1 Introduction
The key aspect of marketing is to influence opinions. By conceiving the political (in a democracy at least), as being yet another application of marketing, politics become little more than campaigning for votes, supporters and financial contributions.
When looking at influencing opinions, two use cases spring to mind: (1) counteracting negative opinions online, and (2) to stop radical political views from spreading to safeguard democracy.
With regard to the first use case, consider a Facebook page of a company producing goods. A dissatisfied customer posts harsh criticism to it. A common enough reaction is the threat of legal action against that customer. If other users find this threat to be too harsh, they will voice their opinions themselves, leading to a phenomenon similar in nature to a more informal flamewar but with much more dire consequences for the involved stakeholders. Clearly, a more measured response in these cases would help to stave off disaster. As Lin et al. (2010) point out, this issue is becoming ever more pressing as the availability of online opinion outlets increases.
In a less mundane example, counteracting radical opinions is equally important to keep any democratic system well and functioning. Here, radical opinions from the left and the right end of the political spectrum threaten the very foundations of democracy. If these opinions were to spread, the political system would crumble and fall. But how to counteract these opinions? Clearly, using violent police crowd control tactics to break up a group of peaceful protesters, will overshoot and is likely to create sympathy for the group. Yet not using police against militants will also lead to an undesired outcome.
Whenever treating political issues, research tends to become political itself. To abstract away from the political, it is useful to consider the spreading of opinions as being similar to the spreading of a disease. This approach to modeling opinion dynamics has a sound basis in the literature. For a direct application, see, for instance, Adar and Adamic (2005); Sobkowicz et al. (2012); Lynch (1998), or see Moore et al. (2011) for feeding back into health sciences. While diseases all have their medical idiosyncrasies, for the sake of studying the spreading of a disease, any disease will do. In order to be able to also abstract away from medical realities, let there be a fictional disease turning otherwise healthy humans into zombies. This somewhat absurd approach to science has, however, been used quite successfully in the past (Calderhead et al., 2010; Munz et al., 2009)
. Their method is somewhat different, as it deals with modeling the characteristics of infectious diseases with ordinary differential equations. While this has its merits, differential equations describing highly volatile, dynamic systems based on stochastic selection quickly become unwieldy. Rather, we follow a trajectory similar to
Crossley and Amos (2011), who modeled the spread of zombies in a society using agent-based simulation.We use this zombie approach with some modifications to model how a disease (i.e. radical opinions) spreads in society. The outbreak in our model, however, can be counteracted using a selection of treatments. Finding the right combinations of treatments is though very far from being trivial. To this end, we offer two fine-tuned evolutionary algorithms that will seek out the optimal combination of treatments during the course of the outbreak.
The remainder of this paper is organized as follows. In the next section, the disease model is introduced. After that we discuss the evolutionary algorithms used for solving the curing puzzle in Section 3. The algorithm is tested under varying underlying assumptions. These results are then described in Section 4 and discussed there after. There we follow along the lines of Cruz et al. (2011) and present appreciations of the algorithms’ accuracy, stability and to a lesser extend reactivity. Finally, we offer some conclusions.
2 Simulating opinion propagation
As described above, when modeling the propagation of opinions, it makes sense to assume a (fictional) disease. In the following we will first describe the specific notions of opinions that need to be considered when modeling them as a disease in a general way. We will pay special attention on how and why opinion propagation differs from classical epidemiological models. After that the employed disease model is described in detail.
2.1 Opinions as disease
When modeling opinions as diseases, some notions apply that set these models apart from classical epidemiological models on disease spreading. Canonically (Teng, 1985; Sokolowski and Banks, 2011; Brauer and Castillo-Chavez, 2011), epidemiology employs models that identifies distinct classes of healthy individuals that are susceptible to a disease (S), the infected carriers of the disease (I) and individuals that are immune to the disease either because of medical treatment, seclusion or death (R). These models then assume that any individual can potentially traverse these classes in turn:
(1) |
Dependent on the classes included in the actual model, the model is named in turn: the complete model is named SIRS, a model allowing only for the move from healthy to infected with no healing or curing is called SI.
