# Interaction Strength Analysis to Model Retweet Cascade Graphs

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## Abstract

**:**

## 1. Introduction

#### Research Objective and Contributions

- We introduce the concept of interaction strength (IS), a metric that indicates the strength of the link between two users;
- We propose two novel approaches based on IS to generate retweet cascade graphs—ISN and ISN-AF;
- ISN aims to maximize IS values for each pair of nodes in the graph;
- ISN-AF is similar to ISN, but the first level in the retweet cascade graph is based on the list of followers of the root user;
- Both of the proposed approaches are mainly based on information contained in the users’ timelines, which can be conveniently retrieved through the free Twitter API service (compared to fetching the list of friends for each node as in the traditional approach, which is a substantially more time-consuming task);
- The source code is freely available on GitHub (https://github.com/paolazola/Interaction-strength-analysis-to-model-retweet-cascade-graphs).

## 2. Related Work

**Study**. The second column named

**Target**describes the paper scope with the acronyms reported in the table footnotes; then the columns

**Dataset**,

**Dataset Size**and

**Dataset Collection Date**describe the datasets used. The following eight columns refer to a series of features which have been evaluated (X) or not (-) in the corresponding work. Finally, the column

**Method**reports the acronym of the model proposed by the paper in the respective row. As Table 1 shows, the majority of works has focused on predicting retweet engagement (REP) in terms of total number of retweets. Other studies conducted a joint analysis between tweet information cascade (TIC) and REP, assuming that the retweet chains can be deducted from tweet content.

## 3. The Interaction Strength-Based Network (ISN) Approach to Generating Retweet Cascade Graphs

- The retweet’s creation time ${t}_{r}$;
- The interaction strength between each couple $(u,w)\in \mathbb{V}$, which reflects the trust between users;
- The friend lists ${L}_{\mathbb{F}}$ for the remaining nodes $\mathbb{F}\subset \mathbb{V}$, for which no interactions were found (e.g., $I{S}_{\mathbb{F}}=Null$).

#### 3.1. Twitter User’s Interaction Strength

- The number of quotes ${Q}_{u,w}$ that user u expressed to the node w;
- The number of replies $R{P}_{u,w}$ that user u did to w;
- The number of retweets $R{T}_{u,w}$ that user u did for w.

#### 3.2. Interaction Weights for Retweet Cascade Graph

#### 3.3. Users without Interactions and Sparse Nodes

## 4. Alternative Approach: Information Strength-Based Network with Author’s Followers Evaluation (ISN-AF)

## 5. Evaluation Metrics for Retweet Cascade Validation

- Cascade average strength (CAS): given the $IS$ assigned to each edge $(u,y)\in \mathbb{E}$ we derive the CAS as the average of the maximum IS between each pair of edges in $\mathbb{E}$ such as:$$CAS=\frac{\sum (max(I{S}_{u,y}\forall u,y\in \mathbb{E}))}{\left|\mathbb{E}\right|}$$The aim is to maximise CAS for each cascade graph.
- Connected components count (CCC): returns the number of the connected components in the network. A connected component is a subgraph in which any two vertices $v\in V$ are connected to each other by paths. This metric provide a description of the graph shape.
- Root fan ratio (RFR): it assesses whether there is a path to the $roo{t}_{author}$ from every other user. In other words, it measures the percentage of nodes directly or indirectly connected to the root. In the ISN-AF model, the RFR asses the percentage of $roo{t}_{author}$ followers. A higher RFR reflects a very concentrated graph around the $roo{t}_{author}$ determining the typical star shape of the cascade graph [25].
- Giant component size (GCS): the size, expressed in percentage of the cascade nodes, of the nodes present in the giant component (GC) which is the connected component with biggest size. The GCS is computed as follow:$$GCS=\frac{|u\in \mathbb{V}\u27f6u\in GC|}{\left|\mathbb{V}\right|}$$This metric provides a description of the graph shape in terms of node dispersion: The lower the GCS, the higher the dispersion of nodes in the graph, which can be sparse or connected. To further investigate the nodes’ dispersion we investigate the global reaching centrality and sparse node incidence.
- Global reaching centrality (GRC): It is the average over all nodes of the difference between the node local reaching centrality and the greatest local reaching centrality of any node in the graph. The local reaching centrality, ${C}_{R}\left(i\right)$, of node i is the proportion of all nodes in the graph that can be reached from node i via outgoing edges [35].$$GRC=\frac{{\sum}_{i\in \mathbb{V}}({C}_{R}^{max}-{C}_{R}\left(i\right))}{\left|\mathbb{V}\right|-1}$$
- Sparse node incidence (SNI): it measures the incidence (in percentage) of sparse nodes (i.e., nodes without links) with respect to the total number of nodes in the cascade. A lower SNI determines more connected and realistic retweet cascades.

