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A DISCRETE-TIME POPULATION DYNAMICS MODEL FOR THE INFORMATION SPREAD UNDER THE EFFECT OF SOCIAL RESPONSE

https://orcid.org/0000-0002-0831-2055

Department of Mathematical and Information Sciences, Graduate School of Information Sciences, Tohoku University, Aramaki-Aza-Aoba 6-3-09, Aoba-ku, Sendai, Miyagi 980-8579, Japan

E-mail Address: seno@math.is.tohoku.ac.jp

Corresponding author.

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REINA UCHIOKE

Department of Mathematical and Information Sciences, Graduate School of Information Sciences, Tohoku University, Aramaki-Aza-Aoba 6-3-09, Aoba-ku, Sendai, Miyagi 980-8579, Japan

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EMMANUEL JESUYON DANSU

https://orcid.org/0000-0002-7831-390X

Department of Mathematical Sciences, School of Physical Sciences, Federal University of Technology Akure, PMB 704, Akure, Ondo State, Nigeria

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https://doi.org/10.1142/S0218339024400072Cited by:0 (Source: Crossref)

This article is part of the issue:

Special Issue on Advances in Mathematical Ecology and Epidemiology at MPDEE 2022
Guest Editors: Ezio Venturino (Université de Franche-Comté, France) and Sergei Petrovskii (University of Leicester, UK)

Abstract

In this paper, we construct and analyze a mathematically reasonable and simplest population dynamics model based on Mark Granovetter’s idea for the spread of a matter (rumor, innovation, psychological state, etc.) in a population. The model is described by a one-dimensional difference equation. Individual threshold values with respect to the decision-making on the acceptance of a spreading matter are distributed throughout the population ranging from low (easily accepts it) to high (hardly accepts). Mathematical analysis on our model with some general threshold distributions (uniform; monotonically decreasing/increasing; unimodal) shows that a critical value necessarily exists for the initial frequency of acceptors. Only when the initial frequency of acceptors is beyond the critical, the matter eventually spreads over the population. Further, we give the mathematical results on how the equilibrium acceptor frequency depends on the nature of threshold distribution.

This article is part of the “Special Issue on Advances in Mathematical Ecology and Epidemiology at MPDEE 2022”, edited by Ezio Venturino (Université de Franche-Comté, France) and Sergei Petrovskii (University of Leicester, UK).

Keywords:

References

1. Granovetter M , Threshold models of collective behavior, Amer J Soc 83(6) :1420–1443, 1978, https://doi.org/10.1086/226707. Crossref, Web of Science, Google Scholar
2. Dietz K , Epidemics and rumours: A survey, J Roy Stat Soc: Ser A (General) 130(4) :505–528, 1967, https://doi.org/10.2307/2982521. Crossref, Google Scholar
3. Granovetter MS , The strength of weak ties, Amer J Soc 78(6) :1360–1380, 1973. Crossref, Web of Science, Google Scholar
4. Granovetter M , The strength of weak ties: A network theory revisited, Soc Theory 1 :201–233, 1983, https://doi.org/10.2307/202051. Crossref, Google Scholar
5. Granovetter M, Soong R , Threshold models of diffusion and collective behavior, J Math Sociol 9(3) :165–179, 1983, https://doi.org/10.1080/0022250X.1983.9989941. Crossref, Web of Science, Google Scholar
6. Leibenstein H , Bandwagon, snob, and veblen effects in the theory of consumers’ demand, Quart J Econ 64(2) :183–207, 1950, https://doi.org/10.2307/1882692. Crossref, Web of Science, Google Scholar
7. Nadeau R, Cloutier E, Guay JH , New evidence about the existence of a bandwagon effect in the opinion formation process, Int Politic Sci Rev 14(2) :203–13, 1993, https://doi.org/10.1177/0192512193014002. Crossref, Web of Science, Google Scholar
8. Corneo G, Jeanne O , Snobs, bandwagons, and the origin of social customs in consumer behavior, J Econ Behav Organ 32(3) :333–47, 1997, https://doi.org/10.1016/S0167-2681(96)00024-8. Crossref, Web of Science, Google Scholar
9. Watts DJ, Dodds PS , Threshold models of social influence, Bearman P, Hedström P (eds.), The Oxford Handbook of Analytical Sociology, Chap 20, pp. 475–497, Oxford University Press, New York, 2009, https://doi.org/10.1093/oxfordhb/9780199215362.013.20. Google Scholar
10. Granovetter M, Soong R , Threshold models of interpersonal effects in consumer demand, J Econ Behav Organ 7(1) :83–99, 1986. Crossref, Web of Science, Google Scholar
11. Kaempfer WH, Lowenberg AD , Using threshold models to explain international relations, Publ Choice 73(4) :419–443, 1992, https://doi.org/10.1007/BF01789560. Crossref, Web of Science, Google Scholar
12. Valente TW , Social network thresholds in the diffusion of innovations, Soc Netw 18(1) :69–89, 1996, https://doi.org/10.1016/0378-8733(95)00256-1. Crossref, Web of Science, Google Scholar
13. Delre SA, Jager W, Janssen MA , Diffusion dynamics in small-world networks with heterogeneous consumers, Comput Math Organ Theory 13(2) :185–202, 2007, https://doi.org/10.1007/s10588-006-9007-2. Crossref, Google Scholar
14. Bischi GI, Merlone U , Global dynamics in binary choice models with social influence, J Math Sociol 33(4) :277–302, 2009, https://doi.org/10.1080/00222500902979963. Crossref, Web of Science, Google Scholar
15. Lehmann S, Ahn YY , eds., Complex Spreading Phenomena in Social Systems: Influence and Contagion in Real-World Social Networks, Springer Cham, Switzerland, 2018, https://doi.org/10.1007/978-3-319-77332-2. Crossref, Google Scholar
16. Wiedermann M, Smith EK, Heitzig J, Donges JF , A network-based microfoundation of Granovetter’s threshold model for social tipping, Sci Rep 10(1) :1–10, 2020, https://doi.org/10.1038/s41598-020-67102-6. Crossref, Web of Science, Google Scholar
17. Jawad M , The dynamic threshold model of bandwagon innovations: Role of organizational attention and legitimacy, Organ Psychol Rev 12(2) :162–180, 2022, https://doi.org/10.1177/20413866211054201. Web of Science, Google Scholar
18. Goldstone RL, Janssen MA , Computational models of collective behavior, Trends Cognit Sci 9(9) :424–430, 2005, https://doi.org/10.1016/j.tics.2005.07.009. Crossref, Web of Science, Google Scholar
19. Goldstone RL, Gureckis TM , Collective behavior, Top Cognit Sci 1(3) :412–438, 2009, https://doi.org/10.1111/j.1756-8765.2009.01038.x. Crossref, Web of Science, Google Scholar
20. Belolipetskii AA, Kozitsin IV , Dynamic variant of mathematical model of collective behavior, J Comput Syst Sci Int 56 :385–396, 2017, https://doi.org/10.1134/S1064230717030054. Crossref, Web of Science, Google Scholar
21. Naldi G, Pareschi L, Toscani G , eds., Mathematical Modeling of Collective Behavior in Socio-economic and Life Sciences, Springer Science & Business Media, LLC, 2010, https://doi.org/10.1007/978-0-8176-4946-3. Crossref, Google Scholar
