Diffusion and contagion in networks with heterogeneous agents and hornophily

Network Science

Volumn 1, Issue 1, 2013, Pages 49-67

  Jackson, Matthew O ^c   Lopez Pintado, Dunia ^a,b  

^a STANFORD UNIVERSITY   (United States)

^b UNIVERSIDAD PABLO DE OLAVIDE   (Spain)

^c NONE

Author keywords

homophily; Keywords diffusion; segregation; social networks

Indexed keywords

EID: 84887619076     PISSN: 20501242     EISSN: 20501250     Source Type: Journal    
DOI: 10.1017/nws.2012.7     Document Type: Article

References (38)

1. For background on homophily and some of its consequences, see McPherson et al. (2001) and Jackson (2008), and for some background on how it can affect the empirics of diffusion processes see Aral et al. (2009).
2. For background on diffusion in networks, see Newman (2002), Jackson and Yariv (2005 2007 2011), Lopez-Pintado (2006, 2008, 2012), Jackson and Rogers (2007), among others.
3. Various media, including the internet, enable individuals to observe and follow others without reciprocation.
4. This would usually require a directed meeting process, since the agents of type j would have equal probabilities of being met, but would have different numbers of meetings, d that they undertook themselves.
5. The t proxies for the state of the system at time t, so captures the various infection rates (pjj,{t))j£ of those who i will be expected to meet. 9 One could further approximate the system by evaluating /and g at the expected values rather than taking their expectations so using fid(dpi(t)) and gi Ci (dpi(t)) in place of Equations (2). These are the same when /and g are linear in a (e.g., for the SIS model and imitation diffusion processes), but otherwise differ.
6. In the case of best-response dynamics, steady states of the system correspond to Nash equilibria of the game where players condition behavior on both their type and degree and those whom they meet from the population in each period. In other cases, as for instance the SIS model, individuals display recurrent phases of infection and susceptibility, but the system has a constant average infection rate. 11 The question of moving away from the population all being infected is analogous, swapping notation between 0 and1throughout the model, and noting the reversed roles of /and g and given appropriate variations of A1-A4.
7. The monotonicity of the dynamics can be shown by checking that Hi(p,., p n) is non-decreasing in pj for each i and j. First note that fjj(a) is non-decreasing in a (A2) and gij{a) is non-increasing in a (A4). Then from (2), the rate l (t) is non-decreasing, and rate lj ~°(t) is non-increasing, in pj, since these are expectations over the corresponding non-decreasing and non-increasing fj and gj functions, respectively, taken under binomial distributions with parameters pj. The monotonicity of H then follows from the definition of Hj, since the last fraction is then non-decreasing in pj given that rate°j~ l (t) is non-decreasing and ratelif(t) is non-increasing.
8. If h(b,.,£) > (e,., e), then since H is non-decreasing and maps the compact space [s, l] m into itself, there exists a (nonempty) complete lattice of fixed points in the set [e, l] m by Tarski’s theorem. We establish that the failure of stability of 0 implies that H(s,., s) > (g,., s) for some s > 0 via Lemma 1.
9. One might also wonder whether there are positive steady states that are stable themselves. That can depend on properties of H. For a more detailed discussion of that issue, sne Jackson (2008 Chapter 9, Section 7). Even if there are no other stable steady states, the process would still be expected to move throughout the positive region. Thus, if 0 is not a stable steady state, then regardless of the stability of positive steady states, the process would live largely in the positive region, given any occasional perturbations. 15 Also note that (1–Pj fd (t)) and (1–pj(t)) d ~ l are both near 1 in small neighborhoods of-fi p = 0. To be careful, we assume that there is an upper bound on degree which is not automatically implied given our continuum of agents setting, but would be implied in any finite setting.
10. This result is partly an artifact of the continuous model approximation. For an analysis of the importance of the specifics of initially activated agents, see Banerjee et al. (2012). 17 Note that the second condition implies that either x > 1 or x2 > 1.
11. This extends a finding obtained by GOmez-Garderies et al. (2008) for the two-type SIS model. 19 This is due to the fact that = 0 in a scale-free network. That presumes an unbounded distribution.Uf If the distribution is truncated at some maximal degree, then this expression will be positive, but can be arbitrarily close to 0 depending on the maximal degree.
