Tencent and Facebook Data Validate Metcalfe’s Law

Abstract

In 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, i.e., V ∝ n 2. 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 ∝ nlog(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 utilizing the actual data of Tencent (China’s largest social network company) and Facebook (the world’s largest social network company). Our results show that: 1) of the four laws of network effect, Metcalfe’s law by far fits the actual data the best; 2) both Tencent and Facebook data fit Metcalfe’s law quite well; 3) the costs of Tencent and Facebook are proportional to the squares of their network sizes, not linear; and 4) the growth trends of Tencent and Facebook monthly active users fit the netoid function well.

Full text

Zhang XZ, Liu JJ, Xu ZW. Tencent and Facebook data validate Metcalfe’s law. JOURNAL OF COMPUTER SCIENCE
AND TECHNOLOGY 30(2): 246–251 Mar. 2015. DOI 10.1007/s11390-015-1518-1
Tencent and Facebook Data Validate Metcalfe’s Law
Xing-Zhou Zhang 1,2(张星洲), Jing-Jie Liu 1,2(刘晶杰), and
Zhi-Wei Xu 1,2(徐志伟), Fellow, CCF, Member, ACM, IEEE
1Institute of Computing Technology, Chinese Academy of Sciences, Beijing 100190, China
2University of Chinese Academy of Sciences, Beijing 100049, China
E-mail: {zhangxingzhou, liujingjie, zxu}@ict.ac.cn
Received November 25, 2014; revised January 17, 2015.
Abstract In 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
Vof a network is proportional to the square of the size nof the network, i.e., 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∝nlog(n)), and Reed’s law (V∝2n). 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 utilizing the actual data of Tencent (China’s largest social network company) and Facebook (the world’s largest social
network company). Our results show that: 1) of the four laws of network effect, Metcalfe’s law by far fits the actual data the
best; 2) both Tencent and Facebook data fit Metcalfe’s law quite well; 3) the costs of Tencent and Facebook are proportional
to the squares of their network sizes, not linear; and 4) the growth trends of Tencent and Facebook monthly active users fit
the netoid function well.
Keywords network effect, Metcalfe’s law, cost, netoid function
1 Introduction
Network effect has become an influential concept
not only in the technology field, but also in economy
and business, social sciences, and even global public
events[1-2]. A network effect is the effect that a net-
work’s value Vis dependent on its size n(the number
of its nodes)[3]. Four laws have been proposed to pro-
vide more precise definitions and characterizations of
network effect. They are
•Sarnoff’s law[3]:V∝n,
•Odlyzko’s law[4]:V∝nlog(n),
•Metcalfe’s law[5]:V∝n2, and
•Reed’s law[6]:V∝2n.
Many papers are published[3-9] arguing for or
against these laws. However, no actual evidence was
available in the literature to validate these laws with
real data until December 2013, when Robert Metcalfe
himself utilized Facebook’s actual data over the past
decade to show a good fit to Metcalfe’s law[8].
There are four key points in Metcalfe’s experiments:
1) Metcalfe reiterated the hypotheses proposed 40 years
ago, i.e., a network has a value of V∝n2but a cost
of C∝n; 2) Facebook’s network size nis defined as
the number of its monthly active users (MAUs), while
Facebook’s network value Vis defined as its revenue
(as a proxy); 3) the Facebook data indeed fit Metcalfe’s
law well, i.e., Facebook’s revenue is proportional to the
square of the number of its MAUs; 4) a function, called
netoid function, is defined to describe the growth trend
of a network.
Several key questions are not answered by Metcalfe’s
paper.
•Is Metcalfe’s law only valid for Facebook, a com-
pany in a developed country serving worldwide users?
Short Paper
Special Section on Applications and Industry
The work was supported by the Guangdong Talents Program of China under Grant No. 201001D0104726115.
©2015 Springer Science + Business Media, LLC & Science Press, China

