# Opinion Diversity and the Resilience of Cooperation in Dynamical Networks

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Computational Model

#### 2.2. Information-Based Decision Making

- If $P\wedge Q$ Then Connect
- If $\neg P\wedge $$\neg Q$ Then Do not Connect
- If $P\wedge \neg Q$ Then Connect with probability p
- If $\neg P\wedge Q$ Then Connect with probability q

## 3. Results

#### 3.1. Private Information Decision Making

#### 3.1.1. Increasing Opinion Diversity Leads to Network Changes

#### 3.1.2. Increasing Selection Strength Leads to Increase in Network Instability

#### 3.2. Private/Public Information Decision Making

#### 3.2.1. Opinion Diversity Affects the Frequency of Information Cascades

#### 3.2.2. Strong Selection Results in Higher Frequency of Information Cascades

#### 3.2.3. Opinion Diversity Can Mitigate Disruption of Networks

#### 3.2.4. Sharp Increases in the Number of Transitions for Strong Selection

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. RStudio Functions

#### Appendix A.1. Standard Error Function

#### Appendix A.2. Graph Initialisation Function

#### Appendix A.3. Cooperation Plot Function

#### Appendix A.4. Prosperity Plot Function

#### Appendix A.5. Degree Plot Function

#### Appendix A.6. Transition Plot Function

#### Appendix A.7. NCascade Count Plot Function

#### Appendix A.8. NCascade Length Plot Function

#### Appendix A.9. PCascade Count Plot Function

#### Appendix A.10. PCascade Length Plot Function

#### Appendix A.11. Specificity and Sensitivity Plot Function

## References

- Easley, D.; Kleinberg, J. Networks, Crowds, and Markets; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Kraft-Todd, G.; Yoeli, E.; Bhanot, S.; Rand, D. Promoting Cooperation in the Field. Curr. Opin. Behav. Sci.
**2015**, 3, 96–101. [Google Scholar] [CrossRef] - Rand, D.G.; Nowak, M.A. Human Cooperation. Trends Cogn. Sci.
**2013**, 17, 413–425. [Google Scholar] [CrossRef] [PubMed] - Hauser, O.P.; Hendriks, A.; Rand, D.G.; Nowak, M.A. Think global, act local: Preserving the global commons. Sci. Rep.
**2016**, 6, 36079. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Levin, S. Crossing Scales, Crossing Disciplines: Collective Motion and Collective Action in the Global Commons. Philos. Trans. R. Soc. B Biol. Sci.
**2010**, 365, 13–18. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lloyd, W.F. WF Lloyd on the Checks to Population. Popul. Dev. Rev.
**1980**, 6, 473–496. [Google Scholar] [CrossRef] - Nowak, M.A. Evolutionary Dynamics: Exploring the Equations of Life; Harvard University Press: Cambridge, MA, USA, 2006. [Google Scholar]
- Leskovec, J.; Singh, A.; Kleinberg, J. Patterns of Influence in a Recommendation Network. In Pacific-Asia Conference on Knowledge Discovery and Data Mining; Springer: Berlin, Germany, 2006; pp. 380–389. [Google Scholar]
- Maharani, W.; Gozali, A.A.; Adiwijaya. Degree Centrality and Eigenvector Centrality in Twitter. In Proceedings of the 2014 8th International Conference on Telecommunication Systems Services and Applications (TSSA), Kuta, Indonesia, 23–24 October 2014; pp. 1–5. [Google Scholar]
- Wooldridge, M. Computation and the Prisoner’s Dilemma. IEEE Intell. Syst.