12. Again, recall that A is primitive and thus has a strictly positive eigenvector corresponding to its largest eigenvalue.
13. For example, Pastor-Satorras and Vespignani (2000 3001), Jackson and Rogers (2007), Lopez-Pintado (2008), Galeotti and Goyal (2009), and Galeotti et al. (2010).
14. An exception are the studies by Golub and Jackson (2012a, 2012b, 2012c), which also examine the impact of homophily on some (very different) learning and diffusion processes. There are important differences between our approach and theirs. We focus on generalizations of the SIS infection model and network games, whereas Golub and Jackson (2012a) analyze models of diffusion based either on shortest paths communication, random walks, or linear updating processes. Second, Golub and Jackson (2012a) study the convergence time to the steady state, whereas we analyze whether or not there is convergence to a state with a positive fraction of infected agents. Aral, S., Muchnik, L., & Sundararajan, A. (2009). Distinguishing influence-based contagion from homophily-driven diffusion in dynamic networks. Proceedings of the National Academy of Sciences 106 (51), 21544–21549.
15. Bailey, N. T J. (1975). The mathematical theory of infectious diseases and its applications. London: Charles Griffin.
16. Banerjee A. Chandrasekhar A. Duflo E. & Jackson, M. O. (2012). The diffusion of microfinance. Preprint.
17. Currarini S. Jackson M. O. & Pin, P. (2009). An economic model of friendship: Homophily, minorities and segregation. Econometrica 77 (4), 1003–1045.
18. Galeotti A. Goyal S. Jackson M. O. Vega-Redondo F. & Yariv, L. (2010). Network games. The Review of Economic Studticcibbies 77 (1), 218–244.
19. Galeotti A. & Rogers, B. (2012). Strategic immunization and group structure. American Economic Journal: Microeconomics (in press).
20. Golub B. & Jackson, M. O. (2012a). How homophily affects the speed of learning and best response dynamics. Quarterly Journal of Economics 127 (3), 1287–1338.
21. Golub B. & Jackson, M. O. (2012b). Does homophily predict consensus times? Testing a model of network structure via a dynamic process. Review of Network Economics 11 (3), Article 9.
22. Golub B. & Jackson, M. O. (2012c). Network structure and the speed of learning: Measuring homophily based on its consequences. Annals of Economics and Statistics 107/108 33–50.
23. Gomez-Gardenes J. Latora V. Moreno Y. & Profumo, E. (2008). Spreading of sexually transmitted diseases in heterosexual populations. Proceedings of the National Academy of Sciences 105 (5), 1399–1404.
24. Jackson, M. O (2008). Social and economic networks. New Jersey: Princeton University Press.
25. Jackson, M. O., & Rogers, B. W. (2007). Relating network structure to diffusion properties through stochastic dominance. The B.E. Press Journal of Theoretical Economics 7 (1) (Advances) 1–13.
26. Jackson, M. O., & Yariv, L. (2005). Diffusion on social networks. Economie Publique 16 (1), 69–82.
27. Jackson, M. O., & Yariv, L. (2007). Diffusion of behavior and equilibrium properties in network Games. The American Economic Review (Papers and Proceedings), 97 (2), 92–98. Jackson, M. O., & Yariv, L. (2011). Diffusion, strategic interaction, euand social structure. In J. Benhabib, A. Bisin, & M. Jackson (Eds.),Handbook of social economics (pp. 33–50). The Netherlands: North Holland.
28. Jackson, M. O., & Zenou, Y. (2012). Games on networks. In Handbook of Game Theory (in press). Retrieved from http://papers. ssrn.com/abstract=213 6179.
29. Lopez-Pintado D. (2006). Contagion and coordination in random networks. International Journal of Game Theory 34 371–381.
30. Lopez-Pintado D. (2008). Diffusion in complex social networks. Games and Economic Behavior 62 573–590.
31. Lopez-Pintado D. (2012). Influence networks. Games and Economic Behavior 75 776–787.
32. McPherson M. Smith-Lovin L. & Cook, J. M. (2001). Birds of a feather: Homophily in social networks. Annual Review of Sociology 27 415–444.
33. Newman, M. E J. (2002). The spread of epidemic disease on networks. Physical Review E 66 01 6128.
34. Pastor-Satorras R. & Vespignani, A. (2000). Epidemic spreading in scale-free networks. Physical Review Letters 86 (14), 3200–3203.
35. Pastor-Satorras R. & Vespignani, A. (2001). Epidemic dynamics and endemic states in complex networks. Physical Review E 63 06 6117.
36. Reluga, T. C (2009). An SIS epidemiology game with two subpopulations. Journal of Biological Dynamics, 3 (5), 515–531. 03 In fact, with two types A is a positive matrix since 0 < n < 1.
37. Note that since A is primitive, its largest eigenvalue is real and positive.
38. The relative positions of the curves are easily checked, and note the plus and minus signs that indicate whether one is above or below 1 for the corresponding colored expression.