22. Zhang S, Ringh A, Hu X, Karlsson J , Modeling collective behaviors: A moment-based approach, IEEE Trans Autom Control 66(1) :33–48, 2020, https://doi.org/10.1109/TAC.2020.2976315. Crossref, Google Scholar
23. Ouellette NT, Gordon DM , Goals and limitations of modeling collective behavior in biological systems, Front Phys 9 :687823, 2021, https://doi.org/10.3389/fphy.2021.687823. Crossref, Google Scholar
24. Loy N, Raviola M, Tosin A , Opinion polarization in social networks, Philos Trans Roy Soc A 380(2224) :20210158, 2022, https://doi.org/10.1098/rsta.2021.0158. Crossref, Google Scholar
25. Li Z, Tang X , Collective threshold model based on utility and psychological theories, Int J Knowl Syst Sci 4(4) :55–63, 2013, https://doi.org/10.4018/ijkss.2013100105. Crossref, Google Scholar
26. Garulli A, Giannitrapani A, Valentini M , Analysis of threshold models for collective actions in social networks, 2015 European Control Conference (ECC), pp. 211–216, IEEE, 2015, https://doi.org/10.1109/ECC.2015.7330547. Crossref, Google Scholar
27. Gavrilets S, Richerson PJ , Collective action and the evolution of social norm internalization, Proc Natl Acad Sci 114(23) :6068–6073, 2017, https://doi.org/10.1073/pnas.1703857114. Crossref, Web of Science, Google Scholar
28. Centola D , How Behavior Spreads: The Science of Complex Contagions, Princeton University Press, Princeton, 2018, https://doi.org/10.23943/9781400890095. Google Scholar
29. Luo H, Ren Y , Analysis of COVID-19 collective irrationalities based on epidemic psychology, Front Psychol 13 :825452, 2022, https://doi.org/10.3389/fpsyg.2022.825452. Crossref, Web of Science, Google Scholar
30. Macy MW, Evtushenko A , Threshold models of collective behavior II: The predictability paradox and spontaneous instigation, Sociol Sci 7(26) :628–48, 2020, https://doi.org/10.15195/v7.a26. Crossref, Google Scholar
31. Palomo-Briones GA, Siller M, Grignard A , An agent-based model of the dual causality between individual and collective behaviors in an epidemic, Comput Biol Med 141 :104995, 2022, https://doi.org/10.1016/j.compbiomed.2021.104995. Crossref, Web of Science, Google Scholar
32. Castellano C, Fortunato S, Loreto V , Statistical physics of social dynamics, Rev Mod Phys 81(2) :591, 2009, https://doi.org/10.1103/RevModPhys.81.591. Crossref, Web of Science, Google Scholar
33. Whitney D , Cascades of rumors and information in highly connected networks with thresholds, Second Int Symp Engineering Systems, MIT, Cambridge, Massachusetts, 2009. Google Scholar
34. Akhmetzhanov AR, Worden L, Dushoff J , Effects of mixing in threshold models of social behavior, Phys Rev E 88(1) :012816, 2013, https://doi.org/10.1103/PhysRevE.88.012816. Crossref, Web of Science, Google Scholar
35. Vallacher RR, Nowak AE (eds.), Dynamical Systems in Social Psychology, Academic Press, San Diego, 1994. Google Scholar
36. Vallacher RR, Read SJ, Nowak A , The dynamical perspective in personality and social psychology, Personal Soc Psychol Rev 6(4) :264–273, 2002, https://doi.org/10.1207/S15327957PSPR0604_01. Crossref, Web of Science, Google Scholar
37. Pathak N, Banerjee A, Srivastava J , A generalized linear threshold model for multiple cascades, 2010 IEEE Int Conf Data Mining, pp. 965–970, IEEE, 2010, https://doi.org/10.1109/ICDM.2010.153. Crossref, Google Scholar
38. Shrestha M, Moore C , Message-passing approach for threshold models of behavior in networks, Phys Rev E 89(2) :022805, 2014, https://doi.org/10.1103/PhysRevE.89.022805. Crossref, Web of Science, Google Scholar
39. Zeppini P, Frenken K, Kupers R , Thresholds models of technological transitions, Environ Innov Soc Trans 11 :54–70, 2014, https://doi.org/10.1016/j.eist.2013.10.002. Crossref, Google Scholar
40. Gao J, Ghasemiesfeh G, Schoenebeck G, Yu FY , General threshold model for social cascades: Analysis and simulations, Proc 2016 ACM Conf Economics and Computation, pp. 617–634, ACM, 2016, https://doi.org/10.1145/2940716.2940778. Crossref, Google Scholar
41. Rossi WS, Como G, Fagnani F , Threshold models of cascades in large-scale networks, IEEE Trans Netw Sci Eng 6(2) :158–172, 2017, https://doi.org/10.1109/TNSE.2017.2777941. Crossref, Google Scholar
42. Marshall JA, Reina A, Bose T , Multiscale modelling tool: Mathematical modelling of collective behaviour without the maths, PLoS One 14(9) :e0222906, 2019, https://doi.org/10.1371/journal.pone.0222906. Crossref, Web of Science, Google Scholar
