# Comparative Agent-Based Simulations on Levels of Multiplicity Using a Network Regression: A Mobile Dating Use-Case

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## Abstract

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## 1. Introduction

#### 1.1. Online Dating

#### 1.2. Our Approach

**M1**) and Multiplicity Level 2 (

**M2**), respectively. Both models were assigned rule-sets with the purpose of replicating their respective real-world counterparts. Our approach combines agent-based modeling and classical social network analysis as our go-to analytical frameworks. As outputs from our simulation models we construct networks containing various link types reflecting agent interactions through the simulated mobile dating application. We define a ‘like’ interaction as a one-sided, directed tie representing interest of a sender agent to some receiving agent, a ‘match’ as dyadic interest of two agents (two-directional ‘like’ links), a ‘dislike’ as a directed rejection, and a ‘message’ as a combination interaction (mechanistically, we model this as a ‘like’ with a higher probability of a response). Through implementation of numerous simulations we investigate the contribution of various mechanisms in generating matches. Figure 1 presents the interplay of various mechanism as a multi-layer network with the visibility network unrecorded by our simulation and the like and match networks being central to our comparisons and overall analysis.

## 2. Materials and Methods

#### 2.1. Principal Model Mechanics

#### 2.2. Agent Preferences

- Female agents threshold for a ‘like’ are higher than male agents [82]. [M1 & M2]
- The probability of a ‘like’ when the compatibility threshold is not achieved ($\left(C\right)$ < ${S}_{Total}$) is non-zero. It is reduced but a reject, like, and like/message may still occur. [M1 & M2]
- A like with a message is still subject to the same aforementioned compatibility thresholds, but increases the probability of a reciprocal like event. [M2]
- Previous positive interactions (reciprocity) increases the probability values of like and like/message. [M2]

## 3. Model Comparison

#### Internal Validation

## 4. Results

#### 4.1. Networks & Regression Models

#### 4.2. Sensitivity Analysis

## 5. Discussion

#### Limitations

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MRQAP | Multiple Regression Quadratic Assignment Procedure |

## References

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**Figure 1.**The different layers of network interactions starting with the visibility network of which likes can occur and then matches. Our approach is to specify the model mechanics, emerge variables of interest and compare using network regressions.

**Figure 2.**Program schematic for M1 and M2 simulations. Agents in both models are allotted attributes at instantiation of the simulation. A compatibility score for each agent i is calculated for each agent j and compared to global threshold. M1 agents can choose to reject or ‘like’ other agents, where M2 agents can choose to reject, like, or message while considering reciprocal interactions.

**Figure 3.**M1 detailed simulation schematic and logic diagram. Note $\sum {P}_{M}(R,L|.)=1$ while maintaining female preference for higher levels of rejection make agents and choice inclusive of compatibility score threshold.

**Figure 4.**M2 detailed simulation schematic and logic diagram. Note $\sum {P}_{M}(R,L|.)=1$ and prior $\sum {P}_{M1}\left(L\right|.)={P}_{M2}(L+L/M|.)$, the overall probability of receiving a like (M1) or a like or a like/Message (M2) verified in Tables 6 and 7 is maintained.

**Figure 5.**Simulation analysis for M1 (solid line distributions) and M2 (dashed line distribution) for both male (blue) and female (salmon) agents. (

**A**) Like distributions highlighting median values conditioned on gender and simulation results. Median like values in all models for female agents were lower, but females received more total likes. (

**B**) Male and female agents received more median matches in M2 when compared to M1 even when considering that the median number of likes combined with likes/messages overall was equitable with M1. Median matches for males was lower than median matches for female agents. (

**C**) Only M2 agents can send messages with a low probability with the majority of agents not receiving any messages and a small minority receiving almost 5 times the median values.

**Figure 6.**Network visualizations of like network (top) for M1 (left) and M2 (right) and the equivalent Match Networks (bottom) where distinctive colors represent community detection by Louvain method [86]. (

**A**) Like network of M1 agents showing roughly even distribution of likes with 3 components. (

**B**) Like network of M2 agents showing many isolated dyads and one single large component. (

**C**) Match network for M1 model with one large component, few smaller components and many isolated dyads. (

**D**) Match network for M2 showing the disappearance of large components and the appearance of many smaller dyadic and triadic matches in small disparate communities.

**Figure 7.**Likelihood functions for each parameter estimate by standardized scores drawn from Quadratic Assignment Procedure regression permutations.

