# Link Prediction in Bipartite Nested Networks

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Link Prediction Methods

#### 2.1.1. Preferential Attachment Index (PrefA)

#### 2.1.2. Number of Local Community Links (LCL)

#### 2.1.3. Probabilistic Spreading (ProbS)

#### 2.1.4. Number of Violations of the Nestedness Property (NViol)

#### 2.2. Evaluation Process

## 3. Data

#### 3.1. Synthetic Data

#### 3.2. Real Data

## 4. Results

- As $\xi $ grows and the networks’ nested structure thus becomes more pronounced, differences between the methods grow.
- NViol is generally the best-performing method with respect to the metrics r and AUC that take the whole link prediction list into account. Upon a closer inspection of link prediction lists produced by the respective methods, the advantage of NViol is due to its ability to place well also links connecting low-degree nodes that the other methods miss due to their general bias towards high-degree nodes. With NViol, though, probe links adjacent to low-degree nodes are not among the top 100 and hence do not contribute to the method’s precision, yet they rank much better than where other methods are used. If we would increase the number of top ranks included in precision evaluation from 100 to 200 or 300, NViol would have an edge also in this metric.
- As the randomization parameter p grows, PrefA eventually outperforms NViol in terms of link prediction precision. High precision improvement with respect to LCL’s precision for $\xi =5$ are due to the generally low precision achieved for the sparse networks produced at $\xi =5$ (which is made further worse by introducing the noise when $p>0$).
- ProbS outperforms LCL but lacks behind PrefA and NViol. This is expected because ProbS is based on a “personalized” recommendation algorithm; with no communities in the data, there is no place for personalization and thus ProbS’s merits cannot manifest themselves. The situation becomes radically different when there is more than one nested block in the data (see below).

## 5. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A toy example of a bipartite network where ${N}_{1}=4$ and ${N}_{2}=5$. Network links are shown with the solid lines. The dashed line shows the possible link between nodes i and $\alpha $ whose likelihood is being evaluated. The thick lines highlight the only existing link between the neighbors of nodes i and $\alpha $ (see the prediction method “Number of Local Community Links”). The preferential attachment index (PrefA) and LCL scores of the link between the highlighted nodes are 6 and 1, respectively.

**Figure 2.**An illustration of synthetic nested networks. (

**A**) Nestedness contours for various values of the $\xi $ parameter. (

**B**–

**D**) Nested networks with ${N}_{1}=30$ and ${N}_{2}=42$: Perfectly nested network (

**B**), nested network with low noise (

**C**), and in-block nested network with three blocks and low noise (parameter values are specified in the panels). The results are averaged over 10 model realizations and 10 independently chosen probe sets for each realization.

**Figure 3.**Link prediction results on model data with ${N}_{1}=100$, ${N}_{2}=200$, and no community structure. To remove the strong dependency of method performance on the randomization parameter p, the shown results are scaled with the results of the simplest PrefA method.

**Figure 4.**Link prediction results on model data with ${N}_{1}=100$, ${N}_{2}=200$, ${N}_{B}=2$, $\mu =0$ (two blocks, no links between the blocks). As in Figure 3, results are again scaled with the results of the simplest PrefA method.

**Table 1.**Basic properties of the real datasets used to evaluate link prediction methods: Number of rows (${N}_{1}$), columns (${N}_{2}$), edges (E), and the density of edges [${\varrho}_{E}:=E/({N}_{1}{N}_{2})$].

Dataset | ${\mathit{N}}_{1}$ | ${\mathit{N}}_{2}$ | E | ${\mathit{\varrho}}_{\mathit{E}}$ |
---|---|---|---|---|

M_SD_022 | 207 | 110 | 1121 | 0.05 |

M_PL_015 | 131 | 666 | 2933 | 0.03 |

M_PL_021 | 91 | 677 | 1193 | 0.02 |

M_PL_044 | 110 | 609 | 1125 | 0.02 |

M_PL_057 | 114 | 883 | 1920 | 0.02 |

M_PL_062 | 456 | 1044 | 15,255 | 0.03 |

CP-2001 | 169 | 781 | 17,639 | 0.13 |

CP-2009 | 168 | 774 | 17,739 | 0.14 |

**Table 2.**Mean link prediction results on real datasets. Best performance values for a given method and metric are highlighted with bold. Results are averaged over 100 independently chosen probe sets. Standard error of the mean is less than 0.005 in all cases. If NViol produces best AUC on transposed data, it is labeled as NViol${}^{\top}$.

