In the 1980s, Robert Metcalfe, the inventor of Ethernet, proposed a formulation of network value in terms of the network size (the number of nodes of the network), which was later named as Metcalfe’s law. The law states that the value V of a network is proportional to the square of the size n of the network, ie. V ∝ n2. Metcalfe’s law has been influential and an embodiment of the network effect concept. It also generated many controversies. Some scholars went so far as to state “Metcalfe’s law is wrong” and “dangerous”. Some other laws have been proposed, including Sarnoff’s law (V ∝ n), Odlyzko’s law (V ∝ n log(n)), and Reed’s law (V ∝ 2 n). Despite these arguments, for 30 years, no evidence based on real data was available for or against Metcalfe’s law.
The situation was changed in late 2013, when Metcalfe himself used Facebook’s data over the past 10 years to show a good fit for Metcalfe’s law. In this paper, we expand Metcalfe’s results by using the actual data of Tencent (China’s largest social network company) and Facebook (the world’s largest social network company). Our results show that:
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of the 4 laws of network effect, Metcalfe’s law by far fits the actual data the best;
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both Tencent and Facebook data fit Metcalfe’s law quite well;
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the costs of Tencent and Facebook are proportional to the squares of their network sizes, not linear; and
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the growth trends of Tencent and Facebook monthly active users (MAUs) fit the netoid function well.
[Keywords: network effect, Metcalfe’s law, cost, netoid function]
…We follow 2013’s methodology to define network size, value, and cost. We use the revenues as proxies for Tencent’s and Facebook’s network values. We define cost as the total business cost (tax included) incurred in generating revenue. In other words, the cost is the revenue minus the net profit.
We use the number of MAUs to represent the network size (number of nodes) of Tencent and Facebook. MAU is a metric to count the number of unique users who use the social networking services over the past 30 days. Facebook’s MAUs numbers are published in its financial reports. Tencent’s MAUs numbers are defined as the sum of QQ MAUs and Weixin (WeChat) MAUs, as all 250 Tencent services use these two user account systems.
We use Metcalfe’s netoid function to represent the growth trend of the network size n with respect to time t.
Netoid = p/(1 + e−v × (t − h)).
The 3 parameters p, v, & h have the following meanings:
- p: the peak value representing the maximum value of the number of MAUs;
- v: the virality or speed with which adoption occurs;
- h: the point in time at which the growth rate is maximum, when the network size reaches half the peak.
…3.1 Value Functions: Table 3 shows the fitting results and corresponding root-mean-square deviations (RMSDs) of the 4 network effect functions, for Tencent data and Facebook data. The corresponding fitting curves graphics are shown in Figure 1 & Figure 2 respectively. The contrast of the actual data and the derived values of the value functions for Tencent and Facebook data is shown in Appendix A2 and Appendix A3 respectively.
…Formulating the cost function is not so straightforward. Metcalfe hypothesized that the cost of a network is proportional to its size. But this linear-cost hypothesis deviates too much from the real data of both Facebook and Tencent, and we have to abandon it and try other formulations. It turns out that a quadratic-cost hypothesis fits Tencent and Facebook data much better. Thus a cost function is used whereby the cost is proportional to the square of the network size, ie. C = a × n2:
