HOW FACEBOOK GROWS? DO TWITTER, WEIBO, LINKEDIN, WECHAT GROW IN DIFFERENT WAYS? - IWCSN 2013, VANCOUVER
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Innovation Tower (2013) at Hong Kong Polytechnic University, by Zaha Hadid
IWCSN 2013, Vancouver
How Facebook grows?
Michael Tse!
Do Twitter, Weibo, LinkedIn, Hong Kong
WeChat grow in different ways?
"1
Acknowledgments
My best thanks are due to all my students and postdocs who have worked !
with me and taught what networks are all about, especially:!
!
Dr Choujun Zhan!
Miss Jiajing Wu!
!
!
and my colleagues and collaborators:!
!
Prof. Francis Lau!
Prof. Michael Small!
!
"2
This talk is about Keywords:!
Dynamics, transitions,
growth, evolution, consensus,
Growth of Networks spreading, learning, …!
"3
A network is defined by
Computer networks: computers and links!
Social networks: People and relations!
Nodes and Edges
Transport networks: cities and routes!
Biological networks: functions and
relations!
Music networks: notes and connections!
"4
The three seminal papers on networks
Paul Erdos:!
Random graph!
1960
❖ Erdos, Paul, and A. Rényi. "On the
evolution of random graphs." Publ. Math.
Inst. Hungar. Acad. Sci 5 (1960): 17-61.!
❖ Watts, Duncan J., and Steven H. Strogatz.
"Collective dynamics of ‘small-
world’networks." nature 393.6684 (1998):
Duncan Watts: !
440-442.!
Small-world network!
❖ Barabási, Albert-László, and Réka Albert. 1998
"Emergence of scaling in random
networks." science 286.5439 (1999): 509-512. Albert Barabasi: !
Scale-free network!
1999
"5
Static Networks Yorkshire
Melbourne Brunel University (Proposed)
"6
Most networks are dynamic.
And they grow!
"7
Real-time capture of chatting individuals
"8
Computer Network Examples
Internet
(IP addresses)
The internet mapping project (since 1997,
Bell lab) and image by Lumeta:
information about the network including
discovery of all IP-connected devices on a
network, and how they are
interconnected.!
!
Routing paths!
[http://en.wikipedia.org/wiki/
Internet_censorship]
"9
Computer Network Examples
Internet
(WWW)
Network of homepages
connected by hyperlinks!
!
!
Example:!
United nation websites network!
[http://www.cugelman.com/]
"10Computer Network Examples
Internet
(physical links)
Optic fibre networks,
wireless networks,
connecting networked
computers and devices.
"11Social Network Examples
Social Networks
EntityCube!
!
Renlifang!
!
!
Microsoft project to track down relationships
of people!
!
http://entitycube.research.microsoft.com/!
http://renlifang.msra.cn!
"12Biological Network Examples
Protein Interaction
Network
Protein–protein interactions occur when
two or more proteins bind together, often to
carry out their biological functions. Their
interactions provide complex biological
functions. They also play important roles in
biological presses like diseases. !
!
[https://parasol.tamu.edu/dreu2013/
Doroschak/]!
"13Biological Network Examples
Disease Network
a. Each node represents a disorder. Disorders are
connected if they share at least one gene. Size of
nodes presents the number of genes participating
in the disorder.!
b. Each node is a gene. Genes are connected if
they participate in the same disease. Size of nodes
represent the number of disease it participates in.!
!
!
[Barabasi et al.: http://www.pnas.org/content/
104/21/8685/F2.expansion.html]
"14Transport Network Examples
Highway Network
USA highway network!
!
[http://en.wikipedia.org/wiki/
File:National_Highway_System.jpg]
"15Finance Network Examples
Stocks Network
Stocks connected by high correlation
links form networks with scale free
property.!
!
!
!
!
!
Tse: IWCSN 2010; J Empr. Fin. 2010
"16Other Examples
Music Network
Musical note co-occurrence networks.!
!
Universal properties found for different
genres.!
!
!
!
!
!
Tse: IWCSN 2009; Physica D, 389, 2010 Bach’s violin sonatas
"17Other Examples
Orchestra
Interaction and consensus!
among players under the !
leadership of conductor.!
!
Problem of consensus in !
finite time.!
!
!
!
Tse: IWCSN 2011
"18Transition Network
A model for studying growth
"19Transition Network
❖ Each node assumes a state.!
❖ There are more than one
possible states a node can
assume.!
❖ The state of a node changes, as
influenced by others connected
to it, and/or other external
factors.
"20User growth phenomenon
❖ Consider a community of users of a
certain product or software
application.!
❖ Assume people are either users or
not users (prospective users).!
❖ Once they become users, they stay
with that status.!
❖ Red nodes: users U!
❖ Gray nodes: prospective users P
"21User growth phenomenon
Single transition channel
"22Epidemic Spreading
❖ Red, gray and yellow nodes
represent infected I, suspected
S and recovered R, respectively.!
