“The lions of the Caliphate roar in New York, we cause you pain inside your house, the carrier of the Cross,” wrote one Twitter user who went by the name “I am ISIS, come and block me.” The account was soon suspended. Another, called “The Lone Wolves,” tweeted with the Arabic hashtag #ExplosionManhattanNewYork “Oh God burn America, take revenge in the name of your oppressed slaves and believers’ blood.”

Others tweeted about “good tidings that keep coming” from New York, and one user affiliated with al-Qaeda in Syria tweeted: “Oh God go against America, the head of infidels. Oh God, count them and kill them one by one, and do not spare anyone of them.” ...