Red haired patricians

I’ve been enjoying Steven Saylor’s Gordianus novels and it struck me that a lot of the patrician characters in his books have red hair.

I was just wondering if there’s any truth in the idea that there were relatively many patricians in Roman times with red hair (and not just the ones called “Rufus”). As there were only a few patrician families who tended to marry within their own circle I suppose the red hair gene could have been passed around. Or is this part of the novels purely invention?
For those of you who know about Romans, but not about Gordianus, he’s a contemporary of Cicero in case that makes a difference.

Marrying for red hair was a practice also followed by the Ahenobarbus (“red beard”) clan, from which the emperor Nero was descended on his father’s side.

A Roman clan (gens) might include any number of interrelated families, each distinguished from the others by the use of a cognomen. For example, Gaius Julius Caesar (the Dictator), would have gone by “Gaius” as his given name. “Julius” applied to his extended clan. The cognomen “Caesar” specified which branch of the Julian clan he belonged to – in his case, the one whose members typically sported a fine head of hair.

Many families earned cognomens for commonly held traits, such as hair color, complexion, and even such things as crossed eyes or bowleggedness. Others took their cognomen from individual nicknames, some of which were even unflattering. “Cicero”, for example, means “chickpea”, and refers to a wart some member of the Tullius family evidently bore in a prominent location.

Additional cognomens might be added to an individual as a mark of distinction. Publius Cornelius Scipio earned the name “Africanus” for defeating Carthage. The full name of the emperor Claudius was Tiberius Claudius Drusus Nero Germanicus – a name that advertised a distinguished and impeccably Roman pedigree.

Having a cognomen was a matter of social pride. Roman society was obsessed with bloodlines and ancestry, and being descended from a famous name or distinguished family was often more important to one’s social standing (not to mention political prospects) than one’s appearance, personality, or abilities. It shouldn’t come as much of a surprise that someone of high birth would strive to maintain his association with his famous name by ensuring that his children carried the same prominent traits.