Directed preferential attachment models

TL;DR

This paper revisits the directed preferential attachment model, providing an exact characterization of degree distributions through independent birth processes observed at a random time, and extends the model to include undirected edges and degree-dependent attachment probabilities.

Contribution

It introduces a new exact characterization of the joint degree distribution using independent birth processes and extends the model to more general network structures.

Findings

Explicit joint degree distribution formulae derived.

Tail probabilities for large degrees confirmed and extended.

New model variants incorporating undirected edges and degree-dependent attachment.

Abstract

The directed preferential attachment model is revisited. A new exact characterization of the limiting in- and out-degree distribution is given by two \emph{independent} pure birth processes that are observed at a common exponentially distributed time $T$ (thus creating dependence between in- and out-degree). The characterization gives an explicit form for the joint degree distribution, and this confirms previously derived tail probabilities for the two marginal degree distributions. The new characterization is also used to obtain an explicit expression for tail probabilities in which both degrees are large. A new generalised directed prefererantial attachment model is then defined and analysed using similar methods. The two extensions, motivated by empirical evidence, are to allow double-directed (i.e.\ undirected) edges in the network, and to allow the probability to connect an ingoing (outgoing) edge to a specified node to also depend on the out-degree (in-degree) of that node.