## 6. Dataset

## 7. Evaluation of IS Weights

#### 7.1. Experimental Results on the Entire Cascade Sample

#### 7.2. Experimental Results on Cascades with at Least Five Nodes

## 8. ISN, ISN-AF and Baseline Comparisons

#### Research Implications

## 9. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Study | Target ^{a} | Dataset ^{b} | Dataset ^{c}Size | Dataset Collection Date | Topic Features | Text Features | Time Variable | Users Features | Users Interactions | Social Network | Location Features | Users Behaviour | Model ^{d} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Szabo et al. [12] | REP | YT, D | YT: 7K, D: 850K U | 2007–2008 | - | - | X | - | - | LR | |||

Yang et al. [13] | TIC-REP | TW | - | - | X | X | X | - | - | - | - | X | F+EM |

Cogan [14] | TIC | TW | 33K T | 2012 | - | - | - | - | - | X | - | - | RCM |

Comarela et al. [7] | RB | TW | 54M U | 2006–2009 | - | X | X | X | - | - | - | X | SVM, NB |

Yang et al. [15] | RB | TW | 22M T | 2009 | X | - | X | - | - | - | - | - | CHR |

Remy et al. [16] | TIC | TW | 362M T | 2011 | - | - | - | - | - | X | - | - | PL |

Zaman et al. [17] | TIC-REP | TW | 52 | - | - | - | X | X | - | - | - | - | HB |

Taxidou et al. [11] | TIC | TW | 11M T | 2012 | - | - | X | - | - | X | - | - | |

Pramanik et al. [18] | TIC | TW | 55K | - | X | - | X | - | - | X | - | - | H |

Yu et al. [19] | TIC-REP | TWB | 320M U | 2011 | - | - | X | X | - | - | - | X | NEWER |

Zhao et al. [20] | REP | TW | 3.2B | 2011 | - | - | X | X | - | - | - | - | SEISMIC |

Gao et al. [21] | TIC-REP | SW | 164 | - | - | - | X | - | - | - | - | - | RPP |

Kobashy et al. [22] | TIC-REP | TW | 166K | 2011 | - | - | X | X | - | X | - | - | TiDeH |

Rodrigues et al. [23] | TIC | TW | 17K | 2013 | - | X | X | - | - | X | X | - | GetMove |

Cao et al. [24] | REP | SW, PC | 50K T, 35K P | 2016 | - | - | X | - | X | - | - | - | DH |

Zhou et al. [25] | TIC-RB | SW | 69.4M | 2013–2014 | X | - | X | X | - | X | - | X | BN |

Stai et al. [26] | TW | 35K | 2014–2016 | X | - | X | - | EpiM | |||||

Bhowmick et al. [27] | TIC-KR | TW | 8M T | 2015–2018 | - | - | X | - | - | X | - | - | SmartInf |

Chen et al. [28] | REP | TW | 20K | 2016 | - | X | X | - | - | - | - | - | NPP |

Liu et al. [29] | TIC | TW, AM | 30K TW, 35K AM | 2016, 1996–2000 | - | - | X | - | - | - | - | - | ANN |