**2012**, 27, 75–80. [Google Scholar] [CrossRef] - Bikhchandani, S.; Hirshleifer, D.; Welch, I. Learning from the Behavior of Others: Conformity, Fads, and Informational Cascades. J. Econ. Perspect.
**1998**, 12, 151–170. [Google Scholar] [CrossRef] [Green Version] - Sánchez, A. Physics of Human Cooperation: Experimental Evidence and Theoretical Models. J. Stat. Mech. Theory Exp.
**2018**, 2018, 024001. [Google Scholar] [CrossRef] [Green Version] - Wu, B.; Zhou, D.; Fu, F.; Luo, Q.; Wang, L.; Traulsen, A. Evolution of cooperation on stochastic dynamical networks. PLoS ONE
**2010**, 5, e11187. [Google Scholar] [CrossRef] [PubMed] - Lee, S.; Son, Y.J. Extended Decision Field Theory with Social-Learning for Long-Term Decision-Making Processes in Social Networks. Inf. Sci.
**2020**, 512, 1293–1307. [Google Scholar] [CrossRef] - Tappin, B.M.; Pennycook, G.; Rand, D.G. Thinking Clearly About Causal Inferences of Politically Motivated Reasoning: Why Paradigmatic Study Designs Often Undermine Causal Inference. Curr. Opin. Behav. Sci.
**2020**, 34, 81–87. [Google Scholar] [CrossRef] - Busu, M. Game Theory in Strategic Management-Dynamic Games. Theoretical and Practical Examples. Manag. Dyn. Knowl. Econ.
**2018**, 6, 645–655. [Google Scholar] [CrossRef] - Yang, G.; Csikász-Nagy, A.; Waites, W.; Xiao, G.; Cavaliere, M. Information Cascades and the Collapse of Cooperation. Sci. Rep.
**2020**, 10, 1–13. [Google Scholar] [CrossRef] [PubMed] - Sood, V.; Redner, S. Voter model on heterogeneous graphs. Phys. Rev. Lett.
**2005**, 94, 178701. [Google Scholar] [CrossRef] [Green Version] - Xue, M.; Roy, S. Averager-copier-voter models for hybrid consensus. IFAC-PapersOnLine
**2016**, 49, 1–6. [Google Scholar] [CrossRef] - Avilés, L. Cooperation and non-linear dynamics: An ecological perspective on the evolution of sociality. Evol. Ecol. Res.
**1999**, 1, 459–477. [Google Scholar] - Traulsen, A.; Shoresh, N.; Nowak, M.A. Analytical results for individual and group selection of any intensity. Bull. Math. Biol.
**2008**, 70, 1410–1424. [Google Scholar] [CrossRef] [Green Version] - Loess: Local Polynomial Regression Fitting—RDocumentation. Available online: https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/loess (accessed on 26 April 2021).
- Tomassini, M.; Pestelacci, E. Coordination games on dynamical networks. Games
**2010**, 1, 242–261. [Google Scholar] [CrossRef] [Green Version] - Liu, S.; Zhao, L.; Zhang, J. Strategy Dynamics with Feedback Control in the Global Climate Dilemma Games. In Proceedings of the 2019 IEEE International Conference on Systems, Man and Cybernetics (SMC), Bari, Italy, 6–9 October 2019; pp. 518–522. [Google Scholar]
- Pejó, B.; Biczók, G. Corona Games: Masks, Social Distancing and Mechanism Design. In Proceedings of the 1st ACM SIGSPATIAL International Workshop on Modeling and Understanding the Spread of COVID-19, Seattle, WA, USA, 3 November 2020; pp. 24–31. [Google Scholar]
- Brodie, B. Strategy in the Missile Age; Princeton University Press: Princeton, NJ, USA, 2015. [Google Scholar]

**Figure 1.**The payoff matrix that is used to implement the prisoner’s dilemma. The choice each player makes during an interaction with each other will affect the payoff received. Here, b represents the benefits distributed by a cooperator, whilst c represents the cost that the act of cooperating will incur. Parameters should be configured to adhere to the condition of $b>c>0$. In the simulations we use $b=10$ and $c=8$.

