# A Continuous-Time Network Evolution Model Describing 2- and 3-Interactions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model

**The survival function of the life-length.**Let ${L}_{3}\left(t\right)$ denote the distribution function of the triangle’s life-length ${\lambda}_{3}$. Then, the survival function of ${\lambda}_{3}$ is

- (1-2-3): is a triangle, the ancestor with birth time $t=0$,
- (1-2-3-4): represents three triangles, i.e., the offspring of (1-2-3) at its first reproduction time $t=0.571$,
- (1-5): an edge, offspring of (1-2-3) with birth time $t=0.847$,
- (1-5-6): a triangle, offspring of (1-5) with birth time $t=1.06$.

## 3. General Results

**The survival functions.**

**Theorem**

**1.**

**Proof.**

**The mean offspring number.**Let us denote by ${m}_{i,j}\left(t\right)=\mathbb{E}{\xi}_{i,j}\left(t\right)$ the expectation of the number of type j offspring of a type i mother until time t.

**Corollary**

**1.**

**Proof.**

**Proposition**

**1.**

**Proof.**

**The Perron root and the Malthusian parameter.**Let

## 4. Asymptotic Theorems on the Number of Triangles and Edges

**The denominator in the limit theorem.**In the following theorem, we need the next formulae. In Section 8, we see that the denominator of ${m}_{\infty}^{\Phi}$ in the limiting expression is independent of $\Phi $, and it is

**Theorem**

**2.**

**Proof.**

## 5. Generating Functions and the Probability of Extinction

**The joint generating function of ${\Pi}_{2}\left({\lambda}_{2}\right)$, ${\xi}_{22}\left({\lambda}_{2}\right)$ and ${\xi}_{23}\left({\lambda}_{2}\right)$.**Recall that ${\Pi}_{2}$ is the Poisson process describing the reproduction times of the generic edge and ${\lambda}_{2}$ is its life length. Thus,

**Proposition**

**2.**

**Corollary**

**2.**

**The joint generating function of ${\Pi}_{3}\left({\lambda}_{3}\right)$, ${\xi}_{32}\left({\lambda}_{3}\right)$ and ${\xi}_{33}\left({\lambda}_{3}\right)$.**Here, we study the offspring of a triangle. To distinguish the notation of this subsection and the previous subsection, but avoid too many subscripts, we use bar. Thus, here ${\overline{w}}_{i,j,k}$, ${\overline{u}}_{i,j,k}$, ${\overline{v}}_{i,j,k}$, $\overline{G}(x,y,z)$ and $\overline{H}(x,y,z)$ denote quantities relating offspring of the generic triangle. Recall that ${\Pi}_{3}$ is the Poisson process describing the reproduction times of the generic triangle and ${\lambda}_{3}$ is the life length of the triangle. Thus,

**Proposition**

**3.**

**Corollary**

**3.**

**The probability of extinction.**In Theorem 3, we give the probability of extinction. To determine the extinction probability of the process, we consider the well-known embedded multi type Galton–Watson process. At time $t=0$, the 0th generation of the Galton–Watson process consists of a single individual, i.e., the ancestor. The first generation consists of all offspring of the ancestor. The offspring of the individuals of the nth generation form the $\left(n+1\right)$th generation. Under some assumptions, the extinction of our original process has the same probability as the extinction of this embedded Galton–Watson process. The reproduction process ${\xi}_{i,j}\left(t\right)$ gives the number of type j offspring of an ancestor of type i up to time t. With $t\to \infty $, we obtain that the total number of offspring is ${\xi}_{i,j}\left(\infty \right)$. Therefore, Corollary 1 gives us the $2\times 2$ matrix of the expected total offspring number as

**Theorem**

**3.**

**Proof.**

## 6. The Asymptotic Behaviour of the Degree of a Fixed Vertex

**The process of the ‘good children’.**To describe the degree of a fixed vertex, we introduce a new branching process that we call the process of ‘good children’. This process contains those objects that contribute to the degree of the fixed vertex. We can see that a newly born vertex can have 1 or 2 edges if its parent is an edge object and 1, 2 or 3 edges if its parent is a triangle object.

**Limit results for the degree.**We have already mentioned that the ‘good children’ and only they can contribute to the degree of the fixed vertex. Thus, its degree is equal to the initial degree plus the number of ‘good children’. Let ${}_{2}\tilde{C}\left(t\right)$ be the degree of a fixed vertex at time t after its birth in the case when the vertex belongs to an edge at its birth. Similarly, ${}_{3}\tilde{C}\left(t\right)$ is its degree in the case when the vertex belongs to triangle at its birth. Up to an additive constant, ${}_{i}\tilde{C}\left(t\right)$ is the number of ‘good children’ offspring of an i type ‘parent’ object at time t. It is the sum of the number of edge type ‘good children’ ${}_{i}\tilde{E}\left(t\right)$ and the triangle type ‘good children’ ${}_{i}\tilde{T}\left(t\right)$. To apply Proposition 4, we can use the same method as in Theorem 2. Thus, for the edges, we can again use the random characteristic ${\Phi}_{x}\left(t\right)=1$ if x is an edge and ${\Phi}_{x}\left(t\right)=0$ if x is a triangle, but the underlying process is the process of ‘good children’. This is similar for triangles.

