Skip to main content
SpringerLink
Log in

Opinions, influence, and zealotry: a computational study on stubbornness

  • Published:
Computational and Mathematical Organization Theory Aims and scope Submit manuscript

Abstract

We present a simple, efficient, and predictive model for opinion dynamics with zealots. Our model captures curvature-driven dynamics (e.g., clear, smooth boundaries separating domains whose curvature decreases over time) through a simple, individual rule, providing a method for rapidly testing basic hypotheses about innovation diffusion, opinion dynamics, and related phenomena. Our model belongs to a class of models called dimer automata, which are asynchronous, graph-based (i.e., non-uniform lattice) variants of cellular automata. Individuals in the model update their states via a dyadic update rule; population opinion dynamics emerge from these pairwise interactions. Zealots are stubborn individuals whose opinion is not susceptible to influence by others. We observe experimentally that a system without zealots usually converges to the majority opinion, but a relatively small number of zealots can sway the opinion of the whole population. The influence of zealots can be further increased by placing zealots at more effective locations within the network. These locations can be determined by rankings from standard social network analysis metrics, or by using a greedy algorithm for influence maximization. We apply the influence maximization technique to a politically polarized social network to explore opinion dynamics in a real-world network and to gain insight about influence and political entrenchment through the zealot model’s ability to sway the entire network to one side or the other.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. In our experience, a rule of thumb for designing individual models is to avoid, as much as possible, rules that move the system towards the desired global outcome. Doing so can produce interesting and sometimes counter intuitive relationships between the local interactions and the global phenomenon.

  2. Thus, the term “dimer” in dimer automaton refers to the manner in which the rule is applied to pairs of adjacent cells.

  3. How long should the simulation be run? We find that performing \(O(|V|^{1.5})\) updates is a reasonable number of steps for the experiments in this paper.

  4. A time epoch consists of \(O(|V|)\) edge updates.

  5. For this experiment the zealot density is relative to the number of unconverted nodes in the initial configuration. When there are l left-leaning vertices and \(r\) right-leaning vertices, if we are trying to convert left-leaning to right-leaning, and we use \(z_r\) right-leaning zealots, then the zealot density is \(z_r/l\) instead of \(z_r/(r+l)\).

References

  • Arendt D (2013) A concise model for innovation diffusion combining curvature-based oinion dynamics and zealotry. In: Behavior representation in modeling and simulation 2013: BRiMS, BRIMS Society, Ottawa, pp. 64–71

  • Arendt D, Cao Y (2012a) Effective GPU acceleration of large scale, asynchronous simulations on graphs. Adv Complex Syst 15(8):253–254

    Google Scholar 

  • Arendt D, Cao Y (2012b) Evolutionary motifs for the automated discovery of self-organizing dimer automata. Adv Complex Syst 15(7):1–18

    Google Scholar 

  • Arendt D, Cao Y (2012c) GPU acceleration of many independent mid-sized simulations on graphs. In: Proceeding of the 4th cellular automata, theory and applications workshop (*A-CSC’12)

  • Castellano C, Fortunato S, Loreto V (2009) Statistical physics of social dynamics. Rev Mod Phys 81(2):591–646

    Article  Google Scholar 

  • Castelló X, Eguíluz V, San Miguel M (2006) Ordering dynamics with two non-excluding options: bilingualism in language competition. New J Phys 8(12):308

    Article  Google Scholar 

  • Centola D, Gonzalez-Avella JC, Eguiluz VM, San Miguel M (2007) Homophily, cultural drift and the co-evolution of cultural groups. J Confl Resolut 51(6):905–929

    Article  Google Scholar 

  • Chen W, Wang Y, Yang S (2009) Efficient influence maximization in social networks. In: Proceedings of the 15th ACM SIGKDD international conference on knowledge discovery and data mining, ACM, New York, pp. 199–208

  • Conover M, Ratkiewicz J, Francisco M, Gonçalves B, Menczer F, Flammini A (2011) Political polarization on twitter. In: International conference on weblogs and social media

  • Domingos P, Richardson M (2001) Mining the network value of customers. In: Proceedings of the seventh ACM SIGKDD international conference on knowledge discovery and data mining, ACM, New York, pp. 57–66

  • Erdős P, Rényi A (1959) On random graphs. Publ Math Debr 6:290–297

    Google Scholar 

  • Fortunato S (2012) Community detection in graphs. Phys Rep 486(3):75–174

    Google Scholar 

  • Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81(25):2340–2361

    Article  Google Scholar 

  • Goldenberg J, Libai B, Muller E (2001) Talk of the network: a complex systems look at the underlying process of word-of-mouth. Mark lett 12(3):211–223

    Article  Google Scholar 

  • Goyal A, Lu W, Lakshmanan LV (2011) Celf++: optimizing the greedy algorithm for influence maximization in social networks. In: Proceedings of the 20th ACM SIGKDD international conference on knowledge discovery and data mining, ACM, New York, pp. 47–48

  • Granovetter M (1978) Threshold models of collective behavior. Am J Sociol 83(6):1420–1443

    Article  Google Scholar 

  • Holley R, Liggett T (1975) Ergodic theorems for weakly interacting infinite systems and the voter model. Ann Probab 3(4):643–663

    Article  Google Scholar 

  • Holme P, Kim BJ (2002) Growing scale-free networks with tunable clustering. Phys Rev E 65(2):026107