Input | Variable Type | Assignment |
---|---|---|

Gender ($\mathcal{G}$) | Discrete Binary | $\{M,F\}:[0.68,0.32]$ |

Ethnicity ($\mathcal{E}$) | Discrete Uniform | $U[0,1,2,3]$ |

Age ($\mathcal{A}$) | Continuous Uniform | $U[18,65]$ |

Physical Attractiveness ($\mathcal{P}$) | Continuous Uniform | $U[0,1]$ |

Compatibility Threshold ($\mathcal{C}$) | Constant | ${\mathcal{C}}_{M}:0.3$ |

${\mathcal{C}}_{F}:0.4$ |

**Table 2.**Rules applied by agents for calculation of overall compatibility (total score). Compatibility is assigned as the additive accumulation of all assigned attributes (age, attractiveness, ethnicity). Second column conveys conditions under which the rule calculation is presumed. All value calculations and parameters are assigned such that they will produce a maximum partial score of 1. Independent compatibility scores are then added to produce overall compatibility score. Description column contains parameter values chosen for a typical run. [Key: ⊤: (same as), ⊥: (different from)].

Model Attribute | Condition | Value | Description | |
---|---|---|---|---|

Compatibility Score (ethnicity) | ${\mathcal{G}}_{M}\top {\mathcal{G}}_{F}$ ${\mathcal{G}}_{M}\perp {\mathcal{G}}_{F}$ | $\begin{array}{l}{S}_{e}=\left({\displaystyle \frac{\mathcal{X}}{2}}\right)+0.5\\ {S}_{e}=\left({\displaystyle \frac{\mathcal{X}}{2}}\right)\end{array}$ | $\mathcal{X}\in [0,1]$ | |

Compatibility Score (age) | $\mathcal{G}==M$ | ${S}_{M}^{\mathcal{A}}=1-{\left(2\right)\left(\right|\mathcal{N}\left(\mathcal{A}\right)}_{cdf}|-0.5)$ $\mathcal{N}{\left(\mathcal{A}\right)}_{cdf}={\displaystyle \frac{1}{2}}[1+erf\left({\displaystyle \frac{x-\mu}{\sigma \sqrt{2}}}\right)]$ | $\begin{array}{l}\mu =-3.996\\ \sigma =3.317\\ x={\mathcal{A}}_{M}-{\mathcal{A}}_{F}\end{array}$ | |

Compatibility Score (age) | $\mathcal{G}==F$ | ${S}_{M}^{\mathcal{A}}=1-{\left(2\right)\left(\right|\mathcal{N}\left(\mathcal{A}\right)}_{cdf}|-0.5)$ $\mathcal{N}{\left(\mathcal{A}\right)}_{cdf}={\displaystyle \frac{1}{2}}[1+erf\left({\displaystyle \frac{x-\mu}{\sigma \sqrt{2}}}\right)]$ | $\begin{array}{l}\mu =2.046\\ \sigma =2.906\\ x={\mathcal{A}}_{F}-{\mathcal{A}}_{M}\end{array}$ | |

Compatibility Score (attractiveness) | ${S}_{M}^{\mathcal{P}}=1-{\left(2\right)\left(\right|\mathcal{N}\left(\mathcal{P}\right)}_{cdf}|-0.5)$ $\mathcal{N}{\left(\mathcal{P}\right)}_{cdf}={\displaystyle \frac{1}{2}}[1+erf\left({\displaystyle \frac{x-\mu}{\sigma \sqrt{2}}}\right)]$ | $\begin{array}{l}\mu =0\\ \sigma =0.341\\ x={\mathcal{A}}_{M}-{\mathcal{A}}_{F}\end{array}$ | ||

Total Score | ${S}_{Total}^{M}={\beta}_{1}^{M}{S}_{e}^{M}+{\beta}_{2}^{M}{S}_{M}^{\mathcal{P}}+{\beta}_{3}^{M}{S}_{M}^{\mathcal{A}}$ ${S}_{Total}^{F}={\beta}_{1}^{F}{S}_{e}^{F}+{\beta}_{2}^{F}{S}_{F}^{\mathcal{P}}+{\beta}_{3}^{F}{S}_{F}^{\mathcal{A}}$ | ${\beta}_{1}^{M}=0.3$ ${\beta}_{1}^{F}=0.4$ ${\beta}_{2}^{M}=0.3$ ${\beta}_{2}^{F}=0.3$ ${\beta}_{3}^{M}=0.4$ ${\beta}_{3}^{F}=0.3$ |

**Table 3.**Validation of simulation inputs for M1 model including type of statistic, mean, and quartiles.

Statistic | N | $\mathit{\mu}$ | $\mathit{\sigma}$ | 0th | 25th | 75th | 100th |
---|---|---|---|---|---|---|---|

