# Semi-Local Integration Measure of Node Importance

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## Abstract

**:**

## 1. Introduction

#### Related Work

## 2. Semi-Local Intregation Centrality

#### 2.1. $SLI$ Definition

- $deg\left(v\right)$—denotes the $degree$ of the node $v\in V$;
- ${d}^{w}\left(v\right)$—denotes the $weighted\phantom{\rule{3.33333pt}{0ex}}degree$ ($strength$) of the node $v\in V$;
- ${e}_{ab}={e}_{ba}$—denotes the edge between the nodes $a,b\in V$;
- $w\left(e\right)$—denotes the weight of the edge $e\in E$;
- ${E}_{v}$—denotes the set of edges incident to $v\in V$, ${E}_{v}:=\{x\mid {e}_{vx}\in E\}$;
- ${P}_{G}$—denotes the cycle basis of the graph G;
- $p\left(e\right)$—denotes the number of cycles in ${P}_{G}$ that contain the edge $e\in E$.

- Find cycle basis of G, ${P}_{G}$;
- Find $p\left(e\right)$ for all $e\in E$;
- Find the set of edges $E\left(a\right)$ for all $a\in V$;
- Find $S\left(G\right)$, according to (5);

#### 2.2. Discussion on $SLI$ Definition

#### 2.3. $SLI$ in Unweighted Graphs

## 3. Application of $\mathit{SLI}$ in Lexical Networks

#### 3.1. Application of $SLI$ in the Analysis of Sense Structure

#### 3.2. Application of $SLI$ in Sentiment Analysis

#### 3.3. Further Areas of Applications

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**ConGraCNet network representation of the sense structure of a seed noun lexeme work. The node size reflects the node $SLI$ score.

**Figure 3.**ConGraCNet network representation of the sense structure of a seed noun lexeme work pruned to top 50% nodes accoring to $SLI\phantom{\rule{0.166667em}{0ex}}$ centrality. The node size reflects the node $SLI\phantom{\rule{0.166667em}{0ex}}$ score.

**Figure 4.**Sentiment potential (SP) of noun lexeme work propagated from SenticNet using betweenness centrality.

**Figure 5.**Sentiment potential (SP) of noun lexeme work propagated from SenticNet using $SLI\phantom{\rule{0.166667em}{0ex}}$ centrality.

**Table 1.**Comparison of different centrality measures of nodes in graph ${G}_{1}$ illustrated in Figure 1.

Node | $\mathit{SLI}\phantom{\rule{0.166667em}{0ex}}$ | $\mathit{Degree}$ | $\mathit{Weigthed}\phantom{\rule{3.33333pt}{0ex}}\mathit{Degree}$ | $\mathit{Betweenness}$ | $\mathit{PageRank}$ |
---|---|---|---|---|---|

${v}_{1}$ | 40.225 | 6 | 12.65 | 0.594 | 0.148 |

${v}_{2}$ | 16.225 | 7 | 9.4 | 0.345 | 0.118 |

${v}_{4}$ | 13.269 | 6 | 7.45 | 0.246 | 0.096 |

${v}_{6}$ | 9.614 | 5 | 6.1 | 0.147 | 0.076 |

${v}_{5}$ | 6.961 | 5 | 5.7 | 0.193 | 0.077 |

${v}_{3}$ | 6.664 | 6 | 5.65 | 0.259 | 0.081 |

${v}_{60}$ | 1.671 | 2 | 2.7 | 0.024 | 0.035 |

${v}_{7}$ | 1.238 | 4 | 3.6 | 0.239 | 0.071 |

${v}_{40}$ | 0.969 | 2 | 2 | 0.0 | 0.029 |

${v}_{21}$ | 0.569 | 1 | 2 | 0.0 | 0.027 |

${v}_{52}$ | 0.546 | 2 | 1.4 | 0.0 | 0.022 |

${v}_{70}$ | 0.39 | 1 | 2 | 0.0 | 0.039 |

${v}_{23}$ | 0.224 | 1 | 1 | 0.0 | 0.017 |

${v}_{31}$ | 0.21 | 1 | 1 | 0.0 | 0.018 |

${v}_{20}$ | 0.166 | 1 | 0.8 | 0.0 | 0.015 |

${v}_{51}$ | 0.16 | 1 | 0.8 | 0.0 | 0.015 |

${v}_{41}$ | 0.15 | 1 | 0.75 | 0.0 | 0.014 |

${v}_{71}$ | 0.149 | 1 | 0.8 | 0.0 | 0.019 |

${v}_{22}$ | 0.139 | 1 | 0.7 | 0.0 | 0.013 |

${v}_{30}$ | 0.11 | 1 | 0.6 | 0.0 | 0.013 |

${v}_{61}$ | 0.088 | 1 | 0.5 | 0.0 | 0.011 |

${v}_{42}$ | 0.067 | 1 | 0.4 | 0.0 | 0.010 |

${v}_{50}$ | 0.067 | 1 | 0.4 | 0.0 | 0.011 |

${v}_{32}$ | 0.067 | 1 | 0.4 | 0.0 | 0.011 |

${v}_{72}$ | 0.065 | 1 | 0.4 | 0.0 | 0.013 |

**Table 2.**Comparison of different centrality measures of nodes in graph ${G}_{2}$, which is the unweighted version of the graph ${G}_{1}$ illustrated in Figure 1.

