Degrees in Preferential Attachment Networks with an Anomaly

TL;DR

This paper analyzes how an introduced anomaly affects the degree evolution and distribution in preferential attachment networks, revealing that early anomalies significantly alter network structure while late anomalies have minimal impact.

Contribution

It introduces a model incorporating an anomaly into preferential attachment networks and studies its impact on degree growth and distribution, providing new insights into network evolution.

Findings

Anomaly's degree increases almost linearly.

Early anomalies significantly change degree distribution.

Late anomalies have minimal impact on network structure.

Abstract

We consider a preferential attachment model that incorporates an anomaly. Our goal is to understand the evolution of the network before and after the occurrence of the anomaly by studying the influence of the anomaly on the structural properties of the network. The anomaly is such that after its arrival it attracts newly added edges with fixed probability. We investigate the growth of degrees in the network, finding that the anomaly's degree increases almost linearly. We also provide a heuristic derivation for the exponent of the limiting degree distributions of ordinary vertices, and study the degree growth of the oldest vertex. We show that when the anomaly enters early, the degree distribution is altered significantly, while a late anomaly has minimal impact. Our analysis provides deeper insights into the evolution of preferential attachment networks with an anomalous vertex.