# Geometric Deep Lean Learning: Evaluation Using a Twitter Social Network

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## Abstract

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## 1. Introduction

## 2. Summary of Geometric Deep Lean Learning Algorithm

- The structure of the graph is described by its adjacency matrix, the Laplacian of the graph ${\mathcal{L}}^{t}$, or any other normalisation of it, as a linear transformation to encode the structure of a graph. As described in [42], its topology is typically featured by a log–log long-tailed degree distribution, a degree exponent $2<\gamma <5$, an average path length in the range $\left[ln\right(ln\left(N\right)),ln(N\left)\right]$ and high clustering coefficients.
- Each node $i\in {N}^{t}$ and edge $i\to j\in {E}^{t}$ can be characterised by a series of signals expressed in the form of tensors ${{T}_{ii}}^{t}$ for the nodes and ${{T}_{ij}}^{t}$ for the edges. If these tensors are empty, i.e., formed by zeros, the node or edge would be considered nonexistent for our purposes. Subsequently, these signals are described by ${\Xi}^{t}$ given by Equation (1).$${\Xi}^{t}=\left[\begin{array}{ccccccc}{{T}_{11}}^{t}& \cdots & {{T}_{1i}}^{t}& \cdots & {{T}_{1j}}^{t}& \cdots & {{T}_{1N}}^{t}\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ {{T}_{i1}}^{t}& \cdots & {{T}_{ii}}^{t}& \cdots & {{T}_{ij}}^{t}& \cdots & {{T}_{iN}}^{t}\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ {{T}_{j1}}^{t}& \cdots & {{T}_{ji}}^{t}& \cdots & {{T}_{jj}}^{t}& \cdots & {{T}_{jN}}^{t}\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ {{T}_{N1}}^{t}& \cdots & {{T}_{Ni}}^{t}& \cdots & {{T}_{Nj}}^{t}& \cdots & {{T}_{NN}}^{t}\end{array}\right]$$

## 3. Data Mining on Twitter

#### 3.1. Experimental Setup

- Programming language: Python 3.7.6 [58];
- Python wrapper for the Twitter API: Tweepy 3.8.0;
- Python package for the creation and study of complex networks: NetworkX 2.4 [59];
- Python plotting library: Matplotlib 3.1.3;
- Python package for data analysis and manipulation: Pandas 1.0.1;
- Network visualisation and exploration: Gephi 0.9.2 [60].

#### 3.2. Specification of Population and Sampling

#### 3.3. Data Collection

#### 3.4. Standardisation Procedure

#### 3.5. Data Analysis

- For most tweets of the dominant hashtags, distinct communities are formed that mostly have interactions with users that also use that hashtag. The strength of the interconnection can be seen by the density of the communities. Users with the #machinelearning tweets are at close distance, which indicates a high level of connectedness. This, in turn, is a sign that a low average path length and a high clustering coefficient exists within this group. On the other hand, #blockchain has a further stretched the patch, which indicates a higher average path length and lower clustering coefficient. These differences can be seen for all dominant tweets.
- Strong points of contact and overlap can be seen between some groups. As could be expected, one of the strongest can be seen between the tweets of #climatechange and #biodiversity, but also between #machinelearning and #blockchain, although the connection is weaker and limited to specific parts of the network, which is probably due to the users discussing the technical side of both. Strong connections can also be seen in other parts. This demonstrates the overlap of some communities.
- Although clear communities for #5g are visible, these are separated. This can be traced to the fact that it can be used in different contexts. More importantly, it can be used in tweets that are written in different languages, which is rarely the case with other hashtags, as they most often have a translation for that language.

- Network StructureAs shown in Figure 6, the dataset presents a typical log–log long-tailed degree distribution which is typical of small-world/scale-free networks with a degree exponent of $\gamma =2.3$ [41].In Figure 7, several other network metrics are shown. Specifically, in Figure 7a, the average path length, which is defined as the average number of steps along the shortest paths for all possible pairs of network nodes of all journals, is proved to be in the range $\left[ln\right(ln\left(N\right)),ln(N\left)\right]$ which, together with the representation in Figure 7b of the high clustering coefficients, which is a measure of the degree to which nodes in a graph tend to cluster together, show a typical behaviour of small-world networks evolving towards a scale-free network topology [41].
- Network SignalsThe information contained in each retweet network about the nodes and edges can be mined in a semi-structured standard given by the platform. A truncated for our purposes example of a retweet is shown in Table 2.After this inspection, it will be easy for the reader to recognise that all information, both for the nodes and for the edges, except the text field, is structured. The information contained in the text field, which is the content of the retweet, can be considered as unstructured since it is given by the user who composes it.