Opinion propagation in the political process of a democracy sets limits on the usability of these classes. A person that does not hold an opinion can be classified as a member of class S, while convinced advocates of that opinion will be in class I. Class R, those that are removed from the population to halt the spreading of a disease or particular opinion, has no real equivalent in any democratic system. The locking-up of individuals holding any particular, perhaps even iconoclastic opinion, is in essence incompatible with our understanding of liberal democracy. Likewise, it is unlikely that any political opinion will cause its holder’s death. Therefore, opinion propagation models for democracies must do without the R class and are limited to be of type SI or perhaps SIS.
^{1}^{1}1A noteworthy exemption are perhaps the Verbotsgesetze enacted in Austria and Germany under the impression of denazification. Here, indeed, prominent advocates of especially despicable opinions, are being held in prison and thus effectively removed from the population.Another factor that is crucial in the simulation of opinion propagation in human populations is the topology of the underlying network. It has been shown, that human interaction is organized in the fashion of small-world networks (Rodríguez Lucatero et al., 2012; Wessels and Miller, 1999; Eubank et al., 2004; Amaral et al., 2000). These networks govern how individuals are linked to each other. Most prominently, the small world property states that most individuals are connected to most other individuals via only a hand full of other individuals. This requires some individuals to function as hubs, that are connecting many people.
These small world networks are on one hand quite fault tolerant and avoid the breaking down of communication if single individuals are removed. More importantly for the case at hand, however, is that they also facilitate the spreading of disease, once central nodes that connect many individuals are infected.
Even though these models are representative of the way humans interact, they add an additional level of complexity to the modeling of disease and opinion propagation. Therefore, the simplest models of disease propagation focus on a relaxed model that ignores these network properties. Rather, the simpler model assumes a population that is well-mixed. Here, every individual is connected to every other with the same probability. This boils down to a mean-fields model in the terminology of physics’ parlance.
2.2 Disease model
As detailed above and elsewhere (Rahmandad and Sterman, 2008; Patlolla et al., 2006), there are compelling reasons to diverge from canonical epidemiological models when dealing with opinions. We therefore employ a SI-model ignorant to complex network topologies. In our model of opinion propagation, the infected are not being secluded from the population, and every individual has an equal, mean connectedness to every other individual. However, further modifications apply that we will describe in the following.
Here we use the Revil zombie outbreak simulator (Waldhauser, 2013)^{2}^{2}2All computations were done in R (R Core Team, 2012); plots were produced using ggplot2 (Wickham, 2009).. This package allows to model not only the conventional dichotomous disease outcome of healthy versus infected. Rather, it assumes a continuous spectrum of infectiousness. Humans carry parts of the disease within them, at all times. But only once they pass a certain threshold, they will turn into zombies and start infecting others.
This notion of a continuously conceptualized disease, aptly named C-Virus, is quite similar to the way opinions work. Take for example xenophobia. While only very convinced xenophobes will start propagating these views, who can honestly say to not have reservations against the unknown? Or while many cannot agree with every point Marx ever made, who would not cherish the overall improvement of labor conditions? Aspects of even the most radical of views are present in everyone of us, to a varying degree.
In its design, the disease model is quite simple. A society of a given size is initialized to a random state of infection. This is another point of departure from the traditional model that always assumes a single^{3}^{3}3or a small number of patient zero. But who could ever identify e.g. the first xenophobe? In addition, there is a number of zombies, i.e. humans that carry so much of the disease within them, that they can corrupt others. From this initial state forward, interaction between humans and zombies follows a stochastic trajectory. Humans that come in contact with zombies have a certain probability of having their infectiousness increased. Once a human crosses a certain threshold, the disease becomes acute and that human turns into a zombie, lusting to attack others. If unchecked, the disease will continue its spread until all humans have been turned into zombies.
