# The Isomorphic Version of Brualdi’s and Sanderson’s Nestedness

^{*}

## Abstract

**:**

## 1. Introduction

**Example**

**1.**

**Example**

**2.**

**Definition**

**1**(nestedness)

**.**

**Example**

**3.**

**Remark**

**1.**

**Remark**

**2.**

**Example**

**4.**

**for each fixed column sum**s a bipartite graph ${G}_{s}$

**separately**. Each edge $\{{u}_{i},{v}_{j}\}$ in a minimum perfect weighted matching of ${G}_{s}$ tells us that column i in matrix M needs to be moved on position $j.$ In the resulting matrix, all columns with sum s are reordered without violating the non-increasing column sums property. We repeat this approach whenever there exist at least two columns with the same column sum. The resulting matrix is a matrix in ${I}_{M}$ with a minimum discrepancy. One of the most important books about matching theory is from Lovász and Plummer [12]. One suitable algorithm for our scenario is from Gabow [13], which achieves an $O\left({n}^{3}\right)$ asymptotic running time, if we follow our approach.

## 2. The Discrepancy Problem for Isomorphic Matrices

**Example**

**5.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Example**

**6.**

**Theorem**

**1.**

**Proof.**

## 3. Discrepancy versus Isomorphic Discrepancy in Ecological Matrices

#### 3.1. Experiment: Possible Range of Discrepancy in One Isomorphic Class

#### 3.2. Experiment: Standardized Discrepancy Difference

## 4. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**A bipartite graph and its bi-adjacency matrix $N.$ Matrix N is nested because each bee pollinates the same plants as all bees which pollinate more or an equal number of plants.

**Figure 3.**(

**a**) each data point is a matrix M (x-axis) versus discrepancy difference $D\left(M\right)$ (y-axis); (

**b**) standardized discrepancy $Sd\left(M\right)$ (x-axis) versus standardized discrepancy difference $SD\left(M\right)$ (y-axis).

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Berger, A.; Schreck, B.
The Isomorphic Version of Brualdi’s and Sanderson’s Nestedness. *Algorithms* **2017**, *10*, 74.
https://doi.org/10.3390/a10030074

**AMA Style**

Berger A, Schreck B.
The Isomorphic Version of Brualdi’s and Sanderson’s Nestedness. *Algorithms*. 2017; 10(3):74.
https://doi.org/10.3390/a10030074

**Chicago/Turabian Style**

Berger, Annabell, and Berit Schreck.
2017. "The Isomorphic Version of Brualdi’s and Sanderson’s Nestedness" *Algorithms* 10, no. 3: 74.
https://doi.org/10.3390/a10030074