That something fishy may involve the mass creation of “bots,” a catch-all term for accounts that are automated, meaning a single individual or a team can run hundreds or thousands at time. This is something Trump’s supporters have a history of doing well, far better than his political opponents, according to work by several researchers.

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“In my expert opinion, something strange is going on,” said Samuel C. Woolley, research director for the Computational Propaganda project at Oxford University. “It’s consistent with other strange things that have gone on before with this politician’s Twitter feed.”

First, a few facts: Trump’s Twitter following, which is one of the largest in the world, has been surging since his inauguration in January, rising this month alone from 28.6 million to more than 31 million, according to Twitter Counter, a tracking site. That’s an increase of 2.4 million in May, for an average of nearly one each second of every day, around the clock.

That would be extremely impressive for most people but less so for one of the world’s most famous men, not to mention one known to use Twitter to convey some of his bluntest and newsiest utterances. Bear in mind, for perspective, that the Twitter feed for Trump’s predecessor, @BarackObama, has more than 89 million followers, including a substantial percentage of bots, according to various reports.

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But here’s the catch: There is a strangely large percentage of Trump’s followers — and especially his newest followers — that have only the most rudimentary account information, with no profile picture, few followers and little sign that they have ever tweeted. These are so-called “egg followers” because instead of a profile photo they traditionally carried the image of a blank egg on Twitter account pages.

And that, say some researchers, is odd.

“This is very, very obvious when you just go and click on the newer followers,” said Jonathan Albright, research director of the Tow Center for Digital Journalism at Columbia University. “The quality of the new followers is pretty bad.”

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SocialRank, a New York-based analytics company that works with Southwest Airlines, L’Oreal and the NFL, reported this week that as Trump’s number of followers surged from 24.1 million in February to 31 million in May, his number of “egg followers” grew sharply as well, from 5 million to 9.1 million. Of that group, more than half have never tweeted and only 4 percent have 25 or more followers; 927,000 of Trump’s egg followers opened new accounts in May, according to SocialRank’s analysis posted Tuesday.

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That doesn’t necessarily make the accounts “fake,” as some reports have claimed. Most academic researchers say that determining what percentage of followers are actual individual humans can be extremely difficult — and almost impossible with an account with as many followers as Trump’s. Twitter itself has acknowledged that as much as 8.5 percent of all of its accounts are likely automated, though independent researchers say the number may be twice as high. (The company also has a team that searches for bots and, when found in violation of Twitter policies, shuts them down.)

There another possible explanation for Trump’s mysterious follower surge. Twitter spokesman Nick Pacilio said that newer users often appear without profile photos because they have not yet developed their accounts fully. The company’s most recent earnings report shows that the number of Twitter accounts overall grew by around 9 million in the first quarter of 2017, up to 328 million, meaning plenty of newcomers may be using the platform merely to browse what others are saying. (In March, Twitter abandoned the egg image for users without profile pictures, but the term "egg followers" has endured among researchers).

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Alexander Taub, chief executive of SocialRank, said that both theories may be true. Trump may be drawing an unusual number of new — but real — egg followers. And he may also be benefitting from an aggressive new campaign of bot creation.

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“It’s probably a combination of both,” Taub said, “but there’s something fishy.”

Here’s where a little history may help sharpen the picture. Last year, during the election campaign, several academic researchers tracked the use of Twitter bots supporting either Trump and his Democratic rival Hillary Clinton. They reported that the bots supporting Trump massively outperformed the bots supporting Clinton, by a margin of 5-to-1 in the final days before the vote.

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Among accounts that researchers had identified as “highly automated” — meaning likely bots — 81.9 percent carried at least some messaging supporting Trump, according to a November paper written by Woolly and two colleagues, Bence Kollyani of Corvinus University and Philip N. Howard of Oxford.

It’s that history, in part, that makes Woolley suspicious that Trump’s surge may benefit from aggressive bot development. “There’s a legacy of this.”

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But even Woolley and other researchers skeptical of Trump’s total say there is no definitive way to determine who is behind making Twitter bots, nor is there any plausible way to determine their motives.

Descriptions of how to build Twitter bots are widely available on the Web, and they can even be purchased en masse from companies that specialize in developing them. Spoofing location, language, profile pictures and other information for Twitter accounts is also easy, making it hard to get clear answers, said University of Southern California researcher Emilio Ferrara.