Three degrees of influence
Three Degrees of Influence is a theory in the realm of social networks, proposed by Nicholas A. Christakis and James H. Fowler in 2007. It has since been explored by scientists in numerous disciplines using diverse statistical, psychological, sociological, and biological approaches.
Christakis and Fowler found that social networks have great influence on individuals' behavior. But social influence does not end with the people to whom a person is directly tied. We influence our friends, who in their turn influence their friends, and so our actions can influence people we have never met, to whom we are only indirectly tied. They posited that diverse phenomena "ripple through our network, having an impact on our friends (one degree), our friends’ friends (two degrees), and even our friends’ friends’ friends (three degrees). Our influence gradually dissipates and ceases to have a noticeable effect on people beyond the social frontier that lies at three degrees of separation." They posited a number of reasons for this decay, including informational, psychological, and biological rationales.
This argument is basically that peer effects need not stop at one degree, and that, if we can affect our friends, then we can (in many cases) affect our friends' friends, and so on. However, across a broad set of empirical settings, using both observational and experimental methods, they observed that the effect seems, in many cases, to no longer be meaningful at a social horizon of three degrees.
Christakis and Fowler examined phenomena from various domains, such as obesity, happiness, cooperation, and voting. Investigations by other groups have subsequently explored many other phenomena in this way (including crime, social learning, etc.).
Influence dissipates after three degrees (to and from friends’ friends’ friends) for three reasons, Christakis and Fowler propose:
- Intrinsic decay -- corruption of information or a kind of "social friction" (like the game telephone).
- Network instability -- social ties become unstable (and are not constant across time) at a horizon of more than three degrees of separation.
- Evolutionary purpose -- we evolved in small groups where everyone was connected by three degrees or fewer (an idea receiving subsequent support ).
Initial studies using observational data by Christakis and Fowler suggested that a variety of attributes (like obesity, smoking, and happiness), rather than being individualistic, are casually correlated by contagion mechanisms that transmit such phenomena over long distances within social networks. Certain subsequent analyses explored limitations to these analyses (subject to different statistical assumptions); or expressed concern that the statistical methods employed in these analyses could not fully control for other environmental factors; or resulted in statistical estimates without straightforward interpretations; or did not fully account for homophily processes in the creation and retention of relationships over time.
But other scholarship using sensitivity analysis has found that the basic estimates regarding the transmissibility of obesity and smoking cessation, for example, are robust, or has otherwise replicated or supported the findings. Additional, detailed modeling work published in 2016 showed that the GEE modeling approach used by Christakis and Fowler (and others) was quite effective at estimating social contagion effects and in distinguishing them from homophily. This paper concluded, "For network influence, we find that the approach appears to have excellent sensitivity, and quite good specificity with regard to distinguishing the presence or absence of such a 'network effect,' regardless of whether or not homophily is present in network formation. This was true for small cohorts (n = 30) and larger cohorts (n = 1000), and for cohorts that displayed lesser and greater realism in their distribution of friendships." Another methodological paper concluded that it is indeed possible to bound estimates of peer effects even given the modeling constraints faced by Christakis and Fowler  -- even if parametric assumptions are otherwise required to identify such effects using observational data (if substantial unobserved homophily is thought to be present).
Additional support for the modeling approach used by Christakis and Fowler provided by other authors has continued to appear, including of the three-degrees-of-influence property. And additional analytic approaches to observational data have also been supportive, including matched sample estimation, and reshuffling techniques (validating the edge directionality test proposed by Christakis and Fowler in their 2007 paper as an identification strategy). The three degrees of influence property has also been noted, by other groups, using observational data regarding criminal networks.
Christakis and Fowler reviewed critical and supportive findings regarding the three degrees of influence phenomenon and the analytic approaches used to discern it in 2013.
In addition, subsequent studies (by many research groups, including Christakis and Fowler) have found strong causal evidence of behavioral contagion processes (including those that spread beyond dyads -- out to two, three, or four degrees) using randomized controlled experiments. including one experiment involving 61,000,000 people that showed spread of voting behavior out to two degrees of separation. A 2014 paper also confirmed the spread of emotions beyond dyads, as proposed in 2008 by Christakis and Fowler, using another massive online experiment.
