The easy conquest of the peninsula is generally assumed by historians to have been followed by a rapid Islamization of the indigenous population, although the evidence for such an assertion is wholly inferential. It must be assumed that the process of conversion to Islam was guided by the same mechanisms that were operative in other societies newly conquered by the Arabs. Based on a study of naming patterns among converts to Islam, Richard Bulliet has described a general process of conversion which he believes to have been the norm in all medieval Islamic societies conquered by the Arabs.(39)
The essence of Bulliet's hypothesis, based upon common notions of innovation diffusion, is that the rate of conversion to Islam is logarithmic, and may be illustrated graphically by a logistic curve. That is, few adopt the innovation at first, but, as more do, the probability of others following suit increases. In the case of conversion to Islam, the greater the number of Muslims, the greater the probability of contact between Muslims and non-Muslims, and hence of the conversion of the latter. This is a selfgenerating process and the rate of conversion increases without the necessity of any specific social or political policies, or of any factor extrinsic to the process.
It follows from this analysis that in Umayyad times Islam was a "smallscale affair" characterized by the rule of vast non-Muslim populations by a tiny Arab elite for whose social and political needs traditional Arab tribal structure was sufficient. Arabs, and therefore Islam itself, was first concentrated in the towns, and the early chronicles reflect this urban Arab milieu.
At the moment when the logistic curve begins to rise precipitously, there begins an explosive period of conversion during which most of the previously unconverted population turns Muslim. When the conversion  process is completed, Bulliet reckons that eighty percent of the original indigenous population converted, with the remaining twenty percent still unconverted, with the status of protected religious minorities.
A number of distinctive social phenomena are associated with this process (although the emergence and sequencing of such phenomena differed from society to society). In the first place, the kind of social movements that attracted converts differed in style and content, depending on whether the converts were a minority or a majority. Millenarian revolutionary movements which attracted converts under the Arab state declined in appeal as the density of the convert population increased and as "old" converts entered the power structure. During the explosive period of conversion, when the composition of society was changing rapidly, abrupt political and social changes occurred within a matter of decades. When the great mass of indigenous people had become Muslim, the kind of society that emerged was radically different from that of the Arab state of the past. Society had become distinctively Muslim, with institutions that reflected the social needs of a majority Muslim population. This was a more self-assured society, able to assert its independence within the Islamic world. Nevertheless, social distinctions arose between "old" and "new" converts, the former typically associating themselves with orthodox religious positions, the latter with movements such as Ash'arism and Sufism.
Bulliet's description of the conversion process, while admittedly a hypothesis, provides a compelling framework for analyzing the dynamics of social, political, and cultural change in the emergent Islamic societies of the middle ages and, at the same time, offers a standard by which to assess such developments in any one Islamic society in comparative perspective.
In the following section, and throughout the book where appropriate, I will attempt to show how Bulliet's hypothesis sharpens and clarifies a number of episodes and phenomena of Andalusi history which are thus far poorly understood. The logistic curve for al-Andalus is reproduced, from Bulliet's data, in Figure 1. The rate of conversion is slow until the tenth century (less than one-quarter of the eventual total number of converts had been converted); the explosive period coincides closely with the reign of 'Abd al-Rahmdn III (912961); the process is completed (eighty percent converted) by around 1100. The curve, moreover, makes possible a reasonable estimate of the religious distribution of the  population. Assuming that there were seven million Hispano-Romans in the peninsula in 711 and that the numbers of this segment of the population remained level through the eleventh century (with population growth balancing out Christian migration to the north), then by 912 there would have been approximately 2.8 million indigenous Muslims (muwalladun) plus Arabs and Berbers. At this point Christians still vastly outnumbered Muslims. By 1100, however, the number of indigenous Muslims would have risen to a majority of 5.6 million.
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