This has serious ramifications for the disease model. While conventional models assume distinct classes of healthy humans and infected. In our model, these distinctions blur. In the employed fuzzy model of disease it is difficult to draw a line, as even healthy individuals carry the disease to a certain extent. If considering the threshold described above as the discriminatory boundary, all those not yet carrying the disease to an extent that crosses the threshold, are to be considered healthy; the remainder, i.e. zombie population has to be counted as infected.
While in canonical zombie lore, there is seldom a cure to the outbreak, the C-Virus can be treated. However, there is a caveat. As in real life opinions, treatments are specific to the seriousness of the infection: a mildly infected individual needs different medications than a raging wild zombie. To simplify the model, we assume that there is a limited set of discrete treatments that form a cure that can be administered to the general public. Each treatment is only effective for a certain range of infected. In the cure mixture, we need to specify the dose of each treatment, i.e. the probability of that treatment being used on someone.
Finding a cure, that over many days manages to keep the zombies at bay, proved rather difficult. The following section describes how evolutionary algorithms help in finding that optimal mixture.
3 Searching for a cure
When looking at societies ravaged by our virus, two key indicators can be analyzed. One is the mean infection rate (), which is the average degree of infectedness in the entire society. A society composed entirely of healthy humans would exhibit while in one where humans had died out and all zombies be fully infected, the maximum of would be reached. Any cure for the C-virus is more successful, if the mean infection rate is kept lower than under application of a different cure.
When it is not possible to effectively root out the disease, it might still be possible to slow its progression. The second quantity of interest is thus the last day a human was encountered in the society (). A cure is successful when it manages to postpone the advent of the zombie apocalypse, i.e. the extinction of healthy humans.
For any disease, one of the key variables is virulence (). That is, according to the medical subject headings database MeSH (Rogers, 1963), the ability of a pathogen to cause disease in its host. In a simplified model, one can think of virulence as the probability of becoming ill after being in contact with the virus.
In the following, two scenarios were assumed: one where virulence was assumed to be relatively low, and one where virulence was set at a dramatically high level, where virtually every human coming in contact with a zombie is being infected automatically.
Both scenarios were computed using different search strategies. Each computation was repeated for a number of times and the mean and standard deviation computed. Additionally, a baseline was established using a simulation where no cure at all was available.
The search strategies employed are (a) a purely genetic algorithm and (b) a cultural algorithm, as proposed by
Reynolds (1994). A detailed description of the used algorithms is given in the following subsections.When searching for an optimal cure, i.e. a mixture of treatments, the algorithm only has a limited set of generations available, before that cure needs to be applied to society. This allowance of generations per day will prove to have a dramatic effect on the algorithms’ ability of finding effective cures early on during the simulation. After application of the cure, the simulation is advanced a day forward, with new infections taking place. This turns the optimization problem into a dynamic version of the knapsack problem (Cruz et al., 2011; Branke, 2002). While in the traditional knapsack problem, an optimal combination of items fitting into a static knapsack is the target of the optimization, here, the features of the knapsack change while the search is still well under way. To properly identify this kind of optimization problem, we propose the name of dynamic continuous disease treatment problem.
3.1 Genetic search
The genetic algorithm seeking out the optimal mixture of treatments follows along traditional lines of genetic computing. The set of ten treatment intensities making up the cure comprises the chromosome, with each treatment being a real-valued gene in the domain of .
The goal of the optimization was to find a cure that minimizes the society’s mean infection rate if applied. Fitness was thus established by simulating the application of each chromosome’s encoded solution to the society at the current state, simulating the progression of the disease for one day, and then estimate the resulting mean infection rate. A chromosome was found to be the fitter the lower the caused mean infection rate was.