Diverse lines of work have also explored the specific biopsychosocial mechanisms for the boundedness of contagion effects, some of which had been theorized by Christakis and Fowler. Experiments by Moussaid et al evaluated the spread of risk perception, and documented inflection at approximately three degrees. Another set of experiments documented the impact of information distortion, noting that "despite strong social influence within pairs of individuals, the reach of judgment propagation across a chain rarely exceeded a social distance of three to four degrees of separation.... We show that information distortion and the overweighting of other people’s errors are two individual-level mechanisms hindering judgment propagation at the scale of the chain." And experiments with fMRI scans in a sociocentrically mapped network of graduate students, published in 2018, showed that neural responses to conceptual stimuli were similar between friends, with a nadir at three degrees of separation, providing further biological evidence for this theory.
The theory has also been used to develop validated algorithms for efficient influence maximization. 
The idea of network influence raises the question of free will, because it suggests that we are influenced by factors which we cannot control and which we are not aware of. Christakis and Fowler claim in their book, Connected, that policy makers should use knowledge about social network effects in order to create a better society with a more efficient public policy. This applies to many aspects of life, from public health to economics. For instance, they note that it might be preferable to immunize individuals located in the center of a network in preference to structurally peripheral individuals. Or, it might be much more effective to motivate clusters of people to avoid criminal behavior than to act upon individuals or than to punish each criminal separately. Their subsequent work has explored how to use social contagion to foster the spread of desirable innovations in rural villages.
If people are connected to everyone by six degrees of separation (according to the social psychologist Stanley Milgram) and influence those up to three degrees (Christakis and Fowler), then people can, in some sense, reach "halfway" to anyone in the world.
- "The hidden influence of social networks Nicholas Christakis on TED.com".
- Connected Preface+chapter1
- Morgan, TJH; et al. (2015). ", Experimental evidence for the co-evolution of hominin tool-making teaching and language". Nature Communications. 6: 6029. doi:10.1038/ncomms7029. PMC . PMID 25585382.
- Christakis, Nicholas A.; Fowler, James H. (2007). "The Spread of Obesity in a Large Social Network over 32 Years". The New England Journal of Medicine. 357 (4): 370–379. doi:10.1056/NEJMsa066082. PMID 17652652.
- Christakis, Nicholas A.; Fowler, James H. (2008). "The Collective Dynamics of Smoking in a Large Social Network". The New England Journal of Medicine. 358: 2249–2258. doi:10.1056/NEJMsa0706154. PMC . PMID 18499567.
- Christakis, Nicholas A.; Fowler, James H. (2008). "Dynamic spread of happiness in a large social network: longitudinal analysis over 20 years in the Framingham Heart Study". British Medical Journal. 337 (337): a2338. doi:10.1136/bmj.a2338. PMC . PMID 19056788.
- Christakis, Nicholas A.; Fowler, James H. (2009). Connected:The Surprising Power of Our Social Networks and How They Shape Our Lives. Little, Brown and Co. ISBN 978-0316036146.
- Cohen-Cole, Ethan; Fletcher, Jason M. (2008). "Detecting implausible social network effects in acne, height, and headaches: longitudinal analysis". British Medical Journal. 337: a2533. doi:10.1136/bmj.a2533.
- Cohen-Cole, Ethan; Fletcher, Jason M. (2008). "Is obesity contagious? Social networks vs. environmental factors in the obesity epidemic". Journal of Health Economics. 27: 1382–1387. doi:10.1016/j.jhealeco.2008.04.005.
- Lyons, Russell (2011). "The Spread of Evidence-Poor Medicine via Flawed Social Network Analysis". Statistics, Politics, and Policy. 2 (1). doi:10.2202/2151-7509.1024.
- Noel, Hans; Nyhan, Brendan (2011). "The 'unfriending problem': The consequences of homophily in friendship retention for causal estimates of social influence". Social Networks. 33 (3): 211–218. doi:10.1016/j.socnet.2011.05.003.
- Shalizi, Cosma R.; Thomas, Andrew C. (2011). "Homphily and Contagion Are Generically Confounded in Observational Social Network Studies". Sociological Methods & Research. 40 (2): 211–239. doi:10.1177/0049124111404820.
- VanderWeele, Tyler J. "Sensitivity Analysis for Contagion Effects in Social Networks". Sociological Methods & Research. 40 (2): 240–255. doi:10.1177/0049124111404821.
- Christakis, NA; Fowler, JH (2013). "Social Contagion Theory: ExaminingDynamic Social Networks and Human Behavior". Statistics in Medicine. 32: 556–577. doi:10.1002/sim.5408.
- Ali, MM; Amialchuk, A; Gao, S; Heiland, F (2012). "Adolescent Weight Gain and Social Networks: Is There a Contagion Effect?". Applied Economics. 44: 2969–2983. doi:10.1080/00036846.2011.568408.
