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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 …
  13. 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. 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!

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