“There’s no one fish saying, ‘Hey, I want everyone to be about five inches away from someone else, and we’re going to have this shape,’ ” he said.
The researchers — including experts in international covert networks who spoke Arabic, Chinese, French, German, Russian and Spanish — analyzed activity on Vkontakte because Facebook shuts down pro-Islamic State discussions very quickly, Dr. Johnson said, and because Vkontakte’s 350 million users are diverse, including many Chechen members who have been targets of Islamic State propaganda. For comparison, the researchers tracked groups promoting civil unrest in Latin America, including strikes and protests, and found both similarities and differences to online activity supporting the Islamic State.
The study focused on small groups of Islamic State supporters that formed online and found about 200 such groups, with more than 100,000 members combined. The groups’ postings included pledges of allegiance to the extremists, fund-raising appeals and survival tips, like how to protect oneself during drone attacks. The pull of small groups is strong. So-called lone wolf sympathizers do not remain alone long, the study said: They usually join a small group within weeks.
Quashing these groups, the study found, can prevent their members from fusing with larger pro-Islamic State groups, which can quickly distribute inciting videos or statements to much broader audiences. But when small groups are blocked by Vkontakte, about a quarter of them reinvent themselves — changing their names, reincarnating, or briefly going dark before re-emerging, the researchers found.
The researchers also said there might be a spike in the formation of small online groups just before an attack takes place. This is based on the one major unexpected attack by the Islamic State that occurred during the months the researchers studied: the attack on Kobani, a Syrian town on the Turkish border, which came under siege in September 2014.
Just before the Kobani incursion, the rate of creation of pro-Islamic State online groups accelerated, Dr. Johnson said. After all the data on small groups was plugged into the team’s equation, the Kobani attack was the only one the formula would have predicted, showing that the equation matched real-world occurrences during the period studied, he said.
Faiza Patel, a director of the Liberty and National Security Program at New York University’s Brennan Center for Justice, said the study showed that “there are ways to look at narrow groups rather than the entire population of internet users.”
But because the Kobani attack was the only example cited, she said, “Frankly, at this point, I don’t think they’re predicting anything.”
“There may be five other independent reasons for the spike in this pro-ISIS propaganda,” she noted, “and I don’t know they have taken into account the other possible explanations.”
Mr. Berger said Kobani, aside from being the sole example cited, was a military-style operation, not a terrorist attack, so the formula, should it hold up, might only apply to large-scale sieges.
“With military action, if their strategy is to use social media to intimidate, then it makes sense” that there would be a spike in online groups’ forming just before, he said. “But with a terrorist attack, we’re often talking about one, two, sometimes a dozen people who are operating out of conditions of secrecy.”
Attacks like the one in Orlando are still rare, Ms. Patel said, with “so many different variables that it’s very, very difficult to come up with any predictive formula.”
Both Mr. Berger and Ms. Patel noted a tricky question raised by the research: When is it best to try to suppress small groups so they do not mushroom into bigger groups, and when should they be left to percolate? Letting them exist for a while might be a way to gather intelligence, Ms. Patel said.
Dr. Johnson said that was one of many questions for further research. “If I break the groups up too quickly,” he said, “I have the risk that I’m liberating out these individuals,” scattering them like a virus to infect other groups.
Other questions involve how fast information travels among small groups, and the role of women — who, the researchers reported in a previous study, accounted for about 40 percent of group members and were likely to communicate about twice as many pieces of information through the social network as men were.
Mr. Berger praised the researchers’ transparency.
“There are a lot of companies that claim to be able to do what this study is claiming, and a lot of those companies seem to me to be selling snake oil,” he said. If the predictive theory holds up, he added, it would be “the silver bullet everybody in the government and everybody in the private sector has been chasing for years.”