- Steeg, A. Galstyan (2012). "Statistical Tests for Contagion in Observational Social Network Studies". Journal of Machine Learning Research: 563–571.
- Zachrison, Kori (2016). "Can Longitudinal Generalized Estimating Equation Models Distinguish Network Influence and Homophily? An Agent-Based Modeling Approach to Measurement Characteristics". BMC Medical Research Methodology. 16: 174. doi:10.1186/s12874-016-0274-4.
- Gonzalez-Pardo, A.; Cajias, R.; Camacho, D. (2014). "An Agent Based Simulation of Christakis-Fowler Social Model". Recent Developments in Computational Collective Intelligence. 513: 69–77.
- Aral, Sinan; Muchnik, Lev; Sunararajan, Arun (2009). "Distinguishing influence-based contagion from homophily-driven diffusion in dynamic networks". Proceedings of the National Academy of Sciences. 106 (51): 21544–21549. doi:10.1073/pnas.0908800106. PMC . PMID 20007780.
- Anagnostopoulos, Aris; Kumar, Ravi; Mahdian, Mohammad (2008). "Influence and Correlation in Social Networks". Proceedings of the 14th ACM SIGKDD Conference on Knowledge Discovery and Data Mining: 7–15. doi:10.1145/1401890.1401897.
- Wildeman, Christopher; Papachristos, Andrew V. "Network Exposure and Homicide Victimization in an African American Community". American Journal of Public Health. 104 (1): 143–150. doi:10.2105/ajph.2013.301441. PMC . PMID 24228655.
- Centola, Damon (2010). "The Spread of Behavior in an Online Social Network Experiment". Science. 329 (5995): 1194–1197. doi:10.1126/science.1185231. PMID 20813952.
- Centola, Damon (2011). "An experimental study of homophily in the adoption of health behavior". Science. 334 (6060): 1269–1272. doi:10.1126/science.1207055. PMID 22144624.
- Fowler, James H.; Christakis, Nicholas A. (2010). "Cooperative behavior cascades in human social networks". Proceedings of the National Academy of Sciences. 107 (12): 5334–5338. doi:10.1073/pnas.0913149107. PMC . PMID 20212120.
- Aral, Sinan; Walker, Dylan (2011). "Creating Social Contagion Through Viral Product Design: A Randomized Trial of Peer Influence in Networks". Management Science. 57 (9): 1623–1639. doi:10.1287/mnsc.1110.1421.
- Rand D, Arbesman S, and Christakis NA,"Dynamic Social Networks Promote Cooperation in Experiments with Humans," PNAS:Proceedings of the National Academy of Sciences 2011; 108: 19193-19198
- Bond, RM; Fariss, CJ; Jones, JJ; Kramer, ADI; Marlow, C; Settle, JE; Fowler, JH (2012). "A 61-million-person experiment in social influence and political mobilization". Nature. 489: 295–298. doi:10.1038/nature11421. PMC . PMID 22972300.
- Kramer, ADI; Guillory, JE; Hancock, JT (2014). "Experimental evidence of massive-scale emotional contagion through social networks" (PDF). Proceedings of the National Academy of Sciences. 111: 8788–8790. doi:10.1073/pnas.1320040111. PMC . PMID 24889601.
- Moussaid, M; Brighton, H; Gaissmaier, W (2015). "The amplification of risk in experimental diffusion chains" (PDF). Proceedings of the National Academy of Sciences. 112: 5631–5636. doi:10.1073/pnas.1421883112. PMC . PMID 25902519.
- Moussaid, M. "Reach and Speed of Judgment Propagation in the Laboratory" (PDF). Proceedings of the National Academy of Sciences. 114: 4117–4122. doi:10.1073/pnas.1611998114.
- Parkinson, Carolyn; Kleinbaum, Adam M.; Wheatley, Thalia (2018-01-30). "Similar neural responses predict friendship". Nature Communications. 9 (1). doi:10.1038/s41467-017-02722-7. ISSN 2041-1723.
- Qin, Yadong; Ma, Jun; Gao, Shuai (2015-06-08). "Efficient Influence Maximization Based on Three Degrees of Influence Theory". Web-Age Information Management. Springer, Cham: 465–468. doi:10.1007/978-3-319-21042-1_42.
- Kim, David A; Hwong, Alison R; Stafford, Derek; Hughes, D Alex; O'Malley, A James; Fowler, James H; Christakis, Nicholas A (2015-07-11). "Social network targeting to maximise population behaviour change: a cluster randomised controlled trial" (PDF). The Lancet. 386 (9989). doi:10.1016/S0140-6736(15)60095-2. ISSN 0140-6736.