Talk Given to the Galois Group, University of Manchester, 2017.

1. The Mathematics
of Memes
Thomas House
School of Mathematics, University of Manchester
Galois Group Talk
Simon 2.39
1pm 5 December 2017

2. How I Ended up Giving This Talk
• Veronica Kelsey Sent me this …

3. What is a Meme?
• The modern usage involves only the internet,
but the word goes back to Dawkins:

4. Modelling literally viral
behaviour: the SIR model
• The ‘SIR model’ has two parameters:
– R0 = β/γ, the average number of secondary cases
produced by an index case early in the epidemic
(more on this later).
– T=1/γ, the average time cases spend infectious.
• As an ODE:
dS
dt
= −βSI ,
dI
dt
= βSI −γI .

5. An SIR epidemic
The SIR model
does
reproduce the
‘up and down’
behaviour seen
in infectious
disease
epidemics
0 20 40 60 80
0
0.2
0.4
0.6
0.8
1
Time (days)
ProportionofPopulation
Susceptible Infectious Recovered
The code to produce this figure and similar output is available on my
website. Parameter choices are R0 = 3; T = 4 days.

6. 2003 SARS, Hong Kong
Source: World Health Organisation

7. 1918-19 H1N1 Influenza, England & Wales
0 5 10 15 20
0
1
2
x 10
4
Reporteddeaths
Week
0 5 10 15 20
0
5
10
x 10
6
Modelledinfluenzacases
Source: House (2012), Cont. Phys.

8. 2002 West Nile Virus, USA
Source: Huhn et al. (2003) AFP.
data through ArboNET, a secure, Web-based
surveillance network comprising 54 state and
local public health departments. Local health
quito. In the United States, the virus is main-
tained in an enzootic mosquito-bird-mos-
quito cycle that primarily involves Culex mos-
FIGURE 2. Human West Nile meningitis and encephalitis cases in 2002, by location and time of illness
onset. As of April 15, 2003, there were 4,156 reported cases. Southern states included Alabama,
Arkansas, California, Delaware, District of Columbia, Florida, Georgia, Kentucky, Louisiana, Mary-
land, Mississippi, North Carolina, Oklahoma, South Carolina, Tennessee, Texas, West Virginia, and Vir-
ginia. Northern states included Colorado, Connecticut, Illinois, Indiana, Iowa, Kansas, Massachusetts,
Michigan, Minnesota, Missouri, Montana, Nebraska, New Jersey, New York, North Dakota, Ohio,
Pennsylvania, Rhode Island, South Dakota, Vermont, Wisconsin, and Wyoming.
Unpublished data compiled by ArboNET. Centers for Disease Control and Prevention, Center for Infectious Dis-
eases, Division of Vector-Borne Infectious Diseases, Fort Collins, Colo.
West Nile Meningitis and Encephalitis Cases
May25
Jun8
Jun22
Jul6
Jul20
Aug3
Aug17
Aug31
Sep14
Sep28
Oct12
Oct26
Nov9
Nov23
Dec7
Dec21
Week ending
Numberofcases
500
400
300
200
100
0
■ North
■■ South

9. Early Behaviour
Feature 1:
Early exponential
growth in
infection
0 20 40 60 80
0
0.2
0.4
0.6
0.8
1
Time (days)
ProportionofPopulation
Susceptible Infectious Recovered
16 18 20 22 24 26
0
0.05
0.1
0.15
0.2
0.25
Time (days)
ProportionofPopulation
Susceptible Infectious Recovered

10. Epidemic Peak
Feature 2:
The epidemic
peaks when herd
immunity is
reached
0 20 40 60 80
0
0.2
0.4
0.6
0.8
1
Time (days)
ProportionofPopulation
Susceptible Infectious Recovered
26 27 28 29 30 31 32
0.2
0.25
0.3
0.35
0.4
Time (days)
ProportionofPopulation
Susceptible Infectious Recovered

11. Late Behaviour
Feature 3:
Every epidemic
leaves a pool of
susceptibles still
vulnerable to new
outbreaks
0 20 40 60 80
0
0.2
0.4
0.6
0.8
1
Time (days)
ProportionofPopulation
Susceptible Infectious Recovered
80 85 90 95
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Time (days)
ProportionofPopulation
Susceptible Infectious Recovered