Xing-Zhou Zhang et al.: Tencent and Facebook Data Validate Metcalfe’s Law 247
This paper provides additional evidence by using
real data from Tencent, a company in a developing
country mostly serving Chinese users.
•Which of the four laws best fits real data?
This paper utilizes the actual data of Tencent and
Facebook to validate the four laws, and shows that Met-
calfe’s law fits the best.
•Is Metcalfe’s linear-cost hypothesis (C∝n) valid?
We show that it does not hold for the Tencent and
the Facebook data.
2 Material and Method
2.1 Data Sources
To validate the four network effect laws with the real
data of Tencent and Facebook network, we need actual
data of more than a decade for the network value Vand
the network size n. In addition, we need actual data
for the network cost Cto validate Metcalfe’s linear-cost
hypothesis. Since Facebook and Tencent are both pub-
lic companies, all actual data are available from their
prospectus and financial reports 1
○∼3
○, and summari-
zed in Appendix A1.
We follow Metcalfe’s methodology to define network
size, value, and cost[8]. 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 net-
work 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 de-
fined as the sum of QQ MAUs and Weixin (WeChat)
MAUs, as all the 250 Tencent services use these two
user account systems.
2.2 Value, Cost, and Trend Functions
Variable definitions are listed in Table 1. The for-
mulations of the value, cost, and trend functions are
listed in Table 2. To maintain a simple and easy-to-use
formulation, we only consider the major term, ignoring
secondary terms.
Table 1. Variable Definitions
Symbol Unit Definition Data Source
VUSD Value of a network Revenue
CUSD Cost of a network Revenue – net profit
nMAU Number of nodes of
a network
MAU
netoid MAU Growth trend of
size n
MAU
Table 2. Models of Network Laws, Cost Function, and
Netoid Function
Model Unit of Parame-
ters
Sarnoff’s function V=a×n a: USD/MAU
Reed’s function V=a×(2n−1) a: USD/MAU
Odlyzko’s function V=a×nlog(n)a: USD/MAU
Metcalfe’s function V=a×n2a: USD/MAU2
Cost function C=a×n2a: USD/MAU2
Netoid function Netoid =
p/(1 + e−v×(t−h))
p: MAU, h: year,
v: year−1
Formulating the value functions for the four net-
work effect laws is straightforward, as specified in Sec-
tion 1. The proportionality constant of the four func-
tions, a, is named as Sarnoff’s coefficient, Odlyzko’s co-
efficient, Metcalfe’s coefficient, and Reed’s coefficient,
respectively. If a network has a larger athan another
network, the former network provides a larger value
per user (per node) than the latter network. When
the number of users of a network is 0, the value of the
network should be 0. Thus we use 2n−1, not 2n, in
Reed’s function, to ensure the curve of Reed’s law can
pass through the origin.
Formulating the cost function is not so straightfor-
ward. Metcalfe hypothesized that the cost of a net-
work is proportional to its size. But this linear-cost hy-
pothesis 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, i.e.,
C=a×n2.
We use Metcalfe’s netoid function[8] to represent the
growth trend of the network size nwith respect to time
t.
Netoid =p/(1 + e−v×(t−h)).
1
○Tencent financial reports. http://www.tencent.com/en-us/ir/reports.shtml, Nov. 2014.
2
○http://www.sec.gov/Archives/edgar/data/1326801/000119312512034517/d287954ds1.htm#toc, Feb. 2012.
3
○Facebook financial reports. http://investor.fb.com/, Feb. 2015.

