tl;dr—depends on place and time, but the relationship between the Safavid central administration and its most outlying provinces was always complex at best.

This is a good question, but a difficult one to answer for a few reasons— (1) "centralized" means different things to different people; (2) ditto for "average persons"; (3) the Safavid era lasted over 200 years and saw decided changes in the level of centralization (according to most definitions) over that period. So my answer has to be rather broad here—though I'm happy to (attempt to) expand on anything further in the comments if you'd like! But, with these caveats, most historians recognize three main periods in Safavid history:

• From the reign of Ismāʿīl I to that of Sulṭān-Muḥāmmad Khudābanda (1501-1587), a period marked by a vague trend towards governmental centralization;

• From the reign of ʿAbbās I to that of ʿAbbās II (1587-1666), the high-water mark of Safavid central power;

• From the reign of Suleymān I to the fall of Isfahan (1666-1722), a period marked by a vague trend away from governmental centralization.

In this schema, the figure of ʿAbbās I stands out as the great reformer and centralizer of the Safavid system. But before we get to ʿAbbās's reforms and how they may have centralized the state, let's start with an important proposition: even at its peak, the Safavid government was nowhere near as "centralized" as its Ottoman and Mughal neighbors at their respective peaks. Unlike the Ottomans and Mughals, the Safavids continued to rely on the power of tribal groups, or uymāqs, and were never able to entirely eliminate clan loyalty as a political force. Moreover, the Safavid state's relationship with its frontier regions and tributaries—largely populated by Sunnis, as opposed to the state religion of Imāmī Shiʿism—was always rocky, and the state could never rely on the loyalty of its periphery. As the shah's central power waned, these tensions were further exacerbated by the rapaciousness of provincial governors and the hardline attitudes of clerics who wanted to convert the frontier regions.

Provincial Administration

As you might expect, the Safavid government was weakest in the furthest provinces of the state. By the late Safavid period, there were five regions governed quasi-independently by appointed valīs: ʿArabistān (modern Khuzestan), Luristān, Georgia, Kurdistān, and the Bakhtīyārī territory. While these provinces were nominally under Safavid control and paid yearly tribute to the court, in practice they enjoyed a considerable degree of autonomy. This began with the choice of valī—the court was often constrained to select governors from the tribes already resident in those areas. In the 1680s, for example, Shah Suleymān's appointment as valī of Kurdestān—a non-Kurd—was run out of town by the local population. While they minted coins in the name of the shah, monetary policy in these outlying provinces—especially in ʿArabistān, as Rudi Matthee has shown—often ran independent of and even counter to that of the central administration.

A relatively strong central government, as under ʿAbbās I, could try to counteract these tendencies by forcing mass migrations and preventing any one tribe from gaining too much political clout in a given area—a process that has been referred to as "cossackization." Another method—and in fact one of the chief innovations with which ʿAbbās has been credited in his attempt to break tribal power over Iran—was the introduction of ghulāms, or crown slaves (largely of Caucasian origin) as a third element in the Turk-Tajik power dynamic that characterized sixteenth-century Iran. ʿAbbās put these slaves who, unlike the tribal leaders, could not draw from independent bases of power, in critical military and administrative positions throughout the state. While the new ghulām administration could not entirely supplant the old tribal powers, and while even the slaves sometimes came into contact with the central authority, this at least cut down heavily on the tendency of tribal military leaders to monopolize governmental power for themselves and their tribes.

Central Authority

Early Safavid power was largely founded on the Qizilbāsh, a tribal confederation devoted to the shah as a religious as well as a political leader. The Qizilbāsh formed the bulk of the Safavid army until the time of Shah ʿAbbās I; provincial governors were largely drawn from their ranks; and while they left civil affairs to the Tajik scribal class, the Qizilbāsh monopolized military administration. There were attempts to break this stranglehold before the time of Shah ʿAbbās—the earliest was Ismāʿīl I's arrogation of supreme civil and military power upon a Tajik minister (Mir Yār-Aḥmad Khūzanī-Isfahānī, Najm-i Sānī), which ended rather poorly when part of the army defected at the Battle of Ghijdavān in 1512. After taking personal control of the reigns of government, Ismāʿīl's son Ṭahmāsb I also pursued a policy of mitigating Qizilbāsh power when possible, playing tribes off against one another, splitting power between multiple figures within a single tribe, and beginning the introduction of Caucasian slaves into the military.

Prior to ʿAbbās's reforms, however, during times when the shah was young or weak (and there were many), governmental control was essentially in the hands of the most powerful member of the military administration—invariably a Qizilbāsh, and often the head of his tribe. These "powers behind the throne" governed not only in their own interest, but also that of their tribe—and when they alienated potential rivals, so too did they alienate their rivals' tribes. This created tensions which expressed themselves in intermittent civil wars carried out by protean Qizilbāsh alliances, without the participation of the shah.

While ʿAbbās's reforms did not entirely eliminate the Qizilbāsh presence in the military or the central administration, his policies (especially that of "cossackization") did make them a less dominant force within the court. This created something of a power gap, one which could be filled by a strong shah but was left to his ministers by a weak one. As a consequence, the post-ʿAbbās period saw the rise of an independent grand vizierate beginning with the administration of Mirza Muḥammad "Sārū" Taqī (1633-45) and reaching its peak with that of Sheykh ʿAlī Khān Zanganah (1669-89). The trend towards a strong vizierate was, however, opposed by other court factions which also sought to fill the power gap—most importantly by the clerical group, which dominated Safavid politics after Sheykh ʿAlī Khān's fall from grace.

Economic Centralization

The Safavid economy was largely decentralized—Iran's geography was a major hindrance to cross-country transport and communication in the pre- and early modern era. Even weights and measures, theoretically imposed by governmental authorities, were never standardized from place to place. While royal orders might intervene in certain aspects of the local economy (especially in Isfahan) from time to time, few economic policies were able to be enacted on a national level.

One major exception to this was the creation of a national monopoly on silk exports. (Silk, grown in the Caspian provinces, was Iran's major export at the time, and had been an important commodity for centuries prior.) Although in practice the monopoly simply amounted to a few extra fees for the Armenian traders who engaged in the trade and did not exempt the court from having to deal with silk producers according to the rules of the market, it still represented a significant effort at instituting a statewide economic policy. "Effort," however, may be the operative term here, since a good deal of silk did manage to be produced and smuggled out of the country despite the monopoly's provisions.

Money supply and mint operation—in short, monetary policy—was the other arena where the central government could and did control economic policy at the national level. Another of ʿAbbās's projects was the consolidation of Safavid mints in order to better deal with the state's recurring money problems. However, this policy could not be evenly enacted throughout the Safavid dominion—in fact, new mints were opened in ʿArabistān. Moreover, the mint at Ḥuveyza, the most active mint in the region, put forth coinage that emphasized local ideas of layout, style, and calligraphy in addition to its conformity to official Safavid patterns and content.