Global digital platforms are software systems designed to serve entire populations, with some already serving billions of people. We propose atomic transactions-based multiagent transition systems and protocols as a formal framework to study them; introduce essential agents -- minimal sets of agents the removal of which makes communication impossible; and show that the cardinality of essential agents partitions all global platforms into four classes: 1. Centralised -- one (the server) 2. Decentralised -- finite $>1$ (bootstrap nodes) 3. Federated -- infinite but not universal (all servers) 4. Grassroots -- universal (all agents but one) Our illustrative formal example is a global social network, for which we provide centralised, decentralised, federated, and grassroots specifications via multiagent atomic transactions, and prove they all satisfy the same basic correctness properties, yet have different sets of essential agents as expected. We discuss informally additional global platforms -- currencies, ``sharing economy'' apps, AI, and more. While this may be the first formal characterisation of centralised, decentralised, and federated global platforms, grassroots platforms have been defined previously, using two incomparable notions. Here, we prove that both definitions imply that all agents are essential, placing grassroots platforms within the broader formal context of all global platforms. This work provides the first mathematical framework for classifying any global platform -- existing or imagined -- by providing a multiagent atomic-transactions specification of it and determining the cardinality of the minimal set of essential agents in the ensuing multiagent protocol. It thus provides a unifying mathematical approach for the study of global digital platforms, perhaps the most important class of computer systems today.

翻译：全球数字平台是旨在服务整个人口的软件系统，其中一些已服务于数十亿用户。我们提出基于原子事务的多智能体变迁系统与协议，作为研究这些平台的形式化框架；引入"必要智能体"概念——即移除后会导致通信不可能的智能体最小集合；并证明必要智能体的基数将全球平台划分为四类：1. 中心化——一个必要智能体（服务器）2. 去中心化——有限且大于1（引导节点）3. 联邦化——无限但不通用（所有服务器）4. 草根化——通用（除一个外的所有智能体）。我们的形式化示例为一个全球社交网络，通过多智能体原子事务为其提供中心化、去中心化、联邦化与草根化规范，并证明它们均满足相同的基本正确性属性，但按预期具有不同的必要智能体集合。我们非正式讨论了其他全球平台——包括货币、"共享经济"应用、人工智能等。尽管这可能是对中心化、去中心化与联邦化全球平台的首次形式化特征描述，但草根平台此前已通过两种不可比较的概念被定义。在此，我们证明这两种定义均暗示所有智能体都是必要的，从而将草根平台置于所有全球平台的更广泛形式化语境中。本研究通过为任意全球平台（现有或设想的）提供多智能体原子事务规范，并确定由此产生的多智能体协议中必要智能体最小集合的基数，提供了首个用于分类其类型的数学框架。由此，本研究为全球数字平台——或许是当今最重要的计算机系统类别——的研究提供了统一的数学方法。