The Latin terms translated as "gangs" in most scholarship of popular violence are extremely hard to translate and rarely work directly into English. Operae ("labors"), exercitus ("army"), multitudo ("crowd"), and plenty other terms I'm forgetting and omitting do not, generally, refer to gangsters as we understand them. These are not the Bloods and the Crips fighting over territory, though "turf wars" might have occurred. These are political bodies raised and organized for the purpose of introducing political violence into public places. The precise identity and organization of these groups eludes us. We simply do not have enough evidence. However, Publius Clodius' gangs are usually taken as the basis from which we may evaluate the others. Clodius' violence is better documented than anything before or since, as Cicero provides us with detailed commentary--short, unfortunately, of giving us an actually accurate picture. These terms are unattested before Cicero, but episodes of organized political violence exist even before the Gracchi. Our sources for earlier political gangster violence describe clients, slaves, freedmen, and hired professional thugs--the enigmatic duces multitudinum that Sallust mentions hired themselves out for political purposes but refused to help Catiline. The identities of these people, varied as they are, are clear enough: these are groups personally attached to individual politicians, either as current or former members of their households, clients, or paid retinues. Such retinues were not uncommon. Every Roman of any significance was surrounded by hundreds, if not thousands (e.g. Tiberius Gracchus) of clients, minor staff, slaves, and other dependents, and by the 60s they were turned towards violence frequently enough as to seem unremarkable. Clodius was murdered by just such a group when his own retinue was assaulted on the Via Appia. When Clodius turned Appius Claudius' gladiators on Quintus Cicero the latter was defended by Pompey's clients. This mention of gladiators is significant. By the 60s at least they were everywhere in private retinues, and they're commonly seen as elements in political violence, and they seem commonly to have been added to such violent groups as extra muscle.

The usual narrative is that Clodius changed political violence by turning the urban plebs into a professional quasi-military body. It is supposed that previously violence had been undertaken largely by the politically interested--Italians in cases of Italian enfranchisement, equites in cases of equestrian rights, etc.--who joined in violence begun by smaller groups or temporarily aligned themselves with violent politicians. Clodius, that view goes, organized the city's collegia and vici into explicitly political groups, which he deployed to mob elections, attack opponents, and command the streets. The argument and counter-argument are too long to describe in full here, but I can see pretty much no evidence that this view is valid. Cicero never once refers to collegia or vici in violence, and he doesn't even seem to imply it, except in that he talks about workers and freedmen fighting for Clodius, which is a bit like saying that since grapes are purple and grapes are fruit all purple fruits are grapes. Instead as far as I can tell the argument rests on a misreading of the lex de collegiis and its significance, extending the description of a single event (re-instituting banned collegia) undertaken specifically to win the electoral support of the plebs media (unusual for a tribune) to Clodius' career as a whole. There was something unprecedented about this movement, but I can only see evidence for a shifting tribunician strategy, and not a change in political violence. Instead it seems like Clodius' political violence worked the same way more or less as earlier instances, with a small group of fighters directly attached to the politician himself beginning the violence, into which more general supporters might join or not.

There is, however, evidence of a sort of unattached kind of gangster violence. I've already mentioned Sallust's duces multitudinum, the meaning of which is not clear. Tacitus talks of what are apparently professional theater gangs during Tiberius' time, and there are echoes of them in some episodes during Clodius' career. It's hard to see how professional theater gangs might have been the loyal retinues of individual politicians--especially since in the Principate the ability of individual politicians to do such things was highly restricted--but as thugs for hire their place seems rather natural. But even then they're described as being politically significant, not just gangs of dudes all getting the same tattoo and running around defending their turf. Plenty of that sort of thing happened among political gangs, but we do not generally hear of such organized violent groups outside the political, or potentially political, sphere

I'm afraid there's not much that can be said here. There's not a whole ton of evidence for these people, and what evidence there is is clouded in most treatments by arguments sometimes outdated by a century and a half. It would take far more than reddit allows to sum up the subject properly (it took me about a hundred pages last time I tried, and to make matters worse I wrote all this on mobile)