The genetic operators implemented are cross-over replication and mutation. Cross-over computes the mean between all gene pairs and returns it as the child’s value. Mutation randomly replaces a gene with a uniformly distributed value. The best solution clones itself to guarantee monotonicity. Breeding is organized in a tournament selection scheme, with tournament size
. The winner of each tournament enters a breeding pool, from which pairs are selected at random to create offspring.3.2 Cultural search
One of the greatest benefits of cultural algorithms is that they can quicken the search for an optimum by retaining a history of previously tried solutions (situational knowledge) and including search-related meta-information (normative knowledge) (Reynolds and Ali, 2008). While in the zombie infection scenario, meta information as in its real world applications is not readily available, the retaining of best solutions was included into the algorithm.
In the traditional purely genetic approach, two extraordinarily fit solutions are selected at random to create offspring. The cultural algorithm changes this behavior drastically. After a generation has been evaluated, the fittest solution writes itself into a meta-storage, called belief space. This belief space then is used to influence all of the offspring created. Influence is implemented by replacing either mother or father of a child solution at random with the information stored in the belief system. Other than the belief space influencing mating partner selection, the algorithm remained unchanged from its genetic brother.
The details of both algorithms’ parametrization can be found in Table 1.
Quantity | Value |
---|---|
Population size | 100 |
Tournament size | 5 |
Mutation probability | 0.05 |
4 Results
To demonstrate the suitability of the evolutionary approaches described above for finding cures to C-Virus outbreaks, a number of scenarios was considered. Pertaining to the real world applications, different degrees of virulence were assumed: a rather mild disease with a low virulence of and an extremely aggressive one (), where every person meeting a zombie becomes infected. Obviously, the former scenario leaves the algorithm more time to react to the outbreak, while the latter requires swift action to effectively combat the threat. All simulation parameters are given in Table 2.
Parameter | Value |
Infection threshold | 0.75 |
Number of treatments | 10 |
Range of treatment effectiveness | |
Number of replications | 10 |
Virulence settings | 0.3, 1 |
Generations/day optimization allowance | 1, 5, 25 |
The second parameter varied was the time the algorithms got to optimize before they were forced to apply their respective optimal solutions to the society and the simulation advanced a day forward. This generations per day allowance, , was simulated at settings of 1, 5 and 25. A value of 50 was tried as well, but omitted from later analysis, as the differences to proved negligible.
All scenarios started from the same society of 50 individuals. The society was initialized with random infection values, ranging for humans from to and for zombies from to . The proportion of zombies in this initial society was set at . This resulted in a society mean infection rate of at the onset of the simulation.
Because the progression of the disease is stochastic in nature, all simulations were replicated 10 times. All reported figures are averages over these replications, with the standard deviation given, when appropriate.
The untreated disease with low virulence will have eradicated the last human after 160 () days. For a highly virulent strand of the disease, 71 () days will be enough to leave nothing but babbling zombies. The mean infection rates for both scenarios were established at 0.59 () and 0.81 (), respectively.
Fortunately for humankind, the algorithms were able to counter the zombie threat successfully. Figure 1 compares the effectiveness of both algorithms in all six scenarios. As the scenario assuming a highly virulent disease produces more pronounced differences and for the sake of brevity, only the results from that scenario will be presented in the following.
The most effective cure translates directly into the lowest mean infection rate achieved during the course of the simulation. Both algorithms’ achievements in that regard are given in Table 3.
GA Mean | GA SD | CA Mean | CA SD | Generation |
---|---|---|---|---|
0.16 | 0.03 | 0.16 | 0.02 | 1 |
0.13 | 0.00 | 0.13 | 0.01 | 5 |
0.13 | 0.01 | 0.13 | 0.01 | 25 |
Besides finding good solutions, also the reliability of solutions produced at any given stage of the simulation are of importance. To assess this stability, the variance of the ten replications for each day was computed. The results across algorithm type and generations per day optimization allowance are presented in Table
4. The lower the variation is, the more stable the results are.Genetic | Cultural | Generation |
---|---|---|
0.03 | 0.03 | 1 |
0.01 | 0.02 | 5 |
0.01 | 0.01 | 25 |
Finally, when combating potentially dangerous political attitudes (and zombie inducing diseases), it is crucial to score good results early on. Table 5 gives the days it took, for the algorithm’s solutions to enter a 5%-neighborhood of their final results. The earlier an algorithm enters this neighborhood, the more suited it will be to combat even highly virulent strands of the disease.
