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# The Mathematics of Memes

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

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### The Mathematics of Memes

1. 1. The Mathematics of Memes Thomas House School of Mathematics, University of Manchester Galois Group Talk Simon 2.39 1pm 5 December 2017
2. 2. How I Ended up Giving This Talk •  Veronica Kelsey Sent me this …
3. 3. What is a Meme? •  The modern usage involves only the internet, but the word goes back to Dawkins:
4. 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. 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. 6. 2003 SARS, Hong Kong Source: World Health Organisation
7. 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. 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 Unpublished data compiled by ArboNET.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. 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. 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. 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. 12. Asch Conformity Experiments •  From Wikipedia article …
14. 14. The problem with these approaches … (meme from knowyourmem e.com)
15. 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. 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. 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
18. 18. 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
19. 19. 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
20. 20. Nomination Challenges
21. 21. Nomination Challenges
22. 22. Future work •  Look at more ‘serious’ examples of social contagion •  Keep having fun with memes
23. 23. Thanks for your Time!