Attractiveness | 1000 | 0.512 | 0.293 | 0.001 | 0.257 | 0.778 | 0.999 |

Age | 1000 | 42.088 | 13.063 | 19 | 31 | 54 | 64 |

Ethnicity | 1000 | 2.462 | 1.122 | 1 | 1 | 3 | 4 |

**Table 4.**Validation of simulation inputs for M2 model including type of statistic, mean, and quartiles.

Statistic | N | $\mathit{\mu}$ | $\mathit{\sigma}$ | 0th | 25th | 75th | 100th |
---|---|---|---|---|---|---|---|

Attractiveness | 1000 | 0.483 | 0.284 | 0.001 | 0.229 | 0.723 | 1.000 |

Age | 1000 | 40.879 | 13.989 | 18 | 28 | 53 | 65 |

Ethnicity | 1000 | 2.495 | 1.101 | 1 | 2 | 3 | 4 |

**Table 5.**Statistical comparison of M1 and M2 models using a paired t-test. Shown are the parameter difference $\Delta b$, the t-statistic, p-value and confidence interval of the parameter estimate. This table shows no mean difference in the number of likes yet a statistically significant mean difference in the number of matches between M1 and M2.

Relation Type | $\mathbf{\Delta}{\mathit{b}}_{\mathit{M}}$ | t | p | 95% CI |
---|---|---|---|---|

‘Like’ | 0.001 | 0.002 | 1.00 | [−0.81, 0.8] |

‘Dislike’ | 3.901 | 1.53 | 0.12 | [−1.07, 8.87] |

‘Match’ | 5.85 | 31.35 | p < 0.001 | [5.48, 6.21] |

‘Like Message’ | −2.874 | −43.57 | p < 0.001 | [−3.00, −2.7] |

**Table 6.**M1 Model Run Summary Statistics including type of statistic, mean, standard deviation, and quartiles.

Statistic | N | $\mathit{\mu}$ | $\mathit{\sigma}$ | 0th | 25th | 75th | 100th |
---|---|---|---|---|---|---|---|

‘Like’ Count | 1000 | 33.030 | 8.504 | 10 | 27 | 39 | 58 |

‘Dislike’ Count | 1000 | 146.600 | 60.624 | 48 | 105 | 207.2 | 264 |

‘Match’ Count | 1000 | 11.694 | 5.029 | 0 | 8 | 15 | 31 |

‘Message’ Count | 1000 | 0.000 | 0.000 | 0 | 0 | 0 | 0 |

**Table 7.**M2 Model Run Summary Statistics including type of statistic, mean, standard deviation, and quartiles.

Statistic | N | $\mathit{\mu}$ | $\mathit{\sigma}$ | 0th | 25th | 75th | 100th |
---|---|---|---|---|---|---|---|

‘Like’ Count | 1000 | 33.029 | 9.983 | 7 | 26 | 41 | 61 |

‘Dislike’ Count | 1000 | 142.699 | 53.134 | 55 | 108.8 | 198.2 | 251 |

‘Match’ Count | 1000 | 5.844 | 3.112 | 0 | 4 | 8 | 18 |

‘Message’ Count | 1000 | 2.874 | 2.086 | 0 | 1 | 4 | 12 |

**Table 8.**Multiple regression model of monadic and dyadic covariates using the quadratic assignment procedure. One model for Tinder-type applications (M1) and two models for Hinge-like applications (M2) are shown. The Message relation covariate is only included in the second model of the M2 model for comparison. Reported are the parameter estimates and p-value estimates for each effect. $Pr(<=)$ conveys the proportion [0, 1] of networks generated/permuted with identical structure to the observed network with a parameter value less than or equal to the reported estimate. $Pr(>=)$ conveys the reverse with the proportion of permuted networks with a parameter value greater than observed. Pr(>|=|) reveals symmetrical properties of the parameter distribution with nil indicating perfect symmetry and other values indicating skewness.

M1 | M2 (False) | M2 (True) | |
---|---|---|---|

like network | 2.55 × 10${}^{-1}$ | −6.05 × 10${}^{-3}$ | −6.06 × 10${}^{-3}$ |

Pr(<=b) | (1.0) | (0.0) | (0.0) |

Pr(>=b) | (0.0) | (1.0) | (1.0) |

Pr(>=|b|) | (0.0) | (0.0) | (0.0) |

message network | −4.35 × 10${}^{-3}$ | ||

Pr(<=b) | (0.0) | ||

Pr(>=b) | (1.0) | ||

Pr(>=|b|) | (0.0) | ||

attractiveness (difference) | 2.04 × 10${}^{-4}$ | −6.87 × 10${}^{-4}$ | 1.64 × 10${}^{-4}$ |

Pr(<=b) | (0.8) | (0.93) | (0.92) |

Pr(>=b) | (0.2) | (0.07) | (0.01) |

Pr(>=|b|) | (0.45) | (0.14) | (0.1) |

age (difference) | 2.02 × 10${}^{-7}$ | 1.64 × 10${}^{-4}$ | −4.96 × 10${}^{-6}$ |