❖ Disease spreads through the
links.
M. Small, C. K. Tse, and D. M. Walker, “Super-spreader and the rate of transmission of the SARS virus”, Physica D, 215 (2006) 146-158.!
M. Small, D. M. Walker and C. K. Tse, “Scale-free distribution of avian influenza outbreaks”, PRL 99, 188702 (2007)
"23Epidemic Spreading
Multiple transition channels
"24Cascade Failure
❖ A simple network : gray nodes represent
functional devices F, while red nodes
represents malfunctioning devices M.!
❖ When a device malfunctions, it’s adjacent
device will have high probability of
malfunctioning.
"25Cascade Failure
"26Transition Network
Each node assumes one of a number
of possible states:!
!
Each time step, nodes may change
their states, under mutual influence.
"27Research Map
We now illustrate a particular and simple case:
Simple transition model!
User Growth Model Single transition channel!
Universal growth equation!
ApplicationsFacebook’s remarkable growth
❖ Facebook!
❖ Listed in the
Fortune 500.!
❖ Placed at position of
462 on the list
published in May
2013.!
❖ Market value
passed $100 billion.
"30Map of the world that slowly
World Map appeared during the
rendering process of Facebook
http://michaellhanson.wordpress.com/2010/12/15/ connection and relationship
facebooks-relational-map-of-the-earth/ data.
"31❖ QQ:!
❖ The most successful
online chatting and
Remarkable user growth
messaging software
for PC in China!
❖ 13 years ago, the
CEO wants to sell
QQ for about
600000 Yuan (less
than 0.1 million US
dollar).!
❖ Now, Tencent’s
market valuation
rose to 101 billion
dollar.
"32Real-time capture of chatting individuals
"33Growth Model in Detail
The Simplest Transition Model Example
"34User Growth Model
❖ Consider a social network with N nodes. Each node
represents an individual which can be one of two
state U (user) and P (prospective user).!
❖ Links mimic relationship among the individuals,
e.g., they are friends, relatives, family members, etc. !
❖ In many cases, our friends and/or family members
use a product, we (prospective users) get the
information of this product from them (users) along
the other side of one/several links in this social
network, e.g., our friends/family members will
“persuade” and/or “infect” us to use the same
product. !
❖ Each link connected a user and a prospective
customer is a connection along which a node can
transit from state P into U.
"35User Growth Transition
"36Transition Channel
❖ Single transition channel:! !
T:P +U !c!
→U!+U .
❖ The conceptual description of the transition process is represented by this chart and
can be understood as a transition channel, with transition rate c.
"37User Growth Model
❖ The number of users can be any positive integer X = 1, 2, … N. Let P(X(t) = m) denotes the
probability that there are m users in the transition network at time t, while P(X(t+∆t) = n |
X(t) = m) is the transition probability of having n users at t + ∆t, conditioned upon there
being m users at the immediately preceding time t. For conciseness, we define !
!
!
❖ Assume that at infinitesimal interval ∆t, at most one prospective transition link undergoes
a transition, namely, at most one prospective user transits into a user. Hence, Pn,n–1(t, ∆t) =
0 for n = 2, 3, … N and Pn,n–1(t, ∆t) ≠ 0 otherwise. Thus, Pn(t+∆t) can be simplified as!
! (*)
#38User Growth Model
❖ Suppose the number of distinct prospective transition links is h(n) with n users.!
❖ Assuming the network is fully connected (but not essential), we have h(n) = n(N–n).!
!
❖ Combining (**) and (*), we get!
! (#)
❖ The expectation of the number of users at t + ∆t is !
!
(##)
#39Universal Growth Equation
❖ Combining (#) and (##), we get!
!
❖ Re-arranging and taking the limit !
!
❖ Define x(t) = E[X(t)]; k1 = cN; and k2 = (c – var(X)/E[X]2).!
!
The growth equation is:!
The solution is:
#40closed form solution!
A surprisingly simple for networks with
single transition
channel12 Datasets
❖ Facebook: One of the most popular online social networking (OSN)
service!
❖ LinkedIn: A social network website for people in professional
occupations and focused on work relationships. !
❖ Evernote: A suite of software and services designed for notetaking and
archiving. !
❖ Tencent QQ: Popularly known as QQ, is an instant messaging software
service developed by Tencent Holding Limited!
❖ Twitter: An online social networking (OSN) and microblogging service
that enables users to send and read “tweets”.!
❖ US Hospital account on Youtube: Some researchers count the number of
account of US hospital on Youtube.!
❖ Line: A Japanese proprietary application for instant messaging on
smartphones and PCs.!
❖ WeChat: A mobile text and voice messaging communication service
developed by Tencent in China.!
❖ Kakao Talk: A free mobile messenger application for smartphones with
free text and free call features. !
❖ World of Warcraft: One of the most popular MMORPG created by
Blizzard Entertainment. !
