Preferential Attachment Random Graphs with General Weight Function

Abstract

Start with graph $G_0 \equiv \{V_1,V_2\}$ with one edge connecting the two vertices $V_1$, $V_2$. Now create a new vertex $V_3$ and attach it (i.e., add an edge) to $V_1$ or $V_2$ with equal probability. Set $G_1 \equiv \{V_1,V_2,V_3\}$. Let $G_n \equiv \{V_1,V_2,\ldots,V_{n+2}\}$ be the graph after $n$ steps, $n \geq 0$. For each $i$, $1 \leq i \leq n+2$, let $d_n(i)$ be the number of vertices in $G_n$ to which $V_i$ is connected. Now create a new vertex $V_{n+3}$ and attach it to $V_i$ in $G_n$ with probability proportional to $w(d_n(i))$, $1 \leq i \leq n+2$, where $w(\cdot)$ is a function from $N \equiv \{1,2,3,\ldots\}$ to $(0,\infty)$. In this paper, some results on behavior of the degree sequence $\{d_n(i)\}_{n\geq 1,i\geq 1}$ and the empirical distribution $\{\pi_n(j) \equiv \frac{1}{n} \sum^n_{i=1} I(d_n(i) = j)\}_{n\geq 1}$ are derived. Our results indicate that the much discussed power-law growth of $d_n(i)$ and power law decay of $\pi(j) \equiv \lim_{n \rightarrow \infty} \pi_n(j)$ hold essentially only when the weight function $w(\cdot)$ is asymptotically linear. For example, if $w(x) = cx^2$ then for $i\geq 1$, $\lim_n d_n(i)$ exists and is finite with probability (w.p.)\ 1 and $\pi(j) \equiv \delta_{j1}$, and if $w(x) = cx^p$, $1/2 <p < 1$ then for $i \geq 1$, $d_n(i)$ grows like $(\log n)^q$ where $q=(1-p)^{-1}$. The main tool used in this paper is an embedding in continuous time of pure birth Markov chains.