# Opinion Formation on Social Networks—The Effects of Recurrent and Circular Influence

## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. Mathematical Formulation

#### 2.2. Microscopic Modelling and Extensibility

## 3. Results

#### 3.1. Features and Interpretation of the Model

#### 3.2. Numerical Demonstrations

#### 3.2.1. Out- and In-Centrality Measures

#### 3.2.2. Betweenness Measure

#### 3.2.3. Demonstration with a Larger Network Structure

#### 3.2.4. Profiling

- If the node’s ratio of the in-centrality to out-centrality is larger than $1.1$, it is classified as ‘Peripheral’ [or $0.7$ as (Peripheral)].
- If the node’s out-centrality value is larger than $0.9$, it is classified as ‘Central’ [or $0.7$ as (Central)].
- If the node’s betweenness value is larger than $0.9$, it is classified as a ‘Mediator’ [or $0.7$ as (Mediator)]. (Sum of betweenness values is normalised to the same value as the out- and in-centrality sums)

## 4. Conclusions

## Funding

## Conflicts of Interest

## Abbreviations

CC | Complex Contagion |

SC | Simple Contagion |

LCP | Longest Common Prefix |

## Appendix A

**Figure A1.**Out- and in-centrality values with a maximum path length ${L}_{max}=10$ and link weights $w=0.025,0.05,0.25,0.5,0.75$, and $0.95$. Nodes on the horizontal axes are arranged in the ascending order of out-centrality values.

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**Figure 1.**The Dutch students’ social network [48].

**Figure 6.**Centrality values for link weight values $w=0.05,0.25,0.5,0.75,$ and $0.95$ in the simple contagion model.

**Figure 9.**Rankings of out-centrality, in-centrality and betweenness values for the Dutch students’ network of Figure 1 ($w=0.25$). Numerical values of the quantities are provided in Figure A2 in the Appendix A.

**Figure 10.**The largest connected component of the collaboration network [49].

**Figure 11.**Out- and in-centrality values with link weights $w=0.05,0.25,0.5$, and $0.75$. Nodes on the horizontal axes are arranged in the ascending order of out-centrality values.

**Figure 12.**Network structure for illustration of centrality measures and profiling of nodes [34].

**Figure 13.**Out- and in-centrality values for the test network of ten nodes in the complex contagion (CC) model. (

**a**) Link weight $w=0.05$ (

**b**) Link weight $w=0.25$ (

**c**) Link weight $w=0.5$ (

**d**) Link weight $w=0.75$.

**Figure 14.**Betweenness values for the test network of ten nodes for link weight values $w=0.05,0.25,0.5,0.75,$ and $0.95$. (

**a**) complex contagion (CC) model. (

**b**) simple contagion (SC) model.

**Figure 16.**Out-centrality, in-centrality and betweenness values of a set of connected nodes in the Collaboration network Figure 10. (Numerical values are normalised equally for the three measures). (

**a**) Link weights of outsider nodes $w=0.0$. (

**b**) Link weights of outsider nodes $w=0.025$.

**Figure 17.**Profiles of the nodes in the Dutch students’ social network of Figure 1 of 32 nodes ($w=0.25$).

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Kuikka, V.
Opinion Formation on Social Networks—The Effects of Recurrent and Circular Influence. *Computation* **2023**, *11*, 103.
https://doi.org/10.3390/computation11050103

**AMA Style**

Kuikka V.
Opinion Formation on Social Networks—The Effects of Recurrent and Circular Influence. *Computation*. 2023; 11(5):103.
https://doi.org/10.3390/computation11050103

**Chicago/Turabian Style**

Kuikka, Vesa.
2023. "Opinion Formation on Social Networks—The Effects of Recurrent and Circular Influence" *Computation* 11, no. 5: 103.
https://doi.org/10.3390/computation11050103