Kong et al. [30] | TIC | TW | 210 K | - | - | - | X | - | - | X | - | - | EP+H |

Wu et al. [31] | KR-TIC | SW | 50K M | - | - | - | X | X | X | - | X | RL2R | |

in this work | TIC | TW | 16K T | 2020 | - | - | X | X | X | X | - | - | W-RCM |

**Target**—key retwetters (KR), tweet information cascade (TIC), retweeting behavior (RB), retweet engagement prediction (REP). ${}^{b}$

**Dataset**—AMiner (AM), Digg (D), paper’s citations (PC), Sina—Weibo (SW), Twitter (TW), Tencent—Weibo (TWB), YouTube (YT). ${}^{c}$ Dataset size—thousand (K), messages (M), papers (P), tweets (T), users (U), YouTube videos (YT). ${}^{d}$

**Model**—attention neural networks (ANN), linear regression (LR), features (F), expectation maximization (EM), epidemic models (EP), support vector machines (SVM), naive Bayes (NB), relation base learning to rank (RL2R), retweet cascade modeling (RCM), Cox proportional hazard regression (CHR), power law (PL), neural popularity prediction (NPP), DeepHawakes (DH), SmartInfluencer (SmartInf), Hawkes process (H), epidemic model (EpiM), reinforcement Poisson process (RPP), networked Weibull regression (NEWER), hierarchical Bayesian approach (HB), 7 Metrics (7M), self exciting point process (SEISMIC), Bayesian networks (BN).

Set nr | Retweet Weight | Quote Weight | Reply Weight |
---|---|---|---|

0.0 | 1.0 | 1.0 | 1.0 |

0.1 | 1.0 | 0.0 | 0.0 |

0.2 | 0.0 | 1.0 | 0.0 |

0.3 | 0.0 | 0.0 | 1.0 |

Set nr | Retweet Weight | Quote Weight | Reply Weight |
---|---|---|---|

1 | 0.0 | 0.3 | 0.6 |

2 | 0.0 | 0.6 | 0.3 |

3 | 0.3 | 0.0 | 0.6 |

4 | 0.3 | 0.6 | 0.0 |

5 | 0.6 | 0.0 | 0.3 |

6 | 0.6 | 0.3 | 0.0 |

7 | 0.1 | 0.4 | 0.5 |

8 | 0.1 | 0.5 | 0.4 |

9 | 0.2 | 0.3 | 0.5 |

10 | 0.2 | 0.5 | 0.3 |

Full Dataset | Sampled Dataset | |
---|---|---|

Tweets Count | 506,147 | 16,304 |

Unique Users Count | 102,468 | 41,592 |

Retweets Count | 683,189 | 112,188 |

Class | Set | Edge Count | CCC | GCS (%) | RFR (%) | GRC (%) | CAS (%) | Depth | SNI (%) |
---|---|---|---|---|---|---|---|---|---|

Guideline | 0.0 | 5.88 | 1.08 | 6.25 | 93.81 | 6.19 | 87.06 ${}^{\u2020,\diamond}$ | 0.53 | 1.98 |

Guideline | 0.1 | 5.88 | 1.08 | 6.29 | 93.75 | 6.18 | 85.39 ${}^{\star ,\diamond ,\u2022}$ | 0.53 | 2.05 |

Guideline | 0.2 | 5.89 | 1.08 | 6.27 | 93.68 | 6.15 | 84.33 ${}^{\u2020}$ | 0.53 | 2.13 |

Guideline | 0.3 | 5.89 | 1.08 | 6.29 | 93.68 | 6.15 | 84.32 ${}^{\star ,\u2020}$ | 0.53 | 2.11 |

