Mr. Trump himself repeatedly seemed to endorse attacks on his detractors, too.
“Maybe he should have been roughed up,” he said of one protester who was reportedly punched and kicked in November 2015. “I’d like to punch him in the face, I’ll tell ya,” he said of another a few months later. He even offered to pay legal fees for his supporters if they became too aggressive.
The supporters also often aimed offensive and violent rhetoric at Mrs. Clinton, suggesting she be killed.
To determine whether those words and news reports corresponded with an actual shift in violence, the researchers compiled a list of 31 Trump rallies and 38 Clinton rallies held in cities with assault data available online.
They compared the number of assaults reported on the day of the rally to the number reported on the corresponding day of the week, for each of the four weeks before and after the event.
On a typical day, cities saw an average of 19.4 assaults, they found. On the day of a Trump rally, that number rose to 21.7.
The pattern held even when the researchers controlled for the influence of factors like population size, data sources and the day used for the comparison.
The researchers offered two explanations for the increase in assaults. Either they were the result of clashes at or near the rallies, or they occurred elsewhere in the cities after the aggressive mood on display by Mr. Trump, his supporters or his opponents had spread through “social contagion.”
There were some limitations to the findings, the authors noted. They may not apply to the rallies or cities that weren’t studied, and a greater police presence during the rallies may have made it more likely for an assault to be reported.