It seems like this question would be incredibly difficult to answer either way. How able are we to track wealth over that timespan? If we don't have specific examples, might a statistician be better suited to answer this since all that's left is probability?

There was a thread about this from a few years ago if you care to read. Many of the same roadblocks regarding record keeping were brought up then.

By the law of large numbers, statistically there's no reason why some families out there wouldn't able to maintain a certain degree of wealth and relevance from the Roman era to modern time. I can guarantee it's not common: It's incredibly difficult to find examples, but it's even more incredibly difficult to prove it was impossible. However, any attempt to answer your question without specific examples would amount only to speculation.

This answer on families that I'm familiar with might interest you. The Dandolo family I mention at the end of the post I linked are especially interesting, as they claim descent from the Roman Gens Ursia; specifically, from Orso Ipato and his son Teodado Ipato, who were Byzantine Military officers when the Northeastern Italian lagoons were a Byzantine protectorate. Although their authorized family tree (which I admittedly haven't seen) evidently finds a way to work around it, I have to point out that there is a two-hundred year gap in the paper trail between Teodado Ipato and the next Dandolo, one Domenico; a Venetian merchant active in Costantinople unremarkable apart from having somehow smuggled the remains of St. Tarasius from Costantinople to the Basilica of St. Zaccaria in Venice.

Because of the way the Venetian Lagoon both collected ethnic Romans fleeing the mainland and had the benefits of a Byzantino-Roman bureaucracy remaining intact through the migration period, you do get some very old families. But the gaps in the early paper trail are big, and too much willingness to stretch lines all the way to the Roman period rapidly leads to rampant speculation. For example, in the early tenth century one Ursus Badovarius applied for a subsidy when raving Maygar hordes devastated his land somewhere near Altino at the edge of the northern lagoon. That area, between the mouth of the river Sile in the northern lagoon and the fortified town of Eraclea, was one of the primary settlement areas for Byzantine soldiers. Could Ursus Badovarious, and consequentially his descendants, the important Badoer family, also be descended from the Gens Ursia? We just don't know (the Badoer themselves are still happily frolicking in the Venetian Countryside to this day. I think old Mrs. Badoer is still speeding up and down the A4 ring road around Padua in her black Lancia but I have to check with my grandma, plus this breaks the twenty-year rule anyway). The examples go on; were the Contarini descended from the Roman Aureli Cotta as they claim? We don't know, the first we hear of them is a jurist in the early ninth century underwriting a will (the Contarini too, are alive and well in a number of villas along the banks of the Brenta river). Heck, the Giustiniani trace their heritage to Emperor Justinian! We have spotty documentation of a Byzantine family called Giustiniani in Istria that arrives in Venice sometime before the eleventh century, but that's it (the Giustiniani themselves are alive and well in the vicinity of San Donà di Piave, near Venice).

So I guess the moral is, it's tough to answer your question with certainty without specific examples that really hard to come by, but it's not like we can outright disprove it either.

Piggybacking off of Op's question, are there examples of people in power carrying wealth and power over from centuries ago? Kind of what /u/kw_Pip asked earlier:

Maybe a better restatement of the question would be, "What is the most prevalent example of old-money we know of in the world today?"

I am speaking specifically of emperors, kings, etc. We do have companies that are ancient that still operate today. Here is a wikipedia page on that (I hope it is okay to link that).

And, my question leans more towards power of influence rather than just wealth alone.

FYI, the genealogical problem you are brushing up against is called "descent from antiquity", and as you imply, none are generally accepted as being proven by genealogical standards.

Just from a general perspective, you've also raised an interesting problem with the intersection of statistics and history. We can do some back-of-the-napkin calculations.

Fall of Rome was about 1600 years ago. Say three generations a century, that gives us 48 generations.

If X is the chance of passing down wealth in each generation to a male heir, then X^48 is the chance the wealth has been successfully passed down today.

A few calculations shows us some boundaries:

The first row is the chance for wealth to be passed down to a male heir. The second row is the calculated chance the wealth survives 48 generations. The third row is the number of wealthy prestigious Roman-era folks needed to exist to get 50/50 odds the wealth has been passed down today.

The fourth row is roughly the ratio of noble and prestigious men to the rest of the population, assuming the world population is 250 million (estimates vary).

65% is below the lower bound - to get a 50% chance of one family's wealth surviving through a male descent to today, you'd need almost 500 million wealthy men.

75% seems doable only one wealthy man for every 500 people. 70% even may be too low - we'd need one wealthy prestigious male in every 18 people.

Any historians want to weigh in?

And hopefully I didn't screw up on the math anywhere.

In Capitalism in the 20th century Thomas Piketty illustrates through tax records and other assorted data sources that the depression and WWII had a huge impact on the concentration of capital on individuals and groups. The loss of capital by the wealthy is said to have been caused by decimating financial asset values, destruction of tangible assets due to war, and another factor or two that i'll edit in when I get home to the book.

According to his research, our economic climate of equality since then is actually an anomaly, and according to most historical records extreme inequality is actually the norm. It's our current regressing egalitarian climate that's unnatural

To give a more concrete question: were there any large-scale purges or rounds of looting of the patricians of Rome after its fall?

Pretty late answer, but it's entirely possible depending on what you consider roman. There were many prominent families in Constantinople and other east roman cities, many of which had been there since the early period. Most of these families remained somewhat influential after the fourth crusade, and some of them retained their wealth after the ottoman conquests. There are probably some wealthy modern Greek families that can nominally trace their ancestry back to the early or middle byzantine period.

Hi, a follow up question. I'm not OP though.

Does this include institutions and monarchy? Does the Papal office count as an entity who maintains its power and wealth? Or the many monarchs of some country count?

With apologies for my lateness and admission that this information does not prove maintenance of wealth and prestige from Roman times, I did come across a recently-published study that found significant long-term wealth retention amongst family dynasties in Florence.

Its authors, Guglielmo Barone and Sauro Mocetti, compared tax records for family dynasties (identified by surname) in Florence, Italy in 1427 and 2011 and found that:

The top earners among the current taxpayers were found to have already been at the top of the socioeconomic ladder six centuries ago. This persistence is identified despite the huge political, demographic, and economic upheavals that occurred between the two dates.

Just an interesting piece of additional information to consider!

I am not sure about the West, but the very wealthy and prestigious Confucius family never fell and still exists today, over 2500 years.