The Peisistratids, as the succession of tyrants were called (after the founder of the dynasty, Peisistratos), strengthened Athenian central administration at the expense of the aristocracy by appointing judges throughout the region.

【分析】as the succession of tyrants were called是非限制性定语从句，as为关系代词，指代the Peisistratids，在从句中作主语补足语（主动语态时为宾语补足语）。

1. as the succession of tyrants were called，把as换为the Peisistratids，那么，结构就是：the succession of tyrants were called the Peisistratids。

2. 转换为主动语态：as people called the succession of tyrants. 把as换为the Peisistratids，那么，结构就是：people called the succession of tyrants the Peisistratids

【翻译】The Peisistratids, as the succession of tyrants were called：

1. 人们称之为“暴君继承人”的Peisistratids

2. 被称为“暴君继承人”的Peisistratids

3. 正所谓是“暴君继承人”的Peisistratids

(Peisistratids is the common term for the three tyrants who ruled in Athens from 546 to 510 BC, namely Peisistratos and his two sons, Hipparchus and Hippias.

--- https://en.wikipedia.org/wiki/Peisistratos)

▲as + 主语+ be + called：正如人们所称；大家都这么叫；正所谓是。

This factory produces China's traditional handcraft: cloisonne,or Beijing enamel, as it is called. 这个厂主要生产我国传统工艺品：景泰蓝制品，也叫北京烧瓷。

These cosmic rays, as they are called, consist mostly of protons. 这些被称为宇宙射线的物质，主要是由质子组成的。