**Figure 2.**An illustration of the process that is undertaken as a newcomer joins the network at each step during the simulation. Firstly, a newcomer will select an existing node to act as its role model, which is influenced by both nodes current payoff and the current selection strength (

**a**). Following this, the newcomer will adopt a strategy, by either copying the role model or mutating (

**b**). Lastly, the newcomer will then proceed to choose which of the role models neighbours to form connections with, which is achieved by making use of available public and private information (

**c**) [17].

**Figure 3.**Truth table for determining which combination of choices made by nodes are considered as contributing to cascades. The first column indicates whether the choice to connect (C) or not connect (NC) is correct. The second and third columns show the indications from public and private information. The forth column is the actual choice made by the newcomer. The fifth column indicates if a cascade has occurred, either P or N cascade, or if none has occurred with 0.

**Figure 4.**An illustration of the two types of cascades that are monitored and and how they can potentially propagate as newcomers join the network. A P cascade is considered to have occurred when a node does not form a connection with another cooperator based on misleading public information. A N cascade is considered to have occurred when a node incorrectly forms a connection with a defector based on misleading public information. Diagram is based on illustration presented in [17] used to describe the progression of cascades in cooperative networks.

**Figure 5.**Specificity and sensitivity of individual decision making. Plots visualise and compare the specificity and sensitivity values that were recorded from simulations. In general, as diversity of opinion diversity is increased, nodes are able to better identify cheaters at higher $\tau $ settings. $\tau $ represents the decision threshold. $\sigma $ represents opinion diversity. Data obtained by running the model for ${10}^{8}$ steps. Plots produced via RStudio utilising code found in Appendix A.

**Figure 6.**Increasing diversity of opinion leads to less transitions and connections, but also less cooperation. The plots visualise and compare the metrics of cooperation, prosperity, average connectivity and number of transitions for differing levels of opinion diversity. As diversity is increased, the number of transitions and connections decreases at various $\tau $ settings. This does appear to come at some cost for the average amount of cooperation and prosperity obtained for higher diversity. Data obtained by running the model for ${10}^{8}$ steps considering only private information and a $\delta $ setting of 0.001. $\tau $ represents the decision threshold. $\sigma $ represents opinion diversity. Shading represents level of standard error calculated for each dataset. Data interpolated utilising RStudio [22]. Plots produced via RStudio utilising code found in Appendix A.

**Figure 7.**Transitions are more frequent in the case of strong selection when only private information is used. The plots visualise and compare the metrics of cooperation, prosperity, average connectivity and number of transitions observed in networks that utilise differing levels of opinion diversity. Although there are some similarities with Figure 6 for how increasing $\sigma $ can affect cooperation, strong selection clearly leads to an increase in transitions. Networks here struggled to engage in long term cooperation for higher values of $\tau $. Data obtained by running the model for ${10}^{8}$ steps with only private information and a $\delta $ setting of 0.1. $\tau $ represents the decision threshold. $\sigma $ represents opinion diversity. Shading represents level of standard error calculated for each dataset. Data interpolated utilising RStudio [22]. Plots produced via RStudio utilising code found in Appendix A.

**Figure 8.**Limited public information leads to a limited decrease in cooperation. The plots visualise and compare the metrics of cooperation, prosperity, average connectivity and number of transitions for differing levels of opinion diversity. Data obtained by running the model for ${10}^{8}$ steps with $p=0.25$ and $q=0.75$ and a $\delta $ setting of 0.001. $\tau $ represents the decision threshold. $\sigma $ represents opinion diversity. Shading represents level of standard error calculated for each dataset. Data interpolated utilising RStudio [22]. Plots produced via RStudio utilising code found in Appendix A.

**Figure 9.**Increased opinion diversity generally leads to more, but shallower, information cascades when private information is prioritised. For both cascade types, each graph represents either the number or average length of each cascade type. Data obtained by running the model for ${10}^{8}$ steps with $p=0.25$ and $q=0.75$ and a $\delta $ setting of 0.001. $\tau $ represents the decision threshold. $\sigma $ represents opinion diversity. Shading represents level of standard error calculated for each dataset. Data interpolated utilising RStudio [22]. Plots produced via RStudio.