248 J. Comput. Sci. & Technol., Mar. 2015, Vol.30, No.2
The three parameters p,v,hhave the following
meanings:
•p: the peak value representing the maximum value
of the number of MAUs;
•v: the virality or speed with which adoption oc-
curs;
•h: the point in time at which the growth rate is
maximum, when the network size reaches half the peak.
2.3 Curve Fitting Method
When validating his law, Metcalfe “fiddled with the
slider parameter” provided by the Python program-
ming language to achieve a good visual fit to the actual
data[8]. Although this method is intuitive and of great
convenience, it is not so accurate and may miss some
important details. We use the least squares method in
curve fitting to fit Tencent and Facebook data to the
value, cost, and trend functions. In particular, we use
the least squares function “leastsq” provided by SciPy,
an open source Python library.
3 Results and Discussions
3.1 Value Functions
Table 3 shows the fitting results and correspond-
ing root-mean-square deviations (RMSDs) of the four
network effect functions, for Tencent data and Face-
book data. The corresponding fitting curves graphics
are shown in Fig.1 and Fig.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.
The fitting RMSDs of Metcalfe’s functions of Ten-
cent and Facebook are significantly smaller than those
of the other network functions. For Tencent, the ra-
tios of the RMSD of Metcalfe’s function to the RMSDs
of Sarnoff’s function, Odlyzko’s function, and Reed’s
function are about 1/11, 1/10, and 1/35 respectively.
For Facebook, the corresponding ratios are about 1/2,
1/2, and 1/8 respectively. These results show that for
both Tencent and Facebook data, Metcalfe’s function
not only fits the real data model well, but also is far
more accurately than the other three laws.
0 0.2 0.4
Sarnoff 's Law
Odlyzko's Law
Metcalfe's Law
Reed's Law
Actual Data
0.6
MAUs (Billion)
Revenue (Billion USD)
0.8 1.0 1.2
10
8
6
4
2
0
Fig.1. Value curves of Tencent.
0 0.2 0.4
Sarnoff 's Law
Odlyzko's Law
Metcalfe's Law
Reed's Law
Actual Data
0.6
MAUs (Billion)
Revenue (Billion USD)
0.8 1.0 1.41.2
14
12
10
8
6
4
2
0
Fig.2. Value curves of Facebook.
3.2 Cost Functions
Fig.3 and Fig.4 show the fitting curves of the cost
functions of Tencent and Facebook respectively. The
cost functions are CTencent = 5.24 ×10−9×n2and
CFacebook = 4.56×10−9×n2. The RMSDs are 0.23 (bil-
lion USD) and 0.39 (billion USD), respectively. Thus
Table 3. Fitting Results of the Four Network Effect Laws
Tencent Data Facebook Data
Value Functions RMSDs Value Functions RMSDs
Sarnoff’s function VTencent = 6.46n1.27 VFacebook = 6.39n1.51
Odlyzko’s function VTencent = 0.22 ×nlog(n) 1.19 VFacebook = 0.21 ×nlog(n) 1.45
Metcalfe’s function VTencent = 7.39 ×10−9×n20.12 VFacebook = 5.70 ×10−9×n20.64
Reed’s function VTencent = 2−1.16×109
×(2n−1) 4.06 VFacebook = 2−1.39×109
×(2n−1) 4.88
Note: V: USD, n: MAU, RMSD: billion USD.

Xing-Zhou Zhang et al.: Tencent and Facebook Data Validate Metcalfe’s Law 249
based on the actual data of Tencent and Facebook, the
assumption that the cost of a network company is pro-
portional to the square of the number of its MAUs is
correct.
0 0.2 0.4 0.6
MAUs (Billion)
Cost (Billion USD)
0.8 1.0 1.2
8
7
6
5
4
3
2
1
0
Tencent Cost Curve
Tencent Actual Cost
Fig.3. Cost curves of Tencent.
0 0.2 0.4 0.6
MAUs (Billion)
Cost (Billion USD)
0.8 1.0 1.2 1.4
10
8
4
2
0
Facebook Cost Curve
Facebook Actual Cost
Fig.4. Cost curves of Facebook.
3.3 Trend Functions
Fig.5 and Fig.6 present the growth trend of MAUs
of Tencent and Facebook respectively. The netoid func-
tions are as follows:
NetoidTencent = 2.61 ×109/(1 + e−0.30×(t−2013.8) ),
NetoidFacebook = 1.45 ×109/(1 + e−0.77×(t−2 010.56)).
The RMSDs of Tencent and Facebook are 0.014 (bil-
lion USD) and 0.028 (billion USD), respectively. We
compare the values of the two companies’ MAUs de-
rived from the netoid functions with the actual data
(as shown in Appendix A4) to validate the netoid func-
tions.
2004 2006 2008 2010
Year
MAUs (Billion)
2012 2014
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
Tencent Netoid Curve
Tencent Actual MAUs
Fig.5. Netoid curves of Tencent.
2004 2006 2008 2010
Year
MAUs (Billion)
2012 2014
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
Facebook Netoid Curve
Facebook Actual MAUs
Fig.6. Netoid curves of Facebook.
4 Related Work
Metcalfe’s law was proposed in the early 1980s,
which states that the value of a network is proportional
to the square of the size of the network.
Recently, a few papers have appeared to argue for or
against Metcalfe’s law. Odlyzko et al. described Met-
calfe’s law as both “wrong” and “dangerous”[4]. They
argued that if Metcalfe’s law is true, then two net-
works ought to interconnect regardless of their relative
sizes. They proposed Odlyzko’s law which states that
the value of a network grows in proportion to nlog(n).
Van Hove proposed that the inference of Odlyzko is
flawed. He argued that Metcalfe’s law is not so wrong
after all[9].
However, all of these arguments do not have evi-
dence that is based on real data. Madureira et al. ex-
ploited the Eurostat data to validate Metcalfe’s law and
concluded that the value of a network can have either
a quadratic or a linear dependency with the size of the

