Emergence of Metcalfe's Law: Mechanism and Model

Abstract: 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.