Pr(<=b) | (0.55) | (0.01) | (0.01) |

Pr(>=b) | (0.45) | (0.99) | (0.99) |

Pr(>=|b|) | (0.99) | (0.03) | (0.03) |

ethnicity (difference) | 1.24 × 10${}^{-4}$ | −4.00 × 10${}^{-6}$ | 2.14 × 10${}^{-4}$ |

Pr(<=b) | (0.70) | (0.78) | (0.8) |

Pr(>=b) | (0.30) | (0.22) | (0.2) |

Pr(>=|b|) | (0.65) | (0.38) | (0.42) |

intercept | 3.15 × 10${}^{-3}$ | 5.88 × 10${}^{-3}$ | 5.90 × 10${}^{-3}$ |

Pr(<=b) | (1.0) | (1.0) | (1.0) |

Pr(>=b) | (0.0) | (0.0) | (0.0) |

Pr(>=|b|) | (0.0) | (0.0) | (0.0) |

Permutation(s) | 100 | 100 | 100 |

R${}^{2}$ | 0.1809 | 0.000205 | 0.000218 |

Adjusted R${}^{2}$ | 0.1809 | 0.00020 | 0.00021 |

Residual Std. Error (df = 998,995) | 0.097 | 0.25 | 0.076 |

F Statistic (df = 4; 998,995) | 55,170 | 51.24 | 43.59 |

**Table 9.**Parameter estimates ordered by contribution to match network (left compartment) from large values to smaller values (including negative values), and by magnitude rank (overall contribution), e.g., while the like relation for the M2-T model was a top contributor to the match dependent variable (rank by magnitude = 1), it was also a negative parameter estimate placing it 5th in the order of contribution.

Order of Contribution | Rank by Magnitude | |||||
---|---|---|---|---|---|---|

M1 | M2-F | M2-T | M1 | M2-F | M2-T | |

like network | ① | ④ | ⑤ | |||

message network | — | — | ④ | — | — | |

attractiveness (difference) | ② | ③ | ② | |||

age (difference) | ④ | ① | ③ | |||

ethnicity (difference) | ③ | ② | ① | |||

intercept | X | X | X | X | X | X |

**Table 10.**M1 sensitivity model run summary statistics including type of statistic, mean, standard deviation, and quartiles.

Statistic | N | $\mathit{\mu}$ | $\mathit{\sigma}$ | 0th | 25th | 75th | 100th |
---|---|---|---|---|---|---|---|

‘Like’ Count | 1000 | 11.365 | 3.900 | 1 | 9 | 14 | 26 |

‘Dislike’ Count | 1000 | 146.581 | 61.168 | 49 | 108 | 203.2 | 268 |

‘Match’ Count | 1000 | 12.810 | 5.558 | 0 | 9 | 16 | 32 |

‘Message’ Count | 1000 | 0.000 | 0.000 | 0 | 0 | 0 | 0 |

**Table 11.**M2 sensitivity model run summary statistics including type of statistic, mean, standard deviation, and quartiles.

Statistic | N | $\mathit{\mu}$ | $\mathit{\sigma}$ | 0th | 25th | 75th | 100th |
---|---|---|---|---|---|---|---|

‘Like’ Count | 1000 | 32.875 | 9.645 | 10 | 26 | 40 | 61 |

‘Dislike’ Count | 1000 | 140.707 | 56.232 | 50 | 103 | 200 | 251 |

‘Match’ Count | 1000 | 5.682 | 3.036 | 0 | 3 | 7 | 16 |

‘Message’ Count | 1000 | 2.841 | 2.068 | 0 | 1 | 4 | 14 |

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**MDPI and ACS Style**

Shaheen, J.A.E.; Henley, C.; McKenna, L.; Hoang, S.; Abdulwahab, F.
Comparative Agent-Based Simulations on Levels of Multiplicity Using a Network Regression: A Mobile Dating Use-Case. *Appl. Sci.* **2022**, *12*, 1982.
https://doi.org/10.3390/app12041982

**AMA Style**

Shaheen JAE, Henley C, McKenna L, Hoang S, Abdulwahab F.
Comparative Agent-Based Simulations on Levels of Multiplicity Using a Network Regression: A Mobile Dating Use-Case. *Applied Sciences*. 2022; 12(4):1982.
https://doi.org/10.3390/app12041982

**Chicago/Turabian Style**

Shaheen, Joseph A. E., Collin Henley, Liam McKenna, Steven Hoang, and Fatma Abdulwahab.
2022. "Comparative Agent-Based Simulations on Levels of Multiplicity Using a Network Regression: A Mobile Dating Use-Case" *Applied Sciences* 12, no. 4: 1982.
https://doi.org/10.3390/app12041982