250 J. Comput. Sci. & Technol., Mar. 2015, Vol.30, No.2
network[7]. In late 2013, Metcalfe used Facebook’s data
over the past 10 years to show a good fit for Metcalfe’s
law[8].
5 Conclusions
Tencent and Facebook have big differences in reve-
nue, cost, business model, and technology. Yet both of
their actual data fit Metcalfe’s law well. The Metcalfe’s
functions of them are VTencent = 7.39 ×10−9×n2and
VFacebook = 5.70 ×10−9×n2respectively.
The relationships between the costs of Tencent and
Facebook and their network size are quadratic, rather
than linear. The cost functions of them are CTencent =
5.24 ×10−9×n2and CFacebook = 4.56 ×10−9×n2,
respectively.
The growth trend of MAUs of Tencent and Face-
book over the past decade can be modeled by the netoid
functions. The netoid functions are N etoidTencent =
2.61 ×109/(1 + e−0.30×(t−2 013.8)) and N etoidFacebook =
1.45 ×109/(1 + e−0.77×(t−2 010.56)), respectively.
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Xing-Zhou Zhang is a Ph.D.
candidate of Institute of Computing
Technology, Chinese Academy of Sci-
ences, Beijing. He received his B.S.
degree in computer science and technol-
ogy from Shandong University in 2014.
His current research interests include
ternary computing and data mining.
Jing-Jie Liu is a Ph.D. candidate
of Institute of Computing Technology,
Chinese Academy of Sciences, Beijing.
He got his B.S. degree in computer
science and technology from University
of Science and Technology of China,
Hefei, in 2009. His research interests
include computational intelligence and sensing theories.
Zhi-Wei Xu received his Ph.D.
degree from the University of Southern
California, USA. He is a professor of
the Institute of Computing Technol-
ogy, Chinese Academy of Sciences,
Beijing. His research areas include
high-performance computer architec-
ture and network computing science.
Appendix A1 Actual Data of Tencent and Facebook
Year Tencent Data Facebook Data
MAUs Revenues Cost MAUs Revenues Cost
(Billion) (Billion USD) (Billion USD) (Billion) (Billion USD) (Billion USD)
2003 0.081 5 0.088 7 0.049 8 N/A N/A N/A
2004 0.134 8 0.138 1 0.084 2 0.001 0.000 382 N/A
2005 0.201 9 0.176 8 0.116 7 0.006 0.009 000 N/A
2006 0.232 6 0.358 6 0.222 4 0.012 0.048 000 N/A
2007 0.300 2 0.523 1 0.308 4 0.058 0.153 000 0.015
2008 0.376 6 1.047 0 0.634 8 0.145 0.272 000 0.216
2009 0.522 9 1.822 0 1.057 2 0.360 0.777 000 0.548
2010 0.647 6 2.967 0 1.741 1 0.608 1.974 000 1.368
2011 0.771 0 4.523 0 2.899 7 0.845 3.711 000 2.711
2012 0.959 0 6.983 0 4.949 3 1.060 5.089 000 5.036
2013 1.163 0 9.913 0 7.360 4 1.230 7.872 000 6.372
Third Quarter 2014 1.288 0 N/A N/A N/A N/A N/A
2014 N/A N/A N/A 1.390 12.470000 09.530