❖ Sina Weibo: One of the most popular Chinese microblogging websites. !
❖ Tencent Weibo: Another Chinese microblogging website launched by
Tencent in Apr/2010.
#42Fitting of datasets
#43Fitting of datasets
#44Fitting of datasets
#45Fitting of datasets
What does this illustrate?!
Competition exists?!
Model does not include.
World of Warcraft: One of the
most popular massively
multiplayer online role-playing
game (MMORPG) created by
Blizzard Entertainment. !
#46More Accurate Fitting
❖ Adding more transition channels to describe the presence of
competition.!
❖ Users have multiple states corresponding to the use of different
products.!
❖ New transition channels describe transiting to a competing product
#47General Model – Sketch
❖ We have also derived the general model where !
❖ nodes assuming general multiple states; !
❖ more than one transition channels exist, each with a stochastic
transition rate ci ;!
❖ individual transition links are considered.!
❖ The model is applicable to all kinds of transition networks,
including disease spreading, consensus, language evolution, etc.
Choujun Zhan, Chi K. Tse and Michael Small, “A General Model for Stochastic Simulation of Time Evolution of
Transition Networks,” to be submitted.
#48General Model – Sketch
❖ Consider a network of n nodes.!
❖ Each node can assume k possible states: x1, x2, …, xk!
❖ There are m transition channels: T1, T2, …, Tm!
Tµ : x! →!
cµ
! p( µ 1)
+ x p( µ 2)
!+ x p( µ L )
! ! xq( µ1) + xq( µ 2) !+ xq( µ L ) , (µ = 1,2,!,m) .
µ µ
❖ Left hand side: transition link before transition. Right hand side: resulting transition
link.!
❖ Stochastic transition rate cµ for each channel. So, cµ∆t is the probability that a
particular selected transition link of channel Tµ at time t will react in the next
infinitesimal time interval (t, t + ∆t).
#49General Model – Sketch
#50General Transition Equation
❖ The state matrix S(t) is required to establish the model. At each time, S(t)
defines the state of each node, i.e., sij(t) = 1 if node j is at state i;
otherwise 0. With this matrix, we can write the general transition
equation:
∂P(S(t) = N s ) m m
= ∑ ∑ cµ P(S(t) = N s− ) − P(S(t) = N s ) ⋅ ∑ cµ hµ (N s ).
∂t µ =1 N − ∈Ω− µ =1
s 1,µ
Prob that S(t) is number of distinct
in state Ns – transition links with state
all state matrices Ns !
that can transit to Ns S(t) for channel Tµ
#51Applying the model to special network and parameters,
❖ we can derive the probability distributions of ∆t, transition
types, and involving transition links.!
❖ We are then able to answer the following questions in
the statistical sense:!
❖ When will the next transition occur? (What is t1?)!
❖ Which particular transition will occur? (x1 to x2; or x2 to x3)!
❖ Which particular transition links are involved? (Which x1
transited?)
#52Snapshots of some results
SIR model of
disease spreading
2 transition channels:
#53SIR Disease Spreading
A profile of disease
propagation in a ER
network with entirely
susceptible population
and a single infectious
individual as initial
condition.
#54SIR Disease Spreading
Disease propagation
dynamics of four
different networks.!
!
Solid lines
represents the
number of S, I and R
nodes, and the gray
regions represent the
dynamics regions.
#55SIR Disease Spreading
Disease propagation
dynamics of four
different networks.!
!
#56Waves of Influenza
Observed weekly mortality in 1918.!
Daihai He, et al. "Inferring the causes of the three waves of the 1918 influenza
pandemic in England and Wales." Proceedings of the Royal Society B: Biological
Sciences 280.1766 (2013).!
#57Waves of Influenza
❖ We can able to explain all the salient features of the flu wave using the transition
model.!
❖ Use the same SIS formulation!
❖ Use the general transition model!
❖ Our trick is to switch network structure!
❖ During school days: small-world (large h(n); same c)!
❖ During vacations: regular (much smaller h(n); same c)
#58Transition model predicts
vacations
school days
#59Language Invasion
X: speaker of language X!
Y: speaker of language Y
#60Scottish Gaelic Speakers in Sutherland
Percentage of population speaker Scottish Gaelic
Our transition model
ODE model
Abrams, Daniel M., and Steven H. Strogatz. "Linguistics: Modelling the dynamics of
language death." Nature 424.6951 (2003): 900-900.
#61Conclusion
❖ We derive a general transition model from the basics, and it
applies universally and seems to be most general.!
❖ Computational algorithms are generally required to
simulate or realise the dynamics “experimentally”.!
❖ For networks with single transition channel, a very simple
and universal closed-form growth equation can be derived.
This equation fits most existing networks. !
❖ The model can be improved with consideration of other
factors like competition, phases of popularity, saturation, etc.
#62“A creative idea plus a fresh network is the best
way to go from zero to millions.”
– www.greenopia.in
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