**Table 6.**Average metrics for each weight set in the experimental setups, computed on the whole dataset.

Set | Edge Count | CCC | GCS (%) | RFR (%) | GRC (%) | CAS (%) | Depth | SNI (%) |
---|---|---|---|---|---|---|---|---|

1 | 5.88 | 1.08 | 6.27 | 93.61 | 6.12 | 83.53 ${}^{\star ,\diamond ,\u2022}$ | 0.53 | 2.04 |

2 | 5.89 | 1.08 | 6.25 | 93.59 | 6.12 | 82.93 ${}^{\star ,\diamond ,\u2022}$ | 0.53 | 2.01 |

3 | 5.89 | 1.08 | 6.29 | 93.77 | 6.17 | 86.32 ${}^{\u2020,\u2022}$ | 0.53 | 2.08 |

4 | 5.89 | 1.08 | 6.25 | 93.81 | 6.19 | 87.23 ${}^{\diamond ,\u2022}$ | 0.52 | 1.98 |

5 | 5.80 | 1.08 | 6.20 | 93.77 | 6.06 | 84.63 ${}^{\star ,\diamond}$ | 0.52 | 2.03 |

6 | 5.80 | 1.08 | 6.20 | 93.74 | 6.06 | 84.21 ${}^{\u2020,\diamond}$ | 0.52 | 2.04 |

7 | 5.80 | 1.08 | 6.22 | 93.78 | 6.07 | 84.80 ${}^{\star ,\u2020,\diamond ,\u2022}$ | 0.52 | 2.11 |

8 | 5.88 | 1.08 | 6.28 | 93.62 | 6.15 | 83.45 ${}^{\star ,\u2020,\diamond ,\u2022}$ | 0.53 | 1.98 |

9 | 5.88 | 1.08 | 6.29 | 95.73269 | 5.76 | 83.67 ${}^{\u2020,\diamond ,\u2022}$ | 0.54 | 1.98 |

10 | 5.88 | 1.08 | 6.31 | 93.79 | 6.18 | 86.63 ${}^{\u2020,\diamond ,\u2022}$ | 0.53 | 2.00 |

Group | Weight | CAS | |||
---|---|---|---|---|---|

Set | Quotes | Replies | Mean | St Dev | |

a | 0.2 | 1.0 | 0.0 | 0.28 | 0.06 |

0.3 | 0.0 | 1.0 | 0.32 * | 0.08 * | |

b | 1 | 0.3 | 0.6 | 0.26 | 0.07 |

2 | 0.6 | 0.3 | 0.27 | 0.07 | |

c | 3 | 0.0 | 0.6 | 0.28 | 0.07 |

4 | 0.6 | 0.0 | 0.30 * | 0.08 | |

d | 5 | 0.0 | 0.3 | 0.29 | 0.07 |

6 | 0.3 | 0.1 | 0.31 | 0.08 | |

e | 7 | 0.4 | 0.5 | 0.26 | 0.06 |

8 | 0.5 | 0.4 | 0.26 | 0.06 | |

f | 9 | 0.3 | 0.5 | 0.28 | 0.09 |

10 | 0.5 | 0.3 | 0.26 | 0.05 | |

g | 0 | 1.0 | 1.0 | 0.28 | 0.07 |

0.1 | 0.0 | 0.0 | 0.28 | 0.06 |

**Table 8.**Average metrics for all the setups (guideline and experimental), computed on the cascades with more than five nodes.

