# Modelling and Simulation of the Formation of Social Networks

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## Abstract

**:**

## 1. Introduction

## 2. A Brief Overview of Existing Results on the Distributions of Node Degree

#### 2.1. The Distribution of Degree of a Node in a Social Network

_{0}. In this case, one may obtain that μ

_{0}= n. To see this, recall that each node is introduced into the network at a point in time, by time t in the network, there are a total of t existing nodes and there are a total of nt edges (since n edges are added to the network each time a new node is introduced); therefore, the average number of edges a node in the random process may acquire in the long run is μ

_{0}= ${\mathrm{lim}}_{t\to \infty}(nt/t)=n$. Bishop further determines that incorporating the initial degree given to a node in the initial process, in this case, n, into the above-mentioned exponential distribution results in a shift to the right by n units of the exponential distribution. Such is true in that each node in the network has at least degree n, the initial degree, and that the degree of each node grows as the node may be chosen by a newborn node in the random process. In sum, letting K be the degree (initial degree plus random degree) of a node in the network, Bishop has found that K is a shifted exponential random variable with the following probability density function,

#### 2.2. The Discrete Analog of the Exponential Distribution

_{0}. Let us imagine slicing the area under the graph of f(x) into vertical bars of width a such that the area of the first bar is equal to F(a) − F(0) = $1-{e}^{-\frac{a}{{\mu}_{0}}}$; the area of the second bar, F(2a) − F(a) = ${e}^{-\frac{a}{{\mu}_{0}}}-{e}^{-\frac{2a}{{\mu}_{0}}}$; the area of the third bar, F(3a) − F(2a) = ${e}^{-\frac{2a}{{\mu}_{0}}}-{e}^{-\frac{3a}{{\mu}_{0}}}$; and so on (Teague 2015; Bain and Engelhardt 1992). In general, the area of the nth bar, n = 0, 1, 2, …, is given by (note that we use n = 0 to mean the first bar, n = 1 to mean the second bar, etc.)

_{0}is the geometric distribution with probability of success $1/(1+{\mu}_{0})$.

#### 2.3. The Discrete Distribution of Degree of a Node in a Social Network

_{0}= n) for the discrete random degree associated with the same random process, is geometrically distributed with probability of success $1/(1+n)$. Furthermore, adding the initial degree n obtained in the initial process to the node moves the geometric distribution to the right by n units. Therefore, if K is the total degree of the node in the network, we now have that K (K = Y + n) is a shifted geometric random variable with the following probability mass function,

## 3. Social Network Formation with a Random Number of Initial Connections

#### 3.1. Our Models

_{1}equally likely chosen existing nodes with probability p

_{1}; to k

_{2}equally likely chosen existing nodes with probability p

_{2}; and so on. We note that k

_{i}, with i = 1, 2, …, n, are positive integers, and that ${\sum}_{i=1}^{n}{p}_{i}}=1$. Of great interest to us then is the number of other nodes to which a node connects as the network becomes sufficiently large.

#### 3.2. Determine the Distribution of Degree of a Node with a Random Number of Initial Connections

_{0}= E(K) = k

_{1}p

_{1}+ k

_{2}p

_{2}+ …... + k

_{n}p

_{n}. Observe that μ

_{0}is the average number of edges a node may acquire in the random process. To see this, each new node at birth adds, on average, (k

_{1}p

_{1}+ k

_{2}p

_{2}+ …... + k

_{n}p

_{n}) edges to the network. By time t, there are a total of t existing nodes and hence there are a total of (k

_{1}p

_{1}+ k

_{2}p

_{2}+ …... + k

_{n}p

_{n}) t edges in the network; thus, the average number of edges a node in the random process may acquire in the long run is ${\mathrm{lim}}_{t\to \infty}(({k}_{1}{p}_{1}\text{}+\text{}{k}_{2}{p}_{2}\text{}+\text{}\dots \dots \text{}+\text{}{k}_{n}{p}_{n})t/t)={k}_{1}{p}_{1}\text{}+\text{}{k}_{2}{p}_{2}\text{}+\text{}\dots \dots \text{}+\text{}{k}_{n}{p}_{n}={\mu}_{0}$. Drawing upon the results presented in Section 2, we are now in a position to further determine the distribution of degree of a node in the network.

_{0}represent the number of edges (random degree) the node acquires in the random process. Then, K

_{0}is a geometric random variable, with probability of success $1/(1+{\mu}_{0})$, having the following probability mass function,

_{i}(i = 1, 2, …, n) connections (initial degree) to equally likely chosen existing nodes in the initial process. Then, if K

_{i}represents the total degree (initial degree plus random degree) the considered node possesses, K

_{i}(K

_{i}= K

_{0}+ k

_{i}) is a shifted geometric random variable resulted from moving K

_{0}by k

_{i}units to the right. Consequently, the probability mass function of K

_{i}is

_{i}(i = 1, 2, …, n). Subsequently, the degree of a node of Type i is represented by K

_{i}. Notably, as the network becomes sufficiently large, one may consider that the proportion of Type i nodes is approximately p

_{i}, and that types of nodes are nearly independent. As it turns out, our results reflect that this consideration is indeed warranted. Therefore, choosing randomly a node from the network is not very much unlike drawing at random an individual from a population comprising multiple subpopulations. Such underlies the basis of determining the distribution of degree of a node in the network.

_{1}, k

_{2}, ..., k

_{n}}; that k = ${k}^{\prime}$, ${k}^{\prime}$ + 1, ${k}^{\prime}$ + 2, ...; and that I

_{i}(k) = 1 if k ≥ k

_{i}and I

_{i}(k) = 0 if k < k

_{i}(i = 1, 2, …, n). Finally, suppose that $\mathcal{K}$ represents the total degree (initial degree plus random degree) of a node in the network. We now state and prove the proposition as follows.

**Proposition**

**1.**

_{I}with P(I = i) = p

_{i}, i = 1, 2, …, n. Furthermore, the probability mass function of $\mathcal{K}$ is $P\left(\mathcal{K}=k\right)=\sum _{i=1}^{n}{I}_{i}\left(k\right)\xb7{p}_{i}\xb7{q}^{k-{k}_{i}}p$ where p = $1/(1+{\mu}_{0})$, q = ${\mu}_{0}/(1+{\mu}_{0})$, and k = ${k}^{\prime}$, ${k}^{\prime}$ + 1, ${k}^{\prime}$ + 2, …

**Proof of Proposition**

**1.**

_{i}; and designates a node of subpopulation i as a node of Type i (i = 1, 2, …, n). The fraction of Type i nodes in the network is approximately p

_{i}; the degree of a Type i node is represented by K

_{i}. Therefore, the degree $\mathcal{K}$ of a node chosen at random from the network is then a mixture $\mathcal{K}$ = K

_{I}with P(I = i) = p

_{i}, i = 1, 2, …, n, since the chosen node is of Type i with probability p

_{i}. Now, to determine the probability mass function of $\mathcal{K}$, we invoke the Law of Total Probability to have

#### 3.3. Numerical Examples

_{0}= E(K) = 1(0.30) + 2(0.70) = 1.7, and hence that p = 1/2.7 and q = 1.7/2.7. We therefore have that P(K

_{1}= k) = (1.7/2.7)

^{k}

^{−1}(1/2.7), k = 1, 2, …, and that P(K

_{2}= k) = (1.7/2.7)

^{k}

^{−2}(1/2.7), k = 2, 3, … Finally, we have, as discussed in Proposition 1, the mixture $\mathcal{K}$ = K

_{I}with P(I = 1) = 0.30 and P(I = 2) = 0.70.

_{0}= 9.55; such is employed to establish the desirable mixture $\mathcal{K}$.

_{0}= 16.66.

## 4. Conclusions

## Acknowledgements

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Probability histograms for networks with Table 2. (

**a**) 100 nodes; (

**b**) 1000 nodes; (

**c**) 10,000 nodes.

**Figure 2.**Probability histograms for networks with Table 3. (

**a**) 1000 nodes; (

**b**) 10,000 nodes; (

**c**) 100,000 nodes.

**Figure 3.**Probability histograms for networks with Table 4. (

**a**) 1000 nodes; (

**b**) 10,000 nodes; (

**c**) 100,000 nodes.

K | k_{1} | k_{2} | …… | k_{n} |

Probability | p_{1} | p_{2} | …… | p_{n} |

K | 1 | 2 |

Probability | 0.30 | 0.70 |

K | 3 | 5 | 9 | 10 | 16 |

Probability | 0.19 | 0.11 | 0.37 | 0.03 | 0.30 |

K | 5 | 7 | 11 | 12 | 19 | 23 | 30 |

Probability | 0.09 | 0.02 | 0.08 | 0.23 | 0.42 | 0.05 | 0.11 |

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**MDPI and ACS Style**

Chew, S.; Metheney, E.; Teague, T.
Modelling and Simulation of the Formation of Social Networks. *Soc. Sci.* **2017**, *6*, 79.
https://doi.org/10.3390/socsci6030079

**AMA Style**

Chew S, Metheney E, Teague T.
Modelling and Simulation of the Formation of Social Networks. *Social Sciences*. 2017; 6(3):79.
https://doi.org/10.3390/socsci6030079

**Chicago/Turabian Style**

Chew, Song, Erica Metheney, and Thomas Teague.
2017. "Modelling and Simulation of the Formation of Social Networks" *Social Sciences* 6, no. 3: 79.
https://doi.org/10.3390/socsci6030079