12. Asch Conformity Experiments
• From Wikipedia article …

13. Centola
Complex
Contation
• More recent
evidence looked
at a controlled
social network
G. Hunt, A. Miller, T. Olszewski, and P. Wagner for their
suggestions; M. Kosnik and A. Miller for reviews; and
M. Foote for verifying that my subsampling algorithms
were programmed correctly. Numerous contributors to the
Paleobiology Database made this study possible, and
I am particularly grateful to M. Clapham, A. Hendy, and
W. Kiessling for recent contributions. Research described
here was funded by donations from anonymous private
individuals having no connection to it. This is Paleobiology
Database publication 117.
Supporting Online Material
www.sciencemag.org/cgi/content/full/329/5996/1191/DC1
Materials and Methods
Figs. S1 to S9
Tables S1 and S2
References
22 March 2010; accepted 30 June 2010
10.1126/science.1189910
The Spread of Behavior in an Online
Social Network Experiment
Damon Centola
How do social networks affect the spread of behavior? A popular hypothesis states that networks
with many clustered ties and a high degree of separation will be less effective for behavioral
diffusion than networks in which locally redundant ties are rewired to provide shortcuts across the
social space. A competing hypothesis argues that when behaviors require social reinforcement, a
network with more clustering may be more advantageous, even if the network as a whole has a
larger diameter. I investigated the effects of network structure on diffusion by studying the spread
of health behavior through artificially structured online communities. Individual adoption was
much more likely when participants received social reinforcement from multiple neighbors
in the social network. The behavior spread farther and faster across clustered-lattice networks than
across corresponding random networks.
M
any behaviors spread through social
contact (1–3). As a result, the network
structure of who is connected to whom
through social networks, an empirical test of
these predictions has not been possible, because
it requires the ability to independently vary the
friends who may have also signed up for the study
(or from trying to contact health buddies outside
the context of the experiment), I blinded the
identifiers that people used. Participants made
decisions about whether or not to adopt a health
behavior based on the adoption patterns of their
health buddies. The health behavior used for this
study was the decision to register for an Internet-
based health forum, which offered access and rat-
ing tools for online health resources (13).
The health forum was not known (or acces-
sible) to anyone except participants in the ex-
periment. This ensured that the only sources of
encouragement that participants had to join the
forum were the signals that they received from their
health buddies. The forum was populated with ini-
tial ratings to provide content for the early adopters.
However, all subsequent content was contributed
by the participants who joined the forum.
Participants arriving to the study were randomly
assigned to one of two experimental conditions—
REPORTS
http://scieDownloadedfrom
The Spread of Behavior in an Online Social Network Experiment
Damon Centola
DOI: 10.1126/science.1185231
(5996), 1194-1197.329Science
each other, as well as yourself).
many other people who have already adopted the behavior (for example, in the circumstances where your friends know
clustered ones. Certain types of behavior within human systems are thus more likely to spread if people are exposed to
that were signed up for the forum. The behavior spread more readily on clustered networks than on random, poorly
individuals choosing to register for a health forum could be influenced by an artificially constructed network of neighbors
(p. 1194) examined whether the number ofCentoladramatically affect the diffusion of behavior through a population.
interventions) and promote behavior change most effectively across a population. The structure of a social network can
An important question for policy-makers is how to communicate information (for example, about public health
Join the Club
ARTICLE TOOLS http://science.sciencemag.org/content/329/5996/1194
MATERIALS
SUPPLEMENTARY http://science.sciencemag.org/content/suppl/2010/08/31/329.5996.1194.DC1
CONTENT
RELATED http://science.sciencemag.org/content/sci/329/5996/1219.2.full
REFERENCES
http://science.sciencemag.org/content/329/5996/1194#BIBL
This article cites 20 articles, 4 of which you can access for free
PERMISSIONS http://www.sciencemag.org/help/reprints-and-permissions

14. The problem
with these
approaches …
(meme from
knowyourmem
e.com)

15. An ODE model of Complex
Contagion
• I considered a model with these ingredients
of
tative
online
more
than
s pro-
atures
epide-
ioural
el;
depends on B(t) in addition to other static parameters).
We assume that individuals with m canvassed neigh-
bours who are engaging in the behaviour commence at
a rate tm or cease at a rate gm as appropriate for their
current behaviour state. The dynamical system for be-
haviour prevalence in the population at time t is then
_BðtÞ ¼
Xn
m¼0
DmðtÞðð1 À BðtÞÞtm À BðtÞgmÞ: ð2:1Þ
To specify an integrable system, it is then necessary
to define a form for the dynamical parameters tm, gm
and a process for the generation of the proportion Dm.
2.2. Dynamical parameters
We now choose a form for the vectors (tm), (gm). It is
2.3. Canvassing method
To complete our model description, we need a form for
the proportion Dm. The simplest assumption is that
there are n independent trials with each trial having
probability B(t), meaning that
Dm ¼ Binðmjn; BðtÞÞ; ð2:3Þ
where Bin() is a binomial probability mass function as
defined in appendix A. This is interpreted as each indi-
2 Report. Modelling behavioural contagion T. House
http://rsif.royalsocieDownloaded from
lation, whereas here dynamics remain Markovian b
the population samples are potentially dependent.
For opinion dynamics, motivated by a comprehe
sive review of the literature and compelling empiric
evidence [2,4], we expect an S-shaped curve for t
response of behavioural transmission probability
the number of encounters with a behaviour. For simp
city, the limiting case of such a curve is taken so tha
tm ¼
t if m ! a;
0 otherwise:
ð2:
This complex form for transmission has not yet be
included in other dynamical systems models of beha
iour spread, and is the main benefit of the modellin
approach considered here. We assume for simplici

