Random Birth-and-Death Networks

Abstract

In this paper, a baseline model termed as random birth-and-death network (RBDN) model is considered, in which at each time step, a new node is added into the network with probability p ($0<p<1$) and connected to m old nodes uniformly, or an existing node is deleted from the network with probability $q=1-p$. This model allows for fluctuations in size, reflecting the behaviour of networks in many different disciplines including physics, ecology and economics. The purpose of this study is to develop the RBDN model and explore its basic statistical properties. For different p, we first discuss the network size of RBDN, then combining the stochastic process rules based Markov chain method and the probability generating function method, we provide the exact solutions of the degree distributions. Finally, the tail characteristics of the degree distributions are explored after simulation verification. Our results show that the tail of the degree distribution for RBDN exhibits a Poisson tail in the case of $0<p\le 1/2$ and an exponential tail as p approaches to 1.

Appendix: One-Step Transition Probability Matrix \({\varvec{P}}\)

Using SPR method [34], the one-step transition probability matrix P has two possibilities:

1.1 Add a Node

• i. To be an isolated node, node v is connected to other networks, the state of node v turns from ( n, k) to $( {n+1,m})$ or $( {n+1,n})$, and the one-step transition probability is

  $$\begin{aligned} p_{( {n,k}),( {n+1,m})}= & {} P\left\{ {NK( {t+1})=( {n+1,m})\left| {NK( t)=( {n,k})} \right. } \right\} \nonumber \\= & {} \frac{p}{n+1},\quad n\ge m,0\le k<n \end{aligned}$$

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  $$\begin{aligned} p_{( {n,k}),( {n+1,n})}= & {} P\left\{ {NK( {t+1})=( {n+1,n})\left| {NK( t)=( {n,k})} \right. } \right\} \nonumber \\= & {} \frac{p}{n+1},\quad n<m,0\le k<n \end{aligned}$$

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• ii. Node v is not connected to the new added node, the state of node v turns from ( n, k) to $( {n+1,k})$, and one-step transition probability is

  $$\begin{aligned} p_{( {n,k}),( {n+1,k})}= & {} P\left\{ {NK( {t+1})=( {n+1,k})\left| {NK( t)=( {n,k})} \right. } \right\} \nonumber \\ {}= & {} \frac{n-m}{n+1}p,\quad n\ge m,0\le k<n \end{aligned}$$

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• iii. Node v is connected to the new added node, the state of node v turns from ( n, k) to $( {n+1,k+1})$ and one-step transition probability is

  $$\begin{aligned} p_{( {n,k}),( {n+1,k+1})}= & {} P\left\{ {NK( {t+1})=( {n+1,k+1})\left| {NK( t)=( {n,k})} \right. } \right\} \nonumber \\ {}= & {} \frac{m}{n+1}p,\quad n\ge m,0\le k<n \end{aligned}$$

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  $$\begin{aligned} p_{( {n,k}),( {n+1,k+1})}= & {} P\left\{ {NK( {t+1})=( {n+1,k+1})\left| {NK( t)=( {n,k})} \right. } \right\} \nonumber \\ {}= & {} \frac{n}{n+1}p,\quad n<m,0\le k<n \end{aligned}$$

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1.2 Delete a Node

Since any node in the network with node v may be deleted with equal probability, we only need to compute the transition probability of nodes not being deleted.

• iv. The degree of node v is decreased by 1, the state of node v turns from ( n, k) to $( {n-1,k-1})$, and the one-step transition probability is

  $$\begin{aligned} p_{( {n,k}),( {n-1,k-1})}= & {} P\left\{ {NK( {t+1})=( {n-1,k-1})\left| {NK( t)=( {n,k})} \right. } \right\} \nonumber \\= & {} \frac{k}{n-1}q,\quad 1\le k<n \end{aligned}$$

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  $$\begin{aligned} p_{( {n,0}),( {n,0})} =P\left\{ {NK( {t+1})=( {n,0})\left| {NK( t) =( {n,0})} \right. } \right\} =q,\quad n=1 \end{aligned}$$

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• v. The degree of node v remains unchanged, the state of node v turns from ( n, k) to $( {n-1,k})$, and the one-step transition probability is

  $$\begin{aligned} p_{( {n,k}),( {n-1,k})}= & {} P\left\{ {NK( {t+1})=( {n-1,k})\left| {NK( t)=( {n,k})} \right. } \right\} \nonumber \\= & {} \frac{n-1-k}{n-1}q,\;\quad n>k+1\ge 1 \end{aligned}$$

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