De-Preferential Attachment Random Graphs

In this work we consider a growing random graph sequence where a new vertex is less likely to join to an existing vertex with high degree and more likely to join to a vertex with low degree. In contrast to the well studied \emph{preferential attachment random graphs} \cite{BarAlb99}, we call such a sequence a \emph{de-preferential attachment random graph model}. We consider two types of models, namely, \emph{inverse de-preferential}, where the attachment probabilities are inversely proportional to the degree and \emph{linear de-preferential}, where the attachment probabilities are proportional to $c-$degree, where $c > 0$ is a constant. For the case when each new vertex comes with exactly one half-edge we show that the degree of a fixed vertex is asymptotically of the order $\sqrt{\log n}$ for the inverse de-preferential case and of the order $\log n$ for the linear case. These show that compared to preferential attachment, the degree of a fixed vertex grows to infinity at a much slower rate for these models. We also show that in both cases limiting degree distributions have exponential tails. In fact we show that for the inverse de-preferential model the tail of the limiting degree distribution is faster than exponential while that for the linear de-preferential model is exactly the $\mbox{Geometric}\left(\frac{1}{2}\right)$ distribution. For the case when each new vertex comes with $m > 1$ half-edges, we show that similar asymptotic results hold for fixed vertex degree in both inverse and linear de-preferential models. Our proofs make use of the martingale approach as well as embedding to certain continuous time age dependent branching processes.