Measuring Reciprocity in a Directed Preferential Attachment Network

TL;DR

This paper analyzes the reciprocity feature in directed preferential attachment networks, deriving asymptotic properties and highlighting discrepancies with empirical data, suggesting the need for alternative models.

Contribution

It provides the first theoretical analysis of reciprocity in directed PA networks and compares it with empirical observations.

Findings

Reciprocity proportion can be close to zero in classical PA models.

Classical PA models may not accurately capture high reciprocity in real networks.

Highlights the need for alternative models to better fit empirical reciprocity data.

Abstract

Empirical studies show that online social networks have not only in- and out-degree distributions with Pareto-like tails but also a high proportion of reciprocal edges. A classical directed preferential attachment (PA) model generates in- and out-degree distribution with power-law tails, but theoretical properties of the reciprocity feature in this model have not yet been studied. We derive the asymptotic results on the number of reciprocal edges between two fixed nodes, as well as the proportion of reciprocal edges in the entire PA network. We see that with certain choices of parameters, the proportion of reciprocal edges in a directed PA network is close to 0, which differs from the empirical observation. This points out one potential problem of fitting a classical PA model to a given network dataset with high reciprocity and indicates alternative models need to be considered.