Genetic | Cultural | Generation |
---|---|---|
141 | 139 | 1 |
122 | 90 | 5 |
28 | 28 | 25 |
When turning to the solutions produced, a dynamic optimization problem not only warrants an examination of the final results. Rather, it is mandated to analyze the development of the solutions during the course of the simulation. Figure 2 displays the development of four treatments chosen to exemplify the overall development. Each panel depicts a single treatment. The lighter lines describe the intensity of the respective treatments as it was applied over the course of the simulation. The darker lines show how large the proportion of the population that was positively affected by that treatment actually was. Note, that these figures was unknown to the optimization algorithms, as would be the case during real world applications.
In this section we described the results of both genetic and cultural algorithms in solving the dynamic disease treatment problem. A crucial parameter proved to be the numbers of generations granted to an algorithm before it has to commit its solution, and the simulation is advanced forward one day. In the following section we will discuss these results and their implications.
5 Discussion
The optimization’s overall goal was to reduce the mean infection rate in the society. Translating the mean infection rate into the realm of real (political) world applications, this means to reduce the overall (negative) attitudes held in a society regarding a specific topic. As demonstrated by the baseline escalation, the spreading of these bad views knows no bounds, if unchecked. Fortunately, both algorithms even with low generations per day optimization allowances were able to keep a balance. Indeed, in this regard, both algorithms proved to be quite indistinguishable. Obviously, the more computing time (in terms of generations per day) an algorithm was granted, the further down the mean infection rate could be pushed. It is also noteworthy, that no algorithm managed to eradicate the disease entirely.
Similarly, both algorithms were on par regarding the stability of their obtained solutions, exhibiting only minuscule differences. If looking at these figures from the point of view, that containing the outbreak is the ultimate goal, these results are encouraging. However, if one wants to treat the disease for good, both algorithms are disappointing. As high stability also means it is unlikely to produce even better results, hopes for curing, i.e. rooting out the disease, are lost.
The perhaps most interesting feature is the time it took to achieve nearly optimal solutions. Here, cultural algorithms truly shine. While the brutal variant, where algorithms are barely allowed to optimize and need to apply their solutions almost instantly, shows little difference, it is the 5 generations per day allowance that exhibits radical differences. Here, the cultural algorithm finds optimal solutions more than a month ahead of its purely genetic brother. Only when providing both algorithms with 25 generations per day of optimization allowance, the differences subside.
This is a crucial observation. Allowing an algorithm to spend that much of computing power is expensive both in the time and money dimensions. Obviously, the detour via a belief space induced culture is worth its price: despite adding complexity to the optimization, it results in better outcomes, early on. In the application of combating negative opinions, these days won by more clever optimization, translate directly into being able to more quickly counteracting views that cost real market share or, more dramatically, democratic quality of a society.
6 Conclusions
In this paper, we aimed at examining the ways of counteracting the spread of negative opinions. To abstract away from the harsh realities of politics and marketing, we used the analogy of a hypothetical zombie-state inducing virus spreading in an artificial society. The disease model diverged significantly from canonical epidemiological literature. By conceiving the disease as a continuous outcome variable, the framework of diseases has been made accessible to the propagation of opinions. The omission of a class of deceased or otherwise immune to an opinion individuals was mandated. The focus on a simple population structure forgoing complex network topologies is adequate for preliminary research.