**Figure 10.**Public information leads to a moderate loss in cooperation when selection is strong and private information is prioritised. The plots visualise and compare the metrics of cooperation, prosperity, average connectivity and number of transitions for differing levels of opinion diversity. Data obtained by running the model for ${10}^{8}$ steps with $p=0.25$ and $q=0.75$ and a $\delta $ setting of 0.1. $\tau $ represents the decision threshold. $\sigma $ represents opinion diversity. Shading represents level of standard error calculated for each dataset. Data interpolated utilising RStudio [22]. Plots produced via RStudio utilising code found in Appendix A.

**Figure 11.**Number of P cascades decreases when selection is strong and private information is prioritised. For both cascade types, each graph represents either the number or average length of each cascade type. Under these circumstances, the number of P cascades that occurred gradually decreased as the level of diversity is increased, although there is some increase in areas for N cascades. Data obtained by running the model for ${10}^{8}$ steps with $p=0.25$ and $q=0.75$ and a $\delta $ setting of 0.1. $\tau $ represents the decision threshold. $\sigma $ represents opinion diversity. Shading represents level of standard error calculated for each dataset. Data interpolated utilising RStudio [22]. Plots produced via RStudio.

**Figure 12.**Prioritising public information leads to a significant drop in cooperation. The plots visualise and compare the metrics of cooperation, prosperity, average connectivity and number of transitions for differing levels of opinion diversity. Data obtained by running the model for ${10}^{8}$ steps with $p=0.75$ and $q=0.25$ and a $\delta $ setting of 0.001. $\tau $ represents the decision threshold. $\sigma $ represents opinion diversity. Shading represents level of standard error calculated for each dataset. Data interpolated utilising RStudio [22]. Plots produced via RStudio utilising code found in Appendix A.

**Figure 13.**Significant increases in cascades occur when public information is prioritised. For both cascade types, each graph represents either the number or average length of each cascade type. Data obtained by running the model for ${10}^{8}$ steps with $p=0.75$ and $q=0.25$ and a $\delta $ setting of 0.001. $\sigma $ represents opinion diversity. Shading represents level of standard error calculated for each dataset. Data interpolated utilising RStudio [22]. Plots produced via RStudio utilising code found in Appendix A.

**Figure 14.**Significant drop in cooperation and increase in transitions occur when public information is prioritised and selection is strong. The plots visualise and compare the metrics of cooperation, prosperity, average connectivity and number of transitions for differing levels of opinion diversity. Data obtained by running the model for ${10}^{8}$ steps with $p=0.75$ and $q=0.25$ and a $\delta $ setting of 0.1. $\tau $ represents the decision threshold. $\sigma $ represents opinion diversity. Shading represents level of standard error calculated for each dataset. Data interpolated utilising RStudio [22]. Plots produced via RStudio utilising code found in Appendix A.

**Figure 15.**Significant decrease in the number of P cascades occurs when public information is prioritised and selection is strong. For both cascade types, each graph represents either the number or average length of each cascade type. With strong selection and public information, significantly more cascades were recorded during simulations; however, increasing diversity of opinion has a significant impact on the number of P cascades that occurs during simulations. Data obtained by running the model for ${10}^{8}$ steps with $p=0.75$ and $q=0.25$ and a $\delta $ setting of 0.1. $\tau $ represents the decision threshold. $\sigma $ represents opinion diversity. Shading represents level of standard error calculated for each dataset. Data interpolated utilising RStudio [22]. Plots produced via RStudio utilising code found in Appendix A.

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Miles, A.L.; Cavaliere, M.
Opinion Diversity and the Resilience of Cooperation in Dynamical Networks. *Mathematics* **2021**, *9*, 1801.
https://doi.org/10.3390/math9151801

**AMA Style**

Miles AL, Cavaliere M.
Opinion Diversity and the Resilience of Cooperation in Dynamical Networks. *Mathematics*. 2021; 9(15):1801.
https://doi.org/10.3390/math9151801

**Chicago/Turabian Style**

Miles, Adam Lee, and Matteo Cavaliere.
2021. "Opinion Diversity and the Resilience of Cooperation in Dynamical Networks" *Mathematics* 9, no. 15: 1801.
https://doi.org/10.3390/math9151801