# Investigating Peer and Sorting Effects within an Adaptive Multiplex Network Model

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## Abstract

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## 1. Introduction

## 2. Literature Review

## 3. Model

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B | (0,0) | (1,1) |

## 4. Results

## 5. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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1 | The literature on the link between group diversity and group performance is vast and not univocal. While it is true that increasing diversity in groups elicits positive outcomes such as enhancing thoughtful decision processes [1], expanding access to social networks and resources [2], promoting innovation [3], and facilitating problem solving [4], it has been also proved that increasing diversity introduces group biases that may contribute to conflict among group members [5,6]. |

2 | In a social network environment homophily means that contact between similar people occurs at a higher rate than among dissimilar people [8]. In static terms homophily and segregation correspond to the same network phenomenon. |

3 | We refer to segregation as the non-random allocation of people who belong to different groups into social positions and the associated social and physical distances between groups [9]. |

4 | The cited works can be also classified by the type of coordination game implemented such as pure coordination, stag-hunt, anti-coordination and battle of the sexes. |

**Figure 1.**Visualization of the adaptive multiplex network model. The coordination with neighbor actions dynamics happens in the evident layer through imitation (according to S-agents) and rewiring (for R-agents), while the learning dynamics occurs in the hidden layer through payoff-driven strategy imitation. Actions are represented as node colors (orange and blue) while strategies are represented as node labels (R for rewiring agents, S for staying ones). Imitation refers to the coordination game according to which agents update their actions on the evident layer. Rewiring indicates the possibility, on the evident layer, for an agent to cut a link with a neighbor of opposite action and then create a link with another randomly chosen agent. Learning denotes the propagation of strategies over the hidden layer.

**Figure 2.**Panel of segregation matrix indexes (SMI) over different values of initial densities of rewirers (R0) for different values of rewiring parameter $\beta $ and learning parameter ${\rho}_{L}$. Every data point is averaged over 50 different replicates. Error bars were not reported since they are of the same size of dots.

**Figure 3.**Final configuration states of the multiplex evident layer. Visualization of the actions (color codes) and link disposition according to model parameters ${\rho}_{L}$ (learning dynamics rate), $\beta $ (hidden layer topology) and ${R}_{0}$ (initial fraction of R-agents). Full consensus (

**a**) can only occur when only S-agents are present in the system, but not always; coexistence (

**b**) is obtained otherwise, while segregation (

**c**) and more profitable social outcomes in terms of game payoffs are reached when R-agents spread in the population. Other panels (

**d**–

**g**) show outcomes having an heterogeneous initial distribution.

**Figure 4.**Segregation results for different hidden layers and R-agents. Segregation matrix indexes (SMI) (

**top row**) and modularity (

**bottom row**) for different values of initial fractions of R-agents (${R}_{0}$), hidden layer topologies ($\beta $) and either for no learning in the system (

**left panels**) or learning rate ${\rho}_{L}=0.5$ (

**right panels**). Every data point is averaged over 50 different replicates. Error bars are represented as gray overlays. When no learning is present in the system, consensus is reduced across all topologies when an optimal number ${R}_{0}^{*}$ of R-agents is initially present in the system. Hence, there is an optimal number of ${R}_{0}$ for reducing segregation. Above that value, the more R-agents in the system the more segregation is present and all curves converge on the same trajectory. The presence of learning destroys this overall convergence so that different hidden topologies display different levels of segregation.

**Figure 5.**Convergence simulation time. Results over the initial fraction of R-agents ${R}_{0}$, for three different hidden layer topologies: (i) a lattice ($\beta =0$, left), (ii) a Watts–Strogatz small-world ($\beta =0.1$) and (iii) a random graph ($\beta =1$). Data points are relative to no social learning ${\rho}_{L}=0$ (orange circles), moderate social learning ${\rho}_{L}=0.25$ (purple squares) and high-rate social learning ${\rho}_{L}=0.5$ (green diamonds). Every data point is averaged over 50 different replicates. Error bars are of the same size of dots.

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**MDPI and ACS Style**

Lipari, F.; Stella, M.; Antonioni, A. Investigating Peer and Sorting Effects within an Adaptive Multiplex Network Model. *Games* **2019**, *10*, 16.
https://doi.org/10.3390/g10020016

**AMA Style**

Lipari F, Stella M, Antonioni A. Investigating Peer and Sorting Effects within an Adaptive Multiplex Network Model. *Games*. 2019; 10(2):16.
https://doi.org/10.3390/g10020016

**Chicago/Turabian Style**

Lipari, Francesca, Massimo Stella, and Alberto Antonioni. 2019. "Investigating Peer and Sorting Effects within an Adaptive Multiplex Network Model" *Games* 10, no. 2: 16.
https://doi.org/10.3390/g10020016