# A Preferential Attachment Paradox: How Preferential Attachment Combines with Growth to Produce Networks with Log-normal In-degree Distributions

## TL;DR

This paper resolves the preferential attachment paradox by demonstrating that previous measurement inaccuracies led to misinterpreting network growth patterns, revealing that preferential attachment can produce log-normal degree distributions.

## Contribution

It identifies and corrects a measurement flaw that previously obscured the relationship between preferential attachment and log-normal degree distributions in networks.

## Key findings

- Corrected measurement procedure resolves the paradox.
- Preferential attachment can produce log-normal distributions.
- Measurement accuracy is crucial for understanding network growth.

## Abstract

Every network scientist knows that preferential attachment combines with growth to produce networks with power-law in-degree distributions. How, then, is it possible for the network of American Physical Society journal collection citations to enjoy a log-normal citation distribution when it was found to have grown in accordance with preferential attachment? This anomalous result, which we exalt as the preferential attachment paradox, has remained unexplained since the physicist Sidney Redner first made light of it over a decade ago. Here we propose a resolution. The chief source of the mischief, we contend, lies in Redner having relied on a measurement procedure bereft of the accuracy required to distinguish preferential attachment from another form of attachment that is consistent with a log-normal in-degree distribution. There was a high-accuracy measurement procedure in use at the time, but it would have have been difficult to use it to shed light on the paradox, due to the presence of a systematic error inducing design flaw. In recent years the design flaw had been recognised and corrected. We show that the bringing of the newly corrected measurement procedure to bear on the data leads to a resolution of the paradox.