16. Fast Growth!
• Such models exhibit very fast growth.initial number I(0) participating in the fad; we will also assume that J(0) = R(0) = 0 and so the
rest of the population is initially in the S compartment so that S(0) = N − I(0).
We can also now make our verbal argument above about ‘excitable’ models more quantita-
tively. Consider the special case of our models in which C = τi = 2 and ✏ = 0. Early in the epi-
demic, for the simple contagion model, making the special choices βi = 1/N and I(0) = 1 for
simplicity, we will be able to make the large-N approximation
dI
dt
⇡ I ) IÖtÜ ⇡ et
; Ö6Ü
i.e. exponential early growth. For the complex contagion model, making the special choices
β = N and I(0) = 1 for simplicity, we will have the large-N approximation
dI
dt
⇡ I2
) IÖtÜ ⇡
1
1 t
; Ö7Ü
which represents super-exponential early growth. In both the simple and complex models I(t)
will eventually stop growing due to non-linear effects as S(t) decreases, but the early growth of
the complex model will be much more ‘explosive’, which is a feature that we will see in real
data.
Evidence for complex contagion

17. Looking for Observational Evidence
• If these are real effects then they ought to be
visible in observational data – i.e. how people
behave ‘in the wild’
• This would have implications for design of
public health interventions (as well as
advertising etc.)
• We sought to do this statistically

19. Testing in the real world – Photo Fads
• Photo-fads like ‘planking’ are spread online
• The involve real-world behaviour
• And as such, they are a ‘pure signal’ for
behaviour
• We looked at ‘Google Trends’ data and fitted
different models to it

20. 2012
11 12 01 02 03 04 05 06 07 08 09
0
1
6nHDky HDt
2014
05 06 07 08 09 10 11 12 01
0
1
CDt %HDrG
2012
07 08 09 10 11 12 01 02 03 04 05
0
1
2wOLng
2013
03 04 05 06 07 08 09 10 11 12 01
0
1
CDt %rHDGLng
2005
05 06 07 08 09 10 11 12 01 02 03
0
1
LynnGLH EngODnG
02 03 04 05 06 07 08 09 10 11 12
0
1
%rDGyLng
2012
09 10 11 12 01 02 03 04 05 06 07
0
1
%DtPDnnLng
2014
04 05 06 07 08 09 10 11 12 01
0
1
HDGokHnLng
2014
04 05 06 07 08 09 10 11 12 01
0
1
9DGHrLng
2012
07 08 09 10 11 12 01 02 03 04 05
0
1
LHLVurH DLvLng
2010
09 10 11 12 01 02 03 04 05 06 07
0
1
LyLng Down GDPH
02 03 04 05 06 07 08 09 10 11 12
0
1
6OHHvHIDcH
2013
09 10 11 12 01 02 03 04 05 06 07
0
1
3HrIHct 6SOLtV
2011
01 02 03 04 05 06 07 08 09 10 11
0
1
241543903
2014
10 11 12 01
0
1
0DPPLng
2012
05 06 07 08 09 10 11 12 01 02 03
0
1
3ODnkLng
2013
05 06 07 08 09 10 11 12 01 02 03
0
1
6kywDOkLng
2012
05 06 07 08 09 10 11 12 01 02 03
0
1
7HDSottLng
08 09 10 11 12
0
1
DuInHrLng
2012
12 01 02 03 04 05 06 07 08 09 10
0
1
7HEowLng
2012
08 09 10 11 12 01 02 03 04 05 06 07
0
1
6tockLng 3ODnkLng
2014
03 04 05 06 07 08 09 10 11 12 01
0
1
CDught 0H 6OHHSLng
2012
07 08 09 10 11 12 01 02 03 04 05
0
1
3ODyLng DHDG
2012
08 09 10 11 12 01 02 03 04 05 06
0
1
HorVHPDnnLng
2012
12 01 02 03 04 05 06 07 08 09 10
0
1
3HoSOH EDtLng 0onHy
2014
04 05 06 07 08 09 10 11 12 01
0
1
3ottHrLng
6LPSOH 0oGHO
95% CI Ior 6LPSOH 0oGHO
CoPSOHx 0oGHO
95% CI Ior CoPSOHx 0oGHO
DDtD
• Complex
contagion
is
massively
preferred
for all
photo fads
found on
knowyour
meme.com

21. Nomination Challenges

22. Nomination Challenges

23. Future work
• Look at more ‘serious’ examples of social
contagion
• Keep having fun with memes

24. Thanks for your Time!