Xing-Zhou Zhang et al.: Tencent and Facebook Data Validate Metcalfe’s Law 251
Appendix A2 Actual Data Versus Derived Values of the Value Functions of Tencent
Year Actual Revenues Sarnoff’s Function Odlyzko’s Function Metcalfe’s Function Reed’s Function
(Billion USD) (Billion USD) (Billion USD) (Billion USD) (Billion USD)
2003 0.088 7 0.526 5 0.471 2 0.049 1 2−1.080×109
2004 0.138 1 0.870 8 0.800 9 0.134 3 2−1.030×109
2005 0.176 8 1.304 0 1.225 0 0.301 1 2−0.961×109
2006 0.358 6 1.503 0 1.422 0 0.399 8 2−0.930×109
2007 0.523 1 1.939 0 1.860 0 0.666 0 2−0.863×109
2008 1.047 0 2.433 0 2.360 0 1.048 0 2−0.786×109
2009 1.822 0 3.378 0 3.332 0 2.021 0 2−0.640×109
2010 2.967 0 4.184 0 4.170 0 3.099 0 2−0.515×109
2011 4.523 0 4.981 0 5.008 0 4.393 0 2−0.392×109
2012 6.983 0 6.195 0 6.295 0 6.796 0 2−0.204×109
2013 9.913 0 7.513 0 7.705 0 9.996 0 20
Appendix A3 Actual Data Versus Derived Values of the Value Functions of Facebook
Year Actual Revenues Sarnoff’s Function Odlyzko’s Function Metcalfe’s Function Reed’s Function
(Billion USD) (Billion USD) (Billion USD) (Billion USD) (Billion USD)
2004 0.000 382 0.006 39 0.004 185 629 0.000 005 7 2−1.389×109
2005 0.009 000 0.038 34 0.028 370 829 0.000 205 2 2−1.384×109
2006 0.048 000 0.076 68 0.059 261 658 0.000 820 8 2−1.378×109
2007 0.153 000 0.370 62 0.314 116 714 0.019 174 8 2−1.332×109
2008 0.272 000 0.926 55 0.825 544 495 0.119 842 5 2−1.245×109
2009 0.777 000 2.300 40 2.148 810 678 0.738 720 0 2−1.030×109
2010 1.974 000 3.885 12 3.725 638 060 2.107 084 8 2−0.782×109
2011 3.711 000 5.399 55 5.262 169 039 4.069 942 5 2−0.545×109
2012 5.089 000 6.747 84 6.647 469 486 6.356 275 2 2−0.334×109
2013 7.872 000 7.846 92 7.786 341 921 8.595 508 8 2−0.160×109
2014 12.470 00008.882 10 8.865 714 575 11.012 970 0020
Appendix A4 Actual Data Versus Derived Values of the Netoid Functions
Year Tencent Data Facebook Data
MAUs Values of Netoid MAUs Values of Netoid
(Billion) Function (Billion) (Billion) Function (Billion)
2003 0.081 5 0.098 37 N/A N/A
2004 0.134 8 0.131 10 0.001 0.009 223 343
2005 0.201 9 0.173 80 0.006 0.019 774 386
2006 0.232 6 0.229 30 0.012 0.042 043 085
2007 0.300 2 0.300 30 0.058 0.087 849 076
2008 0.376 6 0.389 70 0.145 0.177 277 077
2009 0.522 9 0.499 90 0.360 0.335 329 627
2010 0.647 6 0.632 50 0.608 0.571 067 738
2011 0.771 0 0.787 00 0.845 0.846 653 549
2012 0.959 0 0.960 10 1.056 1.090 262 724
2013 1.163 0 1.149 00 1.228 1.257 836 185
2013.75(2014 Q3) 1.288 0 1.295 00 N/A N/A
2014 N/A N/A 1.390 1.354 208 636
Note: This paper uses 2013.75 as a proxy for the third quarter of 2014 in calculation because Tencent has not published its Annual
Financial Report of 2014.