**Checking the conditions of Proposition 4 for the ‘good children’ process.**To complete the previous reasoning, we should check the conditions of Proposition 4. First, we find the the denominator in the limit theorem that is we calculate $\tilde{D}$. By Section 8, we see that

**The extinction of the degree process.**The extinction of the degree process means that the degree of the vertex does not increase after a certain time, that is, the reproduction process of the ‘good children’ dies out. The probability of this kind of extinction is the smallest non-negative root $({\tilde{s}}_{2},{\tilde{s}}_{3})$ of the equation

## 7. Simulations

## 8. Basic Facts on Branching Processes

**Proposition**

**4.**

- (i)
- $\Phi \left(t\right)\ge 0$,
- (ii)
- The trajectories of Φ belong to the Skorohod space D, i.e., they do not have discontinuities of the second kind,
- (iii)
- $\mathbb{E}\left({sup}_{t}\Phi \left(t\right)\right)<\infty $.Assume also
- (iv)
- for some $\epsilon >0$$${\int}_{0}^{\infty}t{(log(1+t))}^{1+\epsilon}{e}^{-\alpha t}{m}_{i,j}\left(dt\right)<\infty ,\phantom{\rule{2.em}{0ex}}i,j=1,\dots ,p$$
- (v)
- for some $\epsilon >0$$$\mathbb{E}\underset{t\ge 0}{sup}\left\{max\left\{t{(log(1+t))}^{1+\epsilon},1\right\}{e}^{-\alpha t}\Phi \left(t\right)\right\}<\infty $$Then,$$\underset{t\to \infty}{lim}{e}^{-\alpha t}{}_{{x}_{0}}{Z}^{\Phi}\left(t\right)={}_{{x}_{0}}{Y}_{\infty}{v}_{i}{m}_{\infty}^{\Phi}$$$${m}_{\infty}^{\Phi}={\displaystyle \frac{{\sum}_{j=1}^{p}{u}_{j}{\int}_{0}^{\infty}{e}^{-\alpha t}\mathbb{E}{\Phi}_{j}\left(t\right)dt}{{\sum}_{l,j=1}^{p}{u}_{l}{v}_{j}{\int}_{0}^{\infty}t{e}^{-\alpha t}{m}_{l,j}\left(dt\right)}},$$If, in addition, we assume that
- (vi)
- $$\mathbb{E}\left[{}_{\alpha}{\xi}_{i,j}(\infty ){log}^{+}{}_{\alpha}{\xi}_{i,j}(\infty )\right]<\infty ,\phantom{\rule{2.em}{0ex}}i,j=1,\dots ,p,$$

## 9. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Example of the graph evolution model with parameter set: ${r}_{1}=0.1,\phantom{\rule{3.33333pt}{0ex}}{p}_{1}=0.4,$ ${p}_{2}=0.2,\phantom{\rule{3.33333pt}{0ex}}b=0.1,\phantom{\rule{3.33333pt}{0ex}}c=0.1$.

${\mathit{r}}_{1}$ | ${\mathit{p}}_{1}$ | ${\mathit{p}}_{2}$ | b | c | $\widehat{\mathit{\alpha}}$ | 2.5% | 97.5% | |
---|---|---|---|---|---|---|---|---|

E | 0.1 | 0.5 | 0.5 | 0.2 | 0.4 | 0.5394 | 0.5393 * | 0.5443 * |

T | 0.5390 * | 0.5440 * | ||||||

$\tilde{E}$ | 0.5410 | 0.5453 | ||||||

$\tilde{T}$ | 0.5395 | 0.5444 | ||||||

E | 0.1 | 0.2 | 0.6 | 0.25 | 0.25 | 0.9133 | 0.9130 * | 0.9141 * |

T | 0.9134 | 0.9142 | ||||||

$\tilde{E}$ | 0.9133 * | 0.9141 * | ||||||

$\tilde{T}$ | 0.9133 * | 0.9148 * | ||||||

E | 0.1 | 0.2 | 0.6 | 0.45 | 0.35 | 0.6622 | 0.6585 * | 0.6659 * |

T | 0.6606 * | 0.6648 * | ||||||

$\tilde{E}$ | 0.6608 * | 0.6647 * | ||||||

$\tilde{T}$ | 0.6597 * | 0.6638 * |

**Table 2.**Comparison of the numeric values of the extinction probabilities and their relative frequencies from ${10}^{5}$ repetitions.

${\mathit{r}}_{1}$ | ${\mathit{p}}_{1}$ | ${\mathit{p}}_{2}$ | b | c | Ancestor | Numeric | Simulation |
---|---|---|---|---|---|---|---|

0.1 | 0.2 | 0.6 | 0.8 | 0.8 | 2 | 0.9095 | 0.9053 |

3 | 0.8855 | 0.8805 | |||||

0.2 | 0.3 | 0.6 | 0.7 | 0.7 | 2 | 0.9247 | 0.9184 |

3 | 0.9141 | 0.9070 | |||||

0.3 | 0.3 | 0.5 | 0.6 | 0.6 | 2 | 0.7371 | 0.7207 |

3 | 0.6896 | 0.6834 |

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Fazekas, I.; Barta, A.
A Continuous-Time Network Evolution Model Describing 2- and 3-Interactions. *Mathematics* **2021**, *9*, 3143.
https://doi.org/10.3390/math9233143

**AMA Style**

Fazekas I, Barta A.
A Continuous-Time Network Evolution Model Describing 2- and 3-Interactions. *Mathematics*. 2021; 9(23):3143.
https://doi.org/10.3390/math9233143

**Chicago/Turabian Style**

Fazekas, István, and Attila Barta.
2021. "A Continuous-Time Network Evolution Model Describing 2- and 3-Interactions" *Mathematics* 9, no. 23: 3143.
https://doi.org/10.3390/math9233143