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In 2014, Stephen Guy of the University of Minnesota and his colleagues described how people move to avoid hitting each other when interacting in large groups. “Human crowds,” they wrote, “bear a striking resemblance to interacting particle systems.” Pedestrians move, the researchers observed, like negatively-charged electrons, which repel each other more strongly as they approach, with one key difference. Unlike electrons, pedestrians anticipate when a collision is imminent and change their motion beforehand by swinging wide to avoid a crash.

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Using this knowledge, the researchers derived a mathematical rule for an electron-like “repulsive force” between any two pedestrians, but based on time-to-collision rather than distance. This allowed the researchers to correctly predict how a moving crowd bunches up when funneled into a narrow passage, or spontaneously forms directional lanes, as when football fans leave a stadium headed for different exits. Other sociophysicists have applied similar principles to auto traffic.

The idea that math and physics can illuminate human conduct dates back to the eighteenth century Scottish philosopher David Hume. Later, the French philosopher Auguste Comte proposed that there are general laws describing human societies, and the Belgian mathematician Adolphe Quetelet began performing statistical analyses of human qualities. Today the sciences of sociophysics and econophysics draw on all these ideas in their attempts to explain human behavior.

Some of this work uses the methods of statistical physics, which studies how swarms of particles interact to produce new effects. For example, individual H2O molecules move randomly in water. But, cooled to zero Celsius, they undergo a “phase transition” and lock together into solid ice. Similarly, whatever the varied qualities of individual people, they can form voting blocs, which may spontaneously reform after a crucial event, such as a debate between political candidates. The French scientist Serge Galam applies physical effects like these to human behavior in his book Sociophysics (2012).

Another approach to sociophysics extracts mathematical rules and behavioral patterns from terabytes of data gleaned from existing records or from digital interactions like social media. Social Physics (2015), by MIT researcher Alex Pentland, presents this data-based method as a source of “reliable, mathematical connections between information and idea flow on the one hand and people’s behavior on the other.”

In truth, meaningful sociophysics necessitates both data and a model or theory. As sociophysicist Frank Schweitzer of ETH Zürich points out, analyzing data may uncover interesting and useful correlations, but it will not produce “an understanding of causal relationships… Successful sociophysics models tend to have interfaces with both empirical data and social theories.”

An early example of this approach in practice came from Princeton astrophysicist turned sociophysicist John Q. Stewart. He noted in 1948 that for many kinds of interactions between groups of people, such as telephone calls between any two cities in the U.S., the number of interactions is proportional to the product of the group populations divided by the physical distance between them—that is, more interactions for bigger groups and smaller distances. This seems intuitive, especially since phone companies charged callers extra for “long distance” dialing; but Stewart also saw that the mathematical equation is like that for Isaac Newton’s expression for the gravitational energy between two objects, which is proportional to the product of their masses divided by the distance between them. From this analogy, Stewart defined a “demographic energy,” an invisible force that structured the distribution of U.S. populations across urban and rural areas.

In Stewart’s time, most scientists found such analyses to be of little value. These interdisciplinary approaches are still less widely accepted than traditional physics, sociology, and economics. But with new analytical tools and new data, researchers are finding convincing results.

In 2011, David Garcia and Frank Schweitzer of ETH Zürich studied how people influence each other by examining nearly two million anonymous Amazon reviews of thousands of books and products. They did this within the “circumflex” model of emotions, where an emotion is defined by: (1) valence, the degree of pleasure or pain associated with it; and (2) arousal, the activity prompted by the emotion. These qualities are believed to arise from two independent human neurophysiological systems.

The researchers assigned a “sentiment” score from -5 to +5, highly negative to highly positive, to the emotional content of each Amazon review. They also examined the actions induced by the sentiments, such as when people rate reviews as “helpful” or “unhelpful” or are moved to write their own reviews, which also carry emotion. The results clearly show that individual reviewers are influenced by other comments, and that unlike negative reviews, positive reviews skew toward highly positive—apparently people who like something want to show that they like it a lot. By comparing reviews of one of the Harry Potter books, which received much media attention, to reviews of another book that did not, the researchers found distinct differences in group emotional patterns, reflecting different emotional responses to external marketing vs. internal word-of-mouth.

Results like these have obvious value for corporate marketers. But on a larger scale, studying how opinions travel through groups is an urgent matter for politics, leading sociophysicists to study “opinion dynamics” with the methods of statistical physics.

