then the corresponding graph of matrix \(F\) (considered now as bi-adjacency matrix) is then a _threshold graph_. For an overview about these problems, we recommend [PERSON]'s book [9] or [PERSON]'s and [PERSON]'s book [10]. In most real world data, a matrix \(M\) is not nested. Hence, the natural question arises of \"how far the ecological matrix is from being nested\". [PERSON] and [PERSON] answered this question with the metrics of _discrepancy disc\((M)\)_[11]. (In ecology, it is common to use the notion BR instead of _disc_[2]). It is based on the fact that every matrix \(M\) has a unique Ferrers matrix \(F\) that can be achieved by shifting \(1\) s in each row from larger to smaller column indices with entry \(0\) (see Remark 2). The discrepancy of a given matrix \(M\) is the minimum number of shifts of \(1\) s to the left required to achieve its Ferrers matrix. A value of \(0\) represents a nested matrix. This means that the _the larger the discrepancy_ is (the more shifts are needed to yield the Ferrers matrix), _the less a matrix is nested_. Hence, the discrepancy metrics measures the strength to be non-nested. Consider matrix \(M\) in Example 4 and its unique Ferrers matrix \(F\). In the first row, it is possible to shift the \(1\) in the third column to the second column, and the \(1\) of the fourth column to the third one. However, one only needs a single shift, i.e., moving the \(1\) of the fourth column to the second one. Following this approach for each row, we find \(disc(M)=4\). The discrepancy can be easily calculated. However, there is a conceptual problem when applied to an ecological data set. Exchanging two different columns \(i\) and \(i^{\prime}\) in \(M\), holding the same column sums, leads to another matrix \(M^{\prime}\) with \(M\
eq M^{\prime}\). The values of the discrepancy for \(M\) and \(M^{\prime}\) can be different. However, both bi-adjacency matrices represent exactly the same relationships between bees and plants. Moreover, these bi-adjacency matrices correspond to two different labelled bipartite graphs. (A graph is labelled if all vertices in vertex set \(V\) are indexed by a unique number, i.e., \(V\) consists of vertices \(v:1,\ldots,v_{n}\) where \(n:=|V|\).) Their structure, however, is exactly the same. Formally, those graphs are called _isomorphic_. (Two labelled graphs \(G=(V,E)\) and \(G^{\prime}=(V,E^{\prime})\) are called isomorphic if there exists a permutation \(\sigma:V\mapsto V\) of vertex labels such that edge \(\{v_{i},v_{j}\}\in E\) if and only if \(\{v_{\sigma(i)},v_{\sigma(j)}\}\in E^{\prime}\).) Consider Example 4. \[\begin{array}{ccccc}A&B&C&D\\ a&b&\left( \begin{array}{cccc}1&\mathbf{0}&1&\mathbf{1}\\ 1&\mathbf{0}&1&\mathbf{1}\\ c&1&1&\mathbf{0}&\mathbf{1}\\ d&1&1&0&0\\ e&1&1&0&0\\ f&1&\mathbf{0}&\mathbf{1}&0\end{array} \right)\text{ and }M^{\prime}:= \begin{array}{ccccc}A&B&D&C\\ a&b&\left(\begin{array}{cccc}1&\mathbf{0}&1&\mathbf{1}\\ 1&\mathbf{0}&1&\mathbf{1}\\ 1&1&1&0\\ d&1&1&0&0\\ e&1&1&0&0\\ f&1&\mathbf{0}&0&\mathbf{1}\end{array}\right)\text{ represent }\\ \end{array} \]  
_two isomorphic bipartite graphs, i.e., both graphs look the same but plants \(\mathcal{C}\) and \(D\) have in \(M\) column indices \(3\) and \(4\), whereas in matrix \(M^{\prime}\) plant \(C\) and \(D\) are labelled with \(4\) and \(3\). Moreover, in \(M\), we need to shift the four bold \(1\)\(s\) to achieve its corresponding Ferrers matrix \(F= \begin{pmatrix}1&1&1&0\\ 1&1&1&0\\ 1&1&1&0\\ 1&1&0&0\\ 1&1&0&0\\ 1&1&0&0\\ 1&1&0&0\end{pmatrix} \), i.e., \(disc(M)=4\). In matrix \(M^{\prime}\), we only need to shift three \(1\)\(s\) to achieve \(F\) leading to \(disc(M^{\prime})=3\)._  
For ecological purposes, these labels are not necessary. They are only assigned to columns and rows due to the mathematical denotation of columns and rows. The value of the discrepancy is dependent on a \"random\" order of bees and pollinators in a data set, and its resulting order of columns and rows in its bi-adjacency matrix.