Asymptotic Dependence of In- and Out-Degrees in a Preferential Attachment Model with Reciprocity

TL;DR

This paper introduces a modified preferential attachment model incorporating reciprocity, demonstrating that in large networks, in- and out-degrees become asymptotically dependent due to reciprocated edges.

Contribution

It proposes a new PA model with a reciprocity parameter and analyzes its asymptotic degree dependence, addressing limitations of classical models.

Findings

Large in- and out-degrees become fully dependent asymptotically.

Reciprocity significantly influences degree dependence in the network.

Modified model better captures high reciprocity observed in real social networks.

Abstract

Reciprocity characterizes the information exchange between users in a network, and some empirical studies have revealed that social networks have a high proportion of reciprocal edges. Classical directed preferential attachment (PA) models, though generating scale-free networks, may give networks with low reciprocity. 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. We give one possible modification of the classical PA model by including another parameter which controls the probability of adding a reciprocated edge at each step. Asymptotic analyses suggest that large in- and out-degrees become fully dependent in this modified model, as a result of the additional reciprocated edges.