In 2015, a team at City College of New York (and other institutions in the U.S., Brazil, and Israel) published its research on extreme opinions, noting the “worldwide trend towards the division of public opinions about… political views, immigration, biotechnology applications, global warming” and more, including cultural matters. “A marked dwindling of moderate voices is found,” they write, “with the concomitant rising of extreme opinions… the opinion or attitude of an initially small group could become the rule.” The authors analyzed hundreds of survey results from many countries, where people gave their opinions about religion, politics, abortion, and other subjects as “very favorable” or “very unfavorable,” classified as extreme; or as “somewhat favorable” or “somewhat unfavorable,” classified as moderate.

Comparing the fraction of respondents fₑ with extreme views to the fraction f who held any opinion at all, the researchers found a surprising pattern. When there were just a few extremists, fₑ was proportional to the total number of opinion holders, as expected for a group of non-interacting members. But at a certain critical value of fₑ, around 20% of the respondents, fₑ plotted against f began rising in a steep nonlinear manner that would lead quickly to a majority of extremists. When there are enough extremists to interact with each other, it amplifies their impact.

This effect is central in statistical physics, where correlations among particles produce nonlinear behavior that cascades into a phase transition, like water turning into ice. When the researchers modeled the group interactions, including an element of “stubbornness,” resistance to changing an opinion, they were able to reproduce the empirical data and also to develop a social “phase diagram.” Analogous to a diagram that shows the conditions of temperature and pressure under which water exists in a solid, liquid, or vapor phase, this diagram shows the conditions for a society to occupy moderate, incipient extremist, or extremist phases.

Quantitative methods have been applied to other social phenomena, such as the spread of rumors and voting patterns in recent European and U.S. elections (though no research I know of predicted Donald Trump’s presidential win in 2016), and are useful in economics as well. According to the authors of Econophysics: An Emerging Discipline, econophysics received a strong push after traditional economics failed to foresee the global economic crisis of 2008. That lack, they claim, shows that rather than rely on theories that may be too simplified to describe reality, economics needs an injection of econophysics, with greater weight given to empirical data.

One current interest of econophysicists is mathematical power laws, where one variable depends on another raised to some power. For example, the area of a square and the volume of a cube are given by the length of a side L raised to the power 2 (L2) and power 3 (L3), respectively. For reasons that are not yet completely understood, power laws describe a range of economic activities, such as stock market transactions and, especially relevant today, income inequality. Since the late nineteenth century, data analysis has shown that the fraction of people with an income greater than some cutoff value follows a power law called the Pareto distribution, which gives a small slice of the population a disproportionately large share of income and wealth. In 2016, for instance, the richest 1% of U.S. households owned 40% of the nation’s wealth, a very high level of inequality.

The question of global inequality was brought into focus through the groundbreaking 2014 book Capital in the Twenty-First Century by the French economist Thomas Piketty, whose analysis includes the Pareto distribution. In 2015, Stanford University economist Charles Jones wrote in his article “Pareto and Piketty” that this distribution is a “key link between data and theory.” If econophysics can provide deeper understanding of the origins and meaning of the Pareto distribution, it would greatly contribute to understanding the important issue of inequality.

Sociophysics and econophysics are answering questions about how people behave, though no one could yet claim that these approaches are uncovering deep truths about human nature. In a 2012 essay for the Guardian, the Nobel Laureate economist and New York Times columnist Paul Krugman wrote about the experience of reading Isaac Asimov’s Foundation trilogy as a teenager. The Foundation books concern a future scientist, Hari Seldon, who invents the science of “psychohistory.” The equations of psychohistory, which describe how human societies evolve, predict the coming fall of the Galactic Empire. Armed with this knowledge, Seldon creates the Foundation, a group devoted to minimizing the dark times, so society can recover and go on. Foundation greatly affected Krugman, apparently. “I grew up wanting to be Hari Seldon, using my understanding of the mathematics of human behavior to save civilization,” he wrote.

We are still a long way from the elegance and power of famous results in physics like Isaac Newton’s equation F = ma or Einstein’s E = mc2. But it took millennia for physicists to derive these insights. Maybe in only a few more centuries, we will become like Hari Seldon, able to better understand ourselves through quantitative science.


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SIAM Review, Vol. 53, No. 3 (September 2011), pp. 409-463
Society for Industrial and Applied Mathematics
Scientific American, Vol. 178, No. 5 (May 1948), pp. 20-23
Scientific American, a division of Nature America, Inc.
Economic and Political Weekly, Vol. 47, No. 32 (AUGUST 11, 2012), pp. 44-65
Economic and Political Weekly
The Journal of Economic Perspectives, Vol. 30, No. 1 (Winter 2016), pp. 185-205
American Economic Association
The Journal of Economic Perspectives, Vol. 29, No. 1 (Winter 2015), pp. 29-46
American Economic Association