(Inside Science) -- Opinions rarely form in a vacuum. People are heavily influenced by the opinions of others in their social networks, whether they be real or virtual. Some people are not open to new ideas. These are the zealots, who proselytize an opinion -- the superiority of Apple products, for example, or skepticism about climate change -- in the hopes of convincing others, while stubbornly resisting being influenced themselves.
Researchers studying the evolution of sentiments in a society, a field called opinion dynamics, have long been interested in the effects of zealots on the dissemination and adoption of ideas. One way to study this is to use a mathematical model such as the so-called naming game.
Originally developed for linguistics to study how a shared vocabulary spontaneously emerges in a population, the naming game has recently been adopted by opinion dynamics researchers to study the propagation of ideas through a society.
"The naming game assumes that individuals come into random contact with a neighbor and they exchange opinions. If you're a zealot, you're never going to change your opinion. Otherwise, there's some probability that you're going to adopt the opinions of your neighbor," said Raissa D'Souza, a computer scientist at the University of California, Davis.
Past naming-game models had a severe limitation, however: they only considered the spread of two opinions at a time. In real life, of course, many opinions about a topic can coexist and jostle for dominance.
In a new study, published this week in the journal Physical Review E, D'Souza and her team modified the naming-game model to capture competition between multiple opinions and the influence of zealots. While it makes the model more accurate, it also adds a great deal of complexity.
"The number of variables we must consider grows dramatically with the number of possible opinions," D'Souza said.
To solve the mathematical equations, Alex Waagen, the study's first author and a graduate student in D'Souza's lab, developed so-called mean-field equations that made the naming-game calculations easier to solve.
"Mean-field equations are used commonly to model disease spreading in human populations, starting back in the late 1920s," D'Souza said. "The technique was invented originally to solve [problems associated with the] properties of materials, but has been applied in many different realms."
Waagen's mean-field equations assume that any person in the social network can interact with any other person.
"This isn't going to happen in real life, but if we make the model too complicated, we won't learn anything about the underlying mechanisms," Waagen said.
The team used the model to solve for the eventual outcome when there are interactions of up to 100 different opinions. What they found was that even though the interactions became more complex, qualitatively, they were the same as when only two opinions were vying with each other.
For example, in the two-opinion naming game, if opinion A had a sufficient number of zealots and opinion B did not, the vast majority of the group would eventually adopt opinion A. Alternatively, if both opinions A and B had a sufficient number of zealots, then over time, the two opinions would reach an equilibrium where each would have an equal number of followers.
The team was surprised to find that these patterns still held even when the number of circulating opinions increased: If one opinion had a sufficient fraction of zealots and others did not, it would eventually influence the entire group, and if all of the opinions equally had a sufficient number of zealots, they would eventually reach a stalemate in which no one opinion dominated.
Boleslaw Szymanski, a computer scientist at the Rensselaer Polytechnic Institute in Troy, New York, said that the team's findings could provide some general guidance for how companies could better manage their brands.
For example, the first result suggests, somewhat counterintuitively, that if a company has enough zealots passionate about its product, it need not worry about emerging competitors that have not attracted enthusiastic followers yet. Instead, it should focus on courting the zealots it already has and increasing their numbers.
"According to this study, when there is a lack of committed customers for other products, the number of competing products is irrelevant," said Szymanski, who was not involved in the study.
The second result -- that multiple opinions vying for dominance will eventually reach a stalemate -- could explain the fate of certain social movements, Szymanski said.
"According to my team's earlier research, if a sufficient fraction of a society is united in its commitment against the current government, the government can be toppled," he said. "But once it is removed, the opposition is likely to split into many competing fractions with different opinions about the future."
According to the second result by D'Souza's group, such divided opposition may fail to create a stable, future government. This has been observed many times in history, Szymanski noted.
For example, the 1917 February Revolution in Russia led to the overthrow of the Czar, but because the provisional government was fractured, it was overthrown during the October Revolution a few months later. The latter revolution allowed the Bolsheviks to take power and set the stage for the Communist Party of the Soviet Union.
The same pattern was seen during the Egyptian Revolution of 2011.
"The revolution included Islamic, liberal, anti-capitalist, nationalist and feminist elements," Syzmanski said. "They won; [Egyptian President Hosni Mubarak] was overthrown, but democracy goals were quickly abandoned when an outside party [the Muslim Brotherhood] took power in 2011 and introduced a religious government."
D'Souza and Waagen caution that all naming-game models, including their own, are still relatively crude.
"But we're making baby steps toward something that is more true to real life," said Waagen.
Ker Than is a freelance writer living in the Bay Area. He tweets at @kerthan.