## 4. Geometric Deep Lean Learning Evaluation

#### 4.1. Experimental Setup

- Programming language: Python 3.7.6 [58];
- Python plotting library: Matplotlib 3.1.3;
- Python package for data analysis and manipulation: Pandas 1.0.1;
- Python package for scientific computing and array calculation: Numpy 1.18.0.

#### 4.2. Data Preprocessing

#### 4.3. Geometric Deep Lean Learning Hyperparameters

#### 4.4. Data Analysis and Results

- As shown in Figure 8a we perform several experiments for different temporal depth $t\in [1,6]$ with a constant spatial depth of the search manifold of $k=2$.
- As shown in Figure 8b,c, we performed several experiments for different spatial depths of the search manifold $k\in [1,4]$ with a constant temporal depth of $t=5$ and $t=6$, respectively.

## 5. Discussion

#### 5.1. Variation of Temporal Depth with Constant Spatial Depth

#### 5.2. Variation of Spatial Depth with Constant Temporal Depth

## 6. Conclusions and Management Implications

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

API | Application programming interface |

AUC | Area under the receiver operating characteristic curve |

MDPI | Multidisciplinary Digital Publishing Institute |

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**Figure 3.**Amount of retweets and other tweets collected for the selected hashtags that contain more than 100 retweets.

**Figure 4.**Retweet network: the edges between two users are coloured according to the associated hashtag of the retweet.

**Figure 5.**Retweet network as shown in Figure 4. The edges between two users are coloured according to the group of hashtags.

Sensors | Sustainability | Biosensors | |
---|---|---|---|

1. | #mdpisensors | #mdpisustainability | #mdpibiosensors |

2. | #sensors | #sustainability | #biosensors |

3. | #iot | #sustainable | #sers |

4. | #deeplearning | #sushighlycitedpaper | #electrochemical |

5. | #biosensors | #climatechange | #fret |

6. | #machinelearning | #susinterestingpaper | #spr |

7. | #internetofthings | #energy | #biomarkers |

8. | #sensor | #sdgs | #i3s2017 |

9. | #wearable | #circulareconomy | #biomarker |

10. | #remotesensing | #sustainabledevelopment | #raman |

11. | #uav | #tourism | #biosensor |

12. | #structuralhealthmonitoring | #callforpapers | #immunoassay |

13. | #wearablesensors | #agriculture | #microfluidic |

14. | #wirelesssensornetworks | #transportation | #i3s2019 |

15. | #i3s2019 | #biodiversity | #microfluidics |

16. | #sensornetworks | #environment | #wearables |

17. | #sensing | #csr | #microarray |

18. | #5g | #management | #bacteria |

19. | #monitoring | #food | #aptamer |

20. | #smartcities | #renewableenergy | #sensors |

21. | #gnss | #openaccess | #flexible |

22. | #biosensor | #innovation | #bret |

23. | #blockchain | #education | #lab_on_chip |

24. | #sensorfusion | #resilience | #immunosensors |

25. | #highlyaccessedpaper | #environmental | #openaccess |

’created at:’ | Tue Jun 09 14:42:30 +0000 2020, |

’id’: | ’1270365661215240192’, |

’username’: | ’MedGIFT group’, |

’user id’: | ’1012268385789513729’, |

’text’: | ’We are hiring! |

’quoted user’: | ’id’: ’1270359799201443840’, ... |

...’user id’: ’2163570636’,... | |

...’username’: ’adepeursinge’ |

Sender | retweeted_user_id |

Receiver | retweeter_user_id |

Evolution | created_at |

Content | hashtag, group and retweet text |

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**MDPI and ACS Style**

Villalba-Diez, J.; Molina, M.; Schmidt, D.
Geometric Deep Lean Learning: Evaluation Using a Twitter Social Network. *Appl. Sci.* **2021**, *11*, 6777.
https://doi.org/10.3390/app11156777

**AMA Style**

Villalba-Diez J, Molina M, Schmidt D.
Geometric Deep Lean Learning: Evaluation Using a Twitter Social Network. *Applied Sciences*. 2021; 11(15):6777.
https://doi.org/10.3390/app11156777

**Chicago/Turabian Style**

Villalba-Diez, Javier, Martin Molina, and Daniel Schmidt.
2021. "Geometric Deep Lean Learning: Evaluation Using a Twitter Social Network" *Applied Sciences* 11, no. 15: 6777.
https://doi.org/10.3390/app11156777