    Article  Google Scholar 

  • Ingerson TE, Buvel RL (1984) Structure in asynchronous cellular automata. Phys D 10(1–2):59–68

    Article  Google Scholar 

  • Kauffman SA (1993) The origins of order: self organization and selection in evolution. Oxford University Press, New York

    Google Scholar 

  • Kempe D, Kleinberg J, Tardos É (2003) Maximizing the spread of influence through a social network. In: Proceedings of the ninth ACM SIGKDD international conference on knowledge discovery and data mining, ACM, New York, pp. 137–146

  • Leskovec J, Huttenlocher D, KLeinberg J (2010a) Predicting positive and negative links in online social networks. In: Proceedings of the 19th international conference on world wide web, ACM, New York, pp. 641–650

  • Leskovec J, Huttenlocher D, Kleinberg J (2010b) Signed networks in social media. In: Proceedings of the SIGCHI conference on human factors in computing systems, ACM, New York, pp. 1361–1370

  • Leskovec J, Kleinberg J, Faloutsos C (2007) Graph evolution: densification and shrinking diameters. ACM Trans Knowl Discov Data 1(1):2

    Article  Google Scholar 

  • Leskovec J, Krause A, Guestrin C, Faloutos C, VanBriesen J, Glance N (2007) Cost-effective outbreak detection in networks. In: Proceedings of the 13th ACM SIGKDD international conference on knowledge discovery and data mining, ACM, New York, pp. 420–429

  • Liggett TM (2010) Stochastic models for large interacting systems and related correlation inequalities. Proc Natl Acad Sci Un States Am 107(38):16413–16419

    Article  Google Scholar 

  • Mahajan V, Peterson R (1985) Models for innovation diffusion, vol 48. Sage Publications, London

    Google Scholar 

  • Mobilia M (2003) Does a single zealot affect an infinite group of voters? Phys Rev Lett 91(2):028701

    Article  Google Scholar 

  • Mobilia M, Georgiev I (2005) Voting and catalytic processes with inhomogeneities. Phys Rev E 71(4):046102

    Article  Google Scholar 

  • Mortveit HS, Reidys CM (2001) Discrete, sequential dynamical systems. Discret Math 226(1–3):281–295

    Article  Google Scholar 

  • Nan N, Zmud R, Yetgin E (2014) A complex adaptive systems perspective of innovation diffusion: an integrated theory and validated virtual laboratory. Comput Math Organ Theory 20:52–88

    Article  Google Scholar 

  • Rouchier J, Tubaro P, Emery C (2013) Opinion transmission in organizations: an agent-based modeling approach. Comput Math Organ Theory 20:1–26

    Google Scholar 

  • Schelling TC (2006) Micromotives and macrobehavior. WW Norton & Company, New York

    Google Scholar 

  • Schönfisch B, Hadeler KP (1996) Dimer automata and cellular automata. Phys D 94(4):188–204

    Article  Google Scholar 

  • Schönfisch B, de Roos A (1999) Synchronous and asynchronous updating in cellular automata. Biosystems 51(3):123–143

    Article  Google Scholar 

  • Sen A, Davulcu H (2010) Influence propagation in adversarial social network-impact of space and time. In: 2010 Specialist meeting-spatio-temporal constraints on social networks

  • Shirazipourazad S, Bogard B, Vachhani H, Sen A, Horn P (2012) Influence propagation in adversarial setting: how to defeat competition with least amount of investment. In: Proceedings of the 21st annual international conference on information and knowledge management CIKM ’12, pp. 585–594. ACM, Maui, HI

  • Turnbull PW, Meenaghan A (1980) Diffusion of innovation and opinion leadership. Eur J Mark 14:3–33

    Article  Google Scholar 

  • Vazquez F, Krapivsky P, Redner S (2003) Constrained opinion dynamics: freezing and slow evolution. J Phys A 36(3):L61–L68

    Article  Google Scholar 

  • Wang Y, Cong G, Song G, Xie K (2010) Community-based greedy algorithm for mining top-k influential nodes in mobile social networks. In: Proceedings of the 16th ACM SIGKDD international conference on knowledge discovery and data mining, ACM, New York, pp. 1039–1048

  • Wasserman S, Faust K (1994) Social network analysis: methods and applications. Cambridge University Press, Mexico

    Book  Google Scholar 

  • Watts D, Strogatz S (1998) Collective dynamics of ‘small-world’ networks. Nature 393(6684):440–442

    Article  Google Scholar 

Download references

Acknowledgments

Distribution A: Approved for public release; distribution unlimited. 88ABW cleared 11/08/2013; 88 ABW-2013-4691. This work was supported by AFOSR LRIR 12RH12COR to L.M.B. This research was performed while D.L.A. held a National Research Council Research Associateship Award at the Air Force Research Laboratory.

Author information

Authors and Affiliations

  1. Pacific Northwest National Laboratory, 902 Battelle Boulevard, Richland, WA, USA

    Dustin L. Arendt

  2. Air Force Research Laboratory, 2255 H St., Wright-Patterson AFB, OH, USA

    Leslie M. Blaha

Authors
  1. Dustin L. Arendt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leslie M. Blaha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dustin L. Arendt.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arendt, D.L., Blaha, L.M. Opinions, influence, and zealotry: a computational study on stubbornness. Comput Math Organ Theory 21, 184–209 (2015). https://doi.org/10.1007/s10588-015-9181-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10588-015-9181-1

Keywords

Search

Navigation