The Islamic State’s attraction, even in the West, was proved when Amedy Coulibaly, one of the gunmen in the Paris terrorist attacks last month, declared allegiance to the group.

In Afghanistan last week, an American drone strike killed a former Taliban commander, Mullah Abdul Rauf Khadim, who had pledged allegiance to the Islamic State and had recently begun recruiting fighters. But that pledge seemed to indicate less a major expansion of the Islamic State than a deepening of internal divisions in the Taliban.

There is no indication that the Islamic State controls territory in Afghanistan, but it has signaled its interest in Afghanistan and Pakistan, and has reportedly sent envoys there to recruit.

Similarly, until recently, leaders of Al Qaeda in the Arabian Peninsula, in Yemen, used nonconfrontational language to mask simmering disagreements with the Islamic State and its head, Abu Bakr al-Baghdadi. But tensions peaked in November, when a faction of Qaeda fighters there swore loyalty to Mr. Baghdadi.

Any authorization to use American military force against the Islamic State could arguably also cover interventions in Egypt and Libya, where active militant organizations have pledged allegiance to the group and have received its public acknowledgment as “provinces” of the putative caliphate.

Although there is little or no public evidence that the Islamic State’s leaders in Syria and Iraq have practical control over its North African provinces, its influence is already apparent in their operations and is destabilizing the countries around them. A publication released by the central group last week included a photograph of fighters in Libya with its affiliate there parading 20 Egyptian Christian captives in the Islamic State’s trademark orange jumpsuits, indicating at least a degree of communication.

In Egypt, the Sinai-based extremist group Ansar Beit al-Maqdis sent emissaries to the Islamic State in Syria last year to seek financial support, weapons and tactical advice, as well as the publicity and recruiting advantages that might come with the Islamic State name, according to Western officials briefed on classified intelligence reports.