As in real life, the goal was to contain the outbreak and thus reduce the overall amount of negative views held within the society. This amounts to a dynamic optimization problem which used to be very hard to solve in the past. By using two different evolutionary approaches, we were able to contain, but not necessarily eradicate the outbreak. Clearly, there remains room for improvement here. If, after simulation day 200, we would have allowed the simulation to continue without the application of constantly updated treatment solutions, the disease would have sprung up again and eventually taken over society at large.
However, the simulation at hand has demonstrated that making the right choices in counteracting negative opinions does matter. Indeed, countering opposite attitudes with maximum (political) force is bound to produce negative consequences. Rather, the situation needs to be analyzed and judged dispassionately. Only then can solutions be found, that keep the opposing force’s proponents at bay. Using evolutionary algorithms shows but one way of achieving this goal.
The complexity added by using cultural algorithms proved to be well invested in producing solutions that were able to contain the outbreak more quickly and at less computational expense than the purely genetic variant. Thus, we are left with two areas that mandate improvement: (1) reduction in computational demand, and (2) providing us with the ability to effectively eradicate the threat of disease now and always.
We would like to point out to a number of issues that would warrant further research. Chiefly, this would concern investigations in evolutionary algorithms that succeed in eradicating the disease for good. Further, applying the simulation to the more prevalent class of complex networks would prove worthwhile. Finally, relaxing the models assumptions, perhaps even (re-)introducing the class of deceased would prove especially intriguing.
The results clearly shows that we are in need of an evolutionary algorithm that has the power to deliver solutions that are able to eradicate the disease once and for all, using only an acceptable amount of computational power. While Hochreiter and Waldhauser (2013) and Guo et al. (2011) both seem to point in the right direction, whether their algorithms are up to the challenge remains to be seen.
References
- Adar and Adamic [2005] E. Adar and L.A. Adamic. Tracking information epidemics in blogspace. In Web Intelligence, 2005. Proceedings. The 2005 IEEE/WIC/ACM International Conference on, pages 207–214. IEEE, 2005.
- Amaral et al. [2000] L.A.N. Amaral, A. Scala, M. Barthélémy, and H.E. Stanley. Classes of small-world networks. Proceedings of the National Academy of Sciences, 97(21):11149–11152, 2000.
- Branke [2002] J. Branke. Evolutionary optimization in dynamic environments. Kluwer, Norwell, 2002.
- Brauer and Castillo-Chavez [2011] F. Brauer and C. Castillo-Chavez. Mathematical models in population biology and epidemiology. Springer, New York, 2011.
- Calderhead et al. [2010] B. Calderhead, M. Girolami, and D.J. Higham. Is it safe to go out yet? statistical inference in a zombie outbreak model. Preprint, University of Strathclyde, Department of Mathematics and Statistics, 2010.
- Crossley and Amos [2011] M. Crossley and M. Amos. Simzombie: a case-study in agent-based simulation construction. Agent and Multi-Agent Systems: Technologies and Applications, pages 514–523, 2011.
- Cruz et al. [2011] Carlos Cruz, Juan R González, and David A Pelta. Optimization in dynamic environments: a survey on problems, methods and measures. Soft Computing-A Fusion of Foundations, Methodologies and Applications, 15(7):1427–1448, 2011.
- Eubank et al. [2004] S. Eubank, H. Guclu, V.S.A. Kumar, M.V. Marathe, A. Srinivasan, Z. Toroczkai, and N. Wang. Modelling disease outbreaks in realistic urban social networks. Nature, 429(6988):180–184, 2004.
- Guo et al. [2011] Yi-nan Guo, Jian Cheng, Yuan-yuan Cao, and Yong Lin. A novel multi-population cultural algorithm adopting knowledge migration. Soft Computing-A Fusion of Foundations, Methodologies and Applications, 15(5):897–905, 2011.
- Hochreiter and Waldhauser [2013] R. Hochreiter and C. Waldhauser. Solving dynamic optimisation problems with revolutionary algorithms. International Journal of Innovative Computing and Applications, 5(1):17–25, 2013.
