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Bear in mind that Rome was the wealthy capital of a multi-racial empire that spanned the entirety of the Middle East and the North African coast. Of course, it would attract grifters, merchants, and economic opportunists from around the empire. This does not mean that the true Romans, the Italians, were not European.
Some intermixing likely occurred between native Italians and non-European immigrant populations, but the lasting impact of this foreign genetic influx is insignificant. The overall genetic contribution of non-Europeans to modern Italians is minuscule; that of Sub-Saharan Africans is less than 1% (in most places 0%), and North Africans less than 2% (in many places 0%). The only outlier is Sicily, the southern tip of Italy that was conquered and occupied by North African Saracens, whose DNA contribution maxes out at 6%. The largest non-Italian genetic legacy found in Italy is that of the Greeks in Sicily, who made a significant genetic contribution of 37%.