Metcalfe's Law captures the relationship between the value of a network and its scale, asserting that a network's value is directly proportional to the square of its size. Over the past four decades, various researchers have proposed different scaling laws on this subject. Remarkably, these seemingly conflicting conclusions have all been substantiated by robust data validation, raising the question of which law holds greater representativeness. Consequently, there remains a need for inherent mechanism to underpin these laws. This study aims to bridge this disparity by offering a theoretical interpretation of Metcalfe's Law and its variations. Based on a certain degree of consensus that "traffic is value", network effects are gauged using network traffic load. A general analytical boundary for network traffic load is deduced by balancing practicality and analytical feasibility through the establishment of a comprehensive network model. From this foundation, the mechanism behind Metcalfe's Law and its variants is elucidated, aligning the theoretical derivations with the previously validated empirical evidence for Metcalfe's Law.

翻译：梅特卡夫定律刻画了网络价值与其规模之间的关系，主张网络价值与其节点数的平方成正比。过去四十年间，不同研究者就该主题提出了多种不同的标度律。值得注意的是，这些看似矛盾的结论均得到了有力数据验证的支持，从而引发了何种定律更具代表性的疑问。因此，这些定律仍需内在机理作为支撑。本研究旨在通过为梅特卡夫定律及其变体提供理论阐释来弥合这一差异。基于“流量即价值”这一一定程度的共识，网络效应通过网络流量负载进行度量。通过建立一个综合网络模型，在实用性与分析可行性之间取得平衡，从而推导出网络流量负载的通用解析边界。在此基础上，阐明了梅特卡夫定律及其变体背后的机理，使得理论推导与先前已验证的梅特卡夫定律经验证据相吻合。