5.4.3 Network Utility
Networks' inherent utility, or "Network Utility," is a foundational principle to understand their dynamics and value (Cao, Sun, and Tsang 2021). Nodes, users, and other independent agents gravitate towards networks primarily to harness the famed "network effect." This effect magnifies the potential to swiftly connect with vast audiences and boost communication efficiency.
Metcalf's Law (Nosovicki 2016) theorizes that the utility of a network grows in proportion to the square of its user or node count. This proposition underscores the potential interconnections amongst nodes, embodying the sheer power of network effects. Nonetheless, a limiting factor of Metcalf's Law is its generalized assumption: each connection or interaction has a uniform value, and each additional node augments the network's utility in a consistent manner.
Our revised metric, which can be perceived as an adaptation of Sarni's Law, prefers a more nuanced depiction of network utility. This methodology weaves in three pivotal dimensions:
Entropy: An emblem of the network's information diversity and complexity, it showcases the spectrum of interactions. A network rich in entropy suggests intricate and potentially invaluable user interactions. In our metric, utility grows with the square root of entropy, denoting diminishing returns from heightened complexity.
User Load: A gauge of the user base's active participation, represented as the ratio of the current user count to the apex user count over a designated period. Elevated user load, suggestive of a denser user-node configuration, hints at an energized user community and resource optimization. Our metric, thus, finely adjusts according to user load, spotlighting networks that adeptly handle their user-node dynamics.
Number of Users: Echoing the essence of Metcalf's Law, our metric also factors in the comprehensive user count, with the utility inflating with the square of this count. Yet, it distinguishes itself by assimilating aspects of user load and entropy, thereby offering a layered perspective of network utility.
To mathematically formalize the network utility, consider the following equation:
Where:
The essence of this metric is to encapsulate a network's utility from the user's standpoint, beyond just its sheer size. It dives deeper, merging its inherent complexity (through entropy) and the users' relative engagement (via user load). In the realm of blockchain networks, the intrinsic worth frequently pivots on the intricate weave and diversity of its transactions, coupled with user fervor. The utility metric, hence, emerges as pivotal, especially when illustrated through a representative curve elucidating network utility's evolution.
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