Node | $\mathit{SLI}\phantom{\rule{0.166667em}{0ex}}$ | $\mathit{Degree}$ | $\mathit{Betweenness}$ | $\mathit{PageRank}$ |
---|---|---|---|---|

${v}_{1}$ | 18.433 | 6 | 0.594 | 0.087 |

${v}_{2}$ | 16.708 | 7 | 0.345 | 0.111 |

${v}_{4}$ | 15.543 | 6 | 0.246 | 0.091 |

${v}_{6}$ | 11.437 | 5 | 0.147 | 0.074 |

${v}_{3}$ | 12.892 | 6 | 0.259 | 0.094 |

${v}_{5}$ | 9.51 | 5 | 0.193 | 0.077 |

${v}_{7}$ | 3.512 | 4 | 0.239 | 0.073 |

${v}_{60}$ | 1.943 | 2 | 0.024 | 0.032 |

${v}_{40}$ | 1.951 | 2 | 0.0 | 0.032 |

${v}_{52}$ | 1.882 | 2 | 0.0 | 0.032 |

${v}_{21}$ | 0.427 | 1 | 0.0 | 0.02 |

${v}_{23}$ | 0.427 | 1 | 0.0 | 0.02 |

${v}_{31}$ | 0.418 | 1 | 0.0 | 0.019 |

${v}_{20}$ | 0.427 | 1 | 0.0 | 0.02 |

${v}_{41}$ | 0.418 | 1 | 0.0 | 0.019 |

${v}_{51}$ | 0.407 | 1 | 0.0 | 0.019 |

${v}_{22}$ | 0.427 | 1 | 0.0 | 0.02 |

${v}_{71}$ | 0.39 | 1 | 0.0 | 0.022 |

${v}_{30}$ | 0.418 | 1 | 0.0 | 0.019 |

${v}_{61}$ | 0.407 | 1 | 0.0 | 0.019 |

${v}_{70}$ | 0.39 | 1 | 0.0 | 0.022 |

${v}_{32}$ | 0.418 | 1 | 0.0 | 0.019 |

${v}_{42}$ | 0.418 | 1 | 0.0 | 0.019 |

${v}_{50}$ | 0.407 | 1 | 0.0 | 0.019 |

${v}_{72}$ | 0.39 | 1 | 0.0 | 0.022 |

**Table 3.**Centrality in lexical dependency graph of noun lexeme work and SenticNet 6 sentiment scores.

Lexeme | $\mathit{Weighted}\phantom{\rule{3.33333pt}{0ex}}\mathit{Degree}$ | $\mathit{Betweenness}$ | $\mathit{SLI}\phantom{\rule{0.166667em}{0ex}}$ | SenticNet6 $\mathit{ODV}$ |
---|---|---|---|---|

work | 96.35 | 6219.4667 | 11.0602 | 0.9 |

family | 143.21 | 1615.5262 | 10.7711 | 0.883 |

time | 115.68 | 1057.2845 | 7.7119 | - |

dedication | 103.43 | 788.0095 | 7.3328 | 0.034 |

research | 117.88 | 1459.5190 | 7.0717 | 0.883 |

determination | 108.08 | 1037.9310 | 7.0436 | 0.231 |

effort | 93.85 | 982.9595 | 6.3767 | 0.037 |

study | 120.33 | 1687.6012 | 6.2252 | - |

commitment | 97.82 | 855.6643 | 6.1272 | 0.704 |

home | 107.3 | 1059.3940 | 6.0986 | - |

project | 115.6 | 1705.4262 | 5.6272 | 0.9 |

school | 107.21 | 1129.0250 | 5.6033 | - |

life | 101.38 | 1188.8881 | 4.6721 | - |

play | 96.1 | 1677 | 2.5525 | - |

passion | 24.54 | 0 | 0.3868 | 1 |

business | 23.7 | 21.0357 | 0.3673 | - |

money | 18.79 | 0 | 0.1937 | 0.065 |

**Table 4.**Sentiment potential (SP): $GSV$ scores of lexical communities of noun lexeme work propagated from SenticNet 6.

SenticNet 6 | Work-n | ||
---|---|---|---|

Community | Lexemes | $\mathit{Betweenness}\phantom{\rule{3.33333pt}{0ex}}\mathit{GSV}$ | $\mathit{SLI}\phantom{\rule{0.166667em}{0ex}}$$\mathit{GSV}$ |

1 | life-n, school-n, home-n, family-n, love-n, property-n, business-n, hospital-n, student-n, community-n, program-n, building-n | 0.37 | 0.61 |

2 | dedication-n, commitment-n, determination-n, passion-n, perseverance-n, enthusiasm-n, patience-n, loyalty-n, persistence-n, courage-n | 0.47 | 0.33 |

3 | work-n, play-n, research-n, study-n, project-n, patient-n, development-n, analysis-n, science-n | 0.76 | 0.88 |

4 | effort-n, time-n, money-n, energy-n, resource-n, cost-n, attention-n, people-n | 0.16 | 0.07 |

$\mathit{ODV}$: 0.9 | $\mathit{betweenness}\phantom{\rule{3.33333pt}{0ex}}\mathit{ADV}$: 0.78 $\mathit{SLI}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathit{ADV}$: 0.65 | ||

$\mathit{betweenness}\phantom{\rule{3.33333pt}{0ex}}\mathit{ASP}$: 0.45 $\mathit{SLI}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathit{ASP}$: 0.56 |

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**MDPI and ACS Style**

Ban Kirigin, T.; Bujačić Babić, S.; Perak, B.
Semi-Local Integration Measure of Node Importance. *Mathematics* **2022**, *10*, 405.
https://doi.org/10.3390/math10030405

**AMA Style**

Ban Kirigin T, Bujačić Babić S, Perak B.
Semi-Local Integration Measure of Node Importance. *Mathematics*. 2022; 10(3):405.
https://doi.org/10.3390/math10030405

**Chicago/Turabian Style**

Ban Kirigin, Tajana, Sanda Bujačić Babić, and Benedikt Perak.
2022. "Semi-Local Integration Measure of Node Importance" *Mathematics* 10, no. 3: 405.
https://doi.org/10.3390/math10030405