43. Nowak A, Vallacher RR , Nonlinear societal change: The perspective of dynamical systems, British J Soc Psychol 58(1) :105–128, 2019, https://doi.org/10.1111/bjso.12271. Crossref, Web of Science, Google Scholar
44. Paul HL, Philips AQ , What goes up must come down: Theory and model specification of threshold dynamics, Soc Sci Quart 103(5) :1273–1289, 2020, https://doi.org/10.1111/ssqu.13191. Crossref, Google Scholar
45. Yamaguti M , Discrete models in social sciences, Comput Math Appl 28(10–12) :263–267, 1994, https://doi.org/10.1016/0898-1221(94)00196-0. Crossref, Web of Science, Google Scholar
46. Watts DJ , A simple model of global cascades on random networks, Proc Natl Acad Sci 99(9) :5766–5771, 2002, https://doi.org/10.1073/pnas.082090499. Crossref, Web of Science, Google Scholar
47. Jalili M, Perc M , Information cascades in complex networks, J Compl Netw 5(5) :665–693, 2017, https://doi.org/10.1093/comnet/cnx019. Web of Science, Google Scholar
48. Ran Y, Deng X, Wang X, Jia T , A generalized linear threshold model for an improved description of the spreading dynamics, Chaos: Interdiscip J Nonlinear Sci 30(8) :083127, 2020, https://doi.org/10.1063/5.0011658. Crossref, Web of Science, Google Scholar
49. Noorazar H , Recent advances in opinion propagation dynamics: A 2020 survey, Eur. Phys. J. Plus 135 :1–20, 2020, https://doi.org/10.1140/epjp/s13360-020-00541-2. Crossref, Web of Science, Google Scholar
50. Bernardo C, Altafini C, Proskurnikov A, Vasca F , Bounded confidence opinion dynamics: A survey, Automatica 159 :111302, 2024, https://doi.org/10.1016/j.automatica.2023.111302. Crossref, Web of Science, Google Scholar
51. Manrubia S, Zanette DH , Individual risk-aversion responses tune epidemics to critical transmissibility ( $R = 1$ ), R Soc Open Sci 9 :211667, 2022, https://doi.org/10.1098/rsos.211667. Crossref, Web of Science, Google Scholar
52. Qiu Z, Espinoza B, Vasconcelos VV, Chen C, Constantino SM, Crabtree SA, Yang L, Vullikanti A, Chen J, Weibull J, Basu K , Understanding the coevolution of mask wearing and epidemics: A network perspective, Proc Natl Acad Sci 119(26) :e2123355119, 2022, https://doi.org/10.1073/pnas.2123355119. Crossref, Web of Science, Google Scholar
53. Della Marca R, Loy N, Menale M , Intransigent vs. volatile opinions in a kinetic epidemic model with imitation game dynamics, Math Med Biol: J IMA 40(2) :111–140, 2023, https://doi.org/10.1093/imammb/dqac018. Crossref, Web of Science, Google Scholar
54. Ventura PC, Aleta A, Rodrigues FA, Moreno Y , Epidemic spreading in populations of mobile agents with adaptive behavioral response, Chaos Solitons Fractals 156 :111849, 2022, https://doi.org/10.1016/j.chaos.2022.111849. Crossref, Web of Science, Google Scholar
55. Dansu EJ, Seno H , A mathematical model for the dynamics of information spread under the effect of social response, Interdiscip Inf Sci 28(1) :75–93, 2022, https://doi.org/10.4036/iis.2022.R.03. Google Scholar
56. Seno H , A Primer on Population Dynamics Modeling: Basic Ideas for Mathematical Formulation, Springer, Singapore, 2022, https://doi.org/10.1007/978-981-19-6016-1. Crossref, Google Scholar
57. Kaplan D, Glass L , Understanding Nonlinear Dynamics, Springer Science & Business Media, New York, 1997, https://doi.org/10.1007/978-1-4612-0823-5. Google Scholar
58. Robinson RC , An Introduction to Dynamical Systems: Continuous and Discrete, 2nd edn., American Mathematical Society, Providence, Rhode Island, 2012. Google Scholar
59. Nyborg K, Anderies JM, Dannenberg A, Lindahl T, Schill C, Schlüter M, Adger WN, Arrow KJ, Barrett S, Carpenter S, Chapin III FS , Social norms as solutions, Science 354(6308) :42–43, 2020, https://doi.org/10.1126/science.aaf8317. Crossref, Google Scholar
60. Kitabayashi K , The Otaku group from a business perspective: Revaluation of enthusiastic consumers, NRI Papers 84(1) :1–10, 2004. Google Scholar
61. Chang CC , What Otaku consumers care about: The factors influential to online purchase intention, AIP Conf Proc 1558(1) :450–454, 2013, https://doi.org/10.1063/1.4825523. Crossref, Google Scholar
62. Lacasa P, Zaballos LM, de la Fuente Prieto J , Fandom, music and personal relationships through media: how teenagers use social networks, IASPM Journal 6(1): 44–67, 2016, https://doi.org/10.5429/2079-3871(2016)v6i1.4en. Crossref, Google Scholar