Set | Edge Count | CCC | GCS (%) | RFR (%) | GRC (%) | CAS (%) | Depth | SNI (%) |
---|---|---|---|---|---|---|---|---|

0.0 | 46.17 | 1.54 | 41.96 | 60.76 | 32.84 | 46.77 ${}^{\u2020,\diamond ,\u2022}$ | 2.37 | 16.33 |

0.1 | 46.17 | 1.56 | 42.31 | 60.29 | 32.71 | 41.45 ${}^{\star ,\diamond ,\u2022}$ | 2.37 | 16.96 |

0.2 | 46.21 | 1.56 | 42.09 | 59.78 | 32.54 | 38.34 ${}^{\star ,\u2020,\u2022}$ | 2.40 | 17.56 |

0.3 | 46.21 | 1.55 | 42.28 | 59.80 | 32.58 | 38.46 ${}^{\star ,\u2020,\diamond}$ | 2.40 | 17.43 |

ine 1 | 46.17 | 1.55 | 42.13 | 59.37 | 32.35 | 36.09 ${}^{\star ,\diamond ,\u2022}$ | 2.43 | 16.81 |

2 | 46.21 | 1.52 | 41.92 | 59.43 | 32.27 | 35.90 ${}^{\star ,\diamond ,\u2022}$ | 2.43 | 16.54 |

3 | 46.21 | 1.55 | 42.28 | 60.51 | 32.65 | 43.65 ${}^{\u2020,\diamond ,\u2022}$ | 2.39 | 17.14 |

4 | 46.21 | 1.52 | 41.95 | 60.70 | 32.82 | 48.31 ${}^{\star ,\u2020,\diamond ,\u2022}$ | 2.36 | 16.33 |

5 | 46.15 | 1.56 | 42.16 | 59.75 | 32.56 | 38.80 ${}^{\star ,\u2020,\diamond ,\u2022}$ | 2.39 | 16.96 |

6 | 46.15 | 1.56 | 42.09 | 59.58 | 32.54 | 37.71 ${}^{\star ,\u2020,\diamond ,\u2022}$ | 2.39 | 17.08 |

7 | 46.15 | 1.56 | 42.31 | 59.83 | 32.58 | 39.34 ${}^{\star ,\u2020,\diamond ,\u2022}$ | 2.39 | 17.64 |

8 | 46.17 | 1.55 | 42.20 | 59.48 | 32.53 | 36.39 ${}^{\star ,\u2020,\diamond ,\u2022}$ | 2.41 | 16.35 |

9 | 46.17 | 1.55 | 42.12 | 55.36 | 31.12 | 45.43 ${}^{\star ,\u2020,\diamond ,\u2022}$ | 2.37 | 16.43 |

10 | 46.17 | 1.55 | 42.46 | 60.63 | 32.73 | 44.74 ${}^{\star ,\u2020,\diamond ,\u2022}$ | 2.38 | 16.47 |

**Table 9.**Comparison between the two proposed approaches (with the best weight set) and the baseline approach.

Edge Count | Depth | CCC | GCS (%) | RFR (%) | GRC (%) | CAS (%) | SNI (%) | |
---|---|---|---|---|---|---|---|---|

ISN (weight set 10) | 6.09 | 0.56 | 1.26 | 6.65 | 91.13 | 6.60 | 85.29 | 2.31 |

ISN-AF (weight set 10) | 6.09 | 0.51 | 1.15 | 6.90 | 94.22 | 7.23 | 94.70 | 2.89 |

Baseline | 5.49 | 0.37 | 2.25 | 4.99 | 92.23 | 6.32 | - | 6.66 |

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**MDPI and ACS Style**

Zola, P.; Cola, G.; Mazza, M.; Tesconi, M. Interaction Strength Analysis to Model Retweet Cascade Graphs. *Appl. Sci.* **2020**, *10*, 8394.
https://doi.org/10.3390/app10238394

**AMA Style**

Zola P, Cola G, Mazza M, Tesconi M. Interaction Strength Analysis to Model Retweet Cascade Graphs. *Applied Sciences*. 2020; 10(23):8394.
https://doi.org/10.3390/app10238394

**Chicago/Turabian Style**

Zola, Paola, Guglielmo Cola, Michele Mazza, and Maurizio Tesconi. 2020. "Interaction Strength Analysis to Model Retweet Cascade Graphs" *Applied Sciences* 10, no. 23: 8394.
https://doi.org/10.3390/app10238394