What are some similarities and differences between the Islamic, Byzantine, and Western European societies? What forces led to the weakening of centralized power in the Islamic, Byzantine and Western European kingdoms from the years 750–1050?

After Mohammed's death in 632 CE, Islam began to spread, initially through military conquests, into Iberia to the north and to the Indus River in the east. Under the Umayyads and Abbasids, Muslim armies conquered many parts of northern Africa and what is now the Middle East, including Persia, Egypt, and Syria. Large numbers of people were converted to Islam because of its promise of salvation, the alms that believers received, and the relief from paying taxes if one were Muslim. In addition, Muslims in conquered lands were rewarded with positions in the imperial bureaucracy and military. However, people of other religions, including Jews and Christians, were tolerated (though they were subject to a tax).
During the 8th through 13th centuries, Islam enjoyed a Golden Age, particularly in Baghdad, the site of the House of Wisdom, an institution devoted to the study of arts and sciences. Architecture reached a highpoint, and the sciences flourished as scholars advanced fields such as geometry, algebra, other mathematical fields, astronomy, and physics, in part through a revival of interest in classical learning. Scholars also developed a system of writing. The Islamic Golden Age largely ended with the Mongol invasion and the sack of Baghdad in 1258 CE.
Like the Islamic Empire, the Byzantine Empire experienced a golden age during the era of late antiquity and the early Middle Ages. After Constantine (who ruled from 324–337 CE) moved the capital of the Roman Empire to Constantinople, Christianity became the preferred religion of the empire. Constantine also tried to unify the church. Christianity became the official religion of the Byzantine Empire under Theodosius (who ruled from 379–395 CE), and pagan religions were not allowed. Jews were tolerated in the Byzantine Empire, but they were subject to a tax. In this sense, the Byzantine Empire was similar to the Islamic Empire in having a state religion that unified the empire but permitting the practice of other religions (while requiring people of other religions to pay a tax).
Justinian I, who ruled from 527–565 CE, developed a legal code that was based on civil law. This legal system was different from that of the Islamic Empire, which ruled based on sharia, or Muslim law. As in the Islamic Empire, there was a flourishing of arts, architecture, and science, particularly under Justinian, who ordered the construction of the Hagia Sophia, a church in Constantinople. The Byzantine Empire also saw the development of domed architecture and the rebirth of classical learning, as in the Islamic Empire. The Cyrillic alphabet was developed during this era. As with the decline of the Islamic Empire, the Byzantine Empire was captured by Ottomans, leading to its demise. It had, like the Islamic Empire, declined after centuries of conquests and costly wars.
Unlike the Islamic Empire and the Byzantine Empire, Western Europe was suffering a decline during this period, in part because of the slow collapse of the Roman Empire. Its decline was caused by fighting external and internal wars and by invasions such as that by the Goths. In addition, religious conflict weakened the Roman Empire. At this time, the Roman religion was in decline and had been replaced by sects such as the Gnostics until Constantine recognized Christianity. Roman and other European cities also suffered a decrease in population at this time. Western Europe, unlike the Islamic Empire and Byzantine Empire, did not experience a flourishing of arts and culture during this era until the rule of Charlemagne, whose Carolingian Empire around 800 CE brought renewed strong leadership to Western Europe and reignited learning by establishing schools.