- Lin et al. [2010] Dazhen Lin, Shaozi Li, and Donglin Cao. Making intelligent business decisions by mining the implicit relation from bloggers’ posts. Soft Computing-A Fusion of Foundations, Methodologies and Applications, 14(12):1317–1327, 2010.
- Lynch [1998] A. Lynch. Thought contagion: How belief spreads through society: The new science of memes. Basic Books, New York, 1998.
- Miller et al. [1999] W.E. Miller, S. Holmberg, and R. Pierce, editors. Policy representation in Western democracies. Oxford University Press, Oxford, 1999.
- Moore et al. [2011] T.W. Moore, P.D. Finley, J.M. Linebarger, A.V. Outkin, S.J. Verzi, N.S. Brodsky, D.C. Cannon, A.A. Zagonel, and R.J. Glass. Extending opinion dynamics to model public health problems and analyze public policy interventions. In Eighth International Conference on Complex Systems, 2011.
- Munz et al. [2009] P. Munz, I. Hudea, J. Imad, and R.J. Smith. When zombies attack!: mathematical modelling of an outbreak of zombie infection. Infectious Disease Modelling Research Progress. Hauppauge NY: Nova Science Publishers, pages 133–150, 2009.
- Patlolla et al. [2006] P. Patlolla, V. Gunupudi, A. Mikler, and R. Jacob. Agent-based simulation tools in computational epidemiology. Innovative Internet Community Systems, pages 212–223, 2006.
- R Core Team [2012] R Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2012. URL http://www.R-project.org/. ISBN 3-900051-07-0.
- Rahmandad and Sterman [2008] H. Rahmandad and J. Sterman. Heterogeneity and network structure in the dynamics of diffusion: Comparing agent-based and differential equation models. Management Science, 54(5):998–1014, 2008.
- Reynolds [1994] R.G. Reynolds. An introduction to cultural algorithms. In Evolutionary Programming — Proceedings of the Third Annual Conference, pages 131–139. World Scientific, 1994.
- Reynolds and Ali [2008] R.G. Reynolds and M. Ali. Computing with the social fabric: The evolution of social intelligence within a cultural framework. Computational Intelligence Magazine, IEEE, 3(1):18–30, 2008.
- Rodríguez Lucatero et al. [2012] C. Rodríguez Lucatero, L. Alarcón, R. Bernal Jaquez, and A. Schaum. Decision dynamics in complex networks subject to mass media and social contact transmission mechanisms. arXiv preprint arXiv:1210.8193, 2012.
- Rogers [1963] F. B. Rogers. Medical subject headings. Bulletin of the Medical Library Association, 51(1):114–116, 1963.
- Sobkowicz et al. [2012] P. Sobkowicz, M. Kaschesky, and G. Bouchard. Opinion formation in the social web: agent-based simulations of opinion convergence and divergence. Agents and Data Mining Interaction, pages 288–303, 2012.
- Sokolowski and Banks [2011] J.A. Sokolowski and C.M. Banks. Principles of modeling and simulation: a multidisciplinary approach. Wiley, Hoboken, 2011.
- Teng [1985] P.S. Teng. A comparison of simulation approaches to epidemic modeling. Annual Review of Phytopathology, 23(1):351–379, 1985.
- Waldhauser [2013] Christoph Waldhauser. Revil: Zombie outbreak simulator for the analysis of opinion propagation, 2013. URL http://knutur.at/Revil. R package version 0.1.
- Wessels and Miller [1999] B. Wessels and W.E. Miller. System characteristics matter: empirical evidence from ten representation studies. In Miller et al. [1999], pages 137–161.
- Wickham [2009] Hadley Wickham. ggplot2: elegant graphics for data analysis. Springer New York, 2009. ISBN 978-0-387-98140-6. URL http://had.co.nz/ggplot2/book.
Comments
There are no comments yet.