method | r | AUC | P | ${F}_{1}$ | method | r | AUC | P | ${F}_{1}$ | |

PrefA | 0.20 | 0.80 | 0.11 | 0.13 | PrefA | 0.38 | 0.61 | 0.12 | 0.10 | |

LCL | 0.19 | 0.80 | 0.14 | 0.15 | LCL | 0.34 | 0.59 | 0.14 | 0.12 | |

ProbS | 0.17 | 0.82 | 0.15 | 0.16 | ProbS | 0.33 | 0.61 | 0.16 | 0.13 | |

NViol${}^{\top}$ | 0.19 | 0.81 | 0.11 | 0.12 | NViol | 0.16 | 0.84 | 0.12 | 0.10 | |

M_PL_015 | M_PL_062 | |||||||||

method | r | AUC | P | ${F}_{1}$ | method | r | AUC | P | ${F}_{1}$ | |

PrefA | 0.23 | 0.77 | 0.12 | 0.09 | PrefA | 0.20 | 0.80 | 0.05 | 0.05 | |

LCL | 0.19 | 0.80 | 0.20 | 0.14 | LCL | 0.17 | 0.83 | 0.12 | 0.07 | |

ProbS | 0.18 | 0.82 | 0.25 | 0.18 | ProbS | 0.16 | 0.84 | 0.15 | 0.08 | |

NViol | 0.24 | 0.75 | 0.13 | 0.08 | NViol${}^{\top}$ | 0.20 | 0.80 | 0.04 | 0.05 | |

M_PL_021 | CP-2001 | |||||||||

method | r | AUC | P | ${F}_{1}$ | method | r | AUC | P | ${F}_{1}$ | |

PrefA | 0.49 | 0.49 | 0.06 | 0.07 | PrefA | 0.23 | 0.78 | 0.05 | 0.10 | |

LCL | 0.41 | 0.46 | 0.09 | 0.09 | LCL | 0.21 | 0.79 | 0.18 | 0.13 | |

ProbS | 0.40 | 0.48 | 0.09 | 0.10 | ProbS | 0.19 | 0.81 | 0.14 | 0.12 | |

NViol | 0.16 | 0.84 | 0.06 | 0.05 | NViol | 0.24 | 0.76 | 0.06 | 0.10 | |

M_PL_044 | CP-2009 | |||||||||

method | r | AUC | P | ${F}_{1}$ | method | r | AUC | P | ${F}_{1}$ | |

PrefA | 0.47 | 0.52 | 0.05 | 0.06 | PrefA | 0.24 | 0.77 | 0.07 | 0.10 | |

LCL | 0.38 | 0.45 | 0.07 | 0.07 | LCL | 0.22 | 0.79 | 0.22 | 0.13 | |

ProbS | 0.37 | 0.46 | 0.06 | 0.07 | ProbS | 0.20 | 0.80 | 0.13 | 0.12 | |

NViol | 0.21 | 0.79 | 0.04 | 0.05 | NViol | 0.25 | 0.75 | 0.08 | 0.10 |

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**MDPI and ACS Style**

Medo, M.; Mariani, M.S.; Lü, L.
Link Prediction in Bipartite Nested Networks. *Entropy* **2018**, *20*, 777.
https://doi.org/10.3390/e20100777

**AMA Style**

Medo M, Mariani MS, Lü L.
Link Prediction in Bipartite Nested Networks. *Entropy*. 2018; 20(10):777.
https://doi.org/10.3390/e20100777

**Chicago/Turabian Style**

Medo, Matúš, Manuel Sebastian Mariani, and Linyuan Lü.
2018. "Link Prediction in Bipartite Nested Networks" *Entropy* 20, no. 10: 777.
https://doi.org/10.3390/e20100777