Disagreement among individuals in a society, even on central questions that have been debated for centuries, is the norm. Agreement is the rare exception. How can disagreement of this sort persist for so long? The question is even more acute in the Age of Social Networks. With half a billion people using Facebook such disagreement is clearly not a consequence of lack of communication, nor some other factors leading to fixed opinions. Disagreement remains even as individuals communicate and sometimes change their opinions in what cognitive philosophers call “Belief Revision.” What would the late Isaac Asimov have said about this?
At Massachusetts Institute of Technology and Politecnico di Torino, four experts on Federal research grants, Daron Acemoglu, Giacomo Como, Fabio Fagnani, and Asuman Ozdaglar, studied a stochastic gossip model of continuous opinion dynamics in a society consisting of two types of agents: regular agents, who update their beliefs according to information that they receive from their social neighbors, and stubborn agents, who never update their opinions and might represent leaders, political parties or media sources attempting to influence the beliefs in the rest of the society.
In the technical field of Applied Probability (since the four experts submitted their results to the Journal of Applied Probability) research has deepened the paradox. Non-Bayesian updating mechanisms typically lead to consensus, provided that communication takes place over a strongly connected network.
The MIT group reached the following conclusion, published in their 52-page submitted paper: “When the society contains stubborn agents with different opinions, opinion dynamics never lead to a consensus (among the regular agents). Instead, beliefs in the society almost surely fail to converge, and the belief of each regular agent converges in law to a non-degenerate random variable. The model thus generates long-run disagreement and continuous opinion fluctuations. The structure of the social network and the location of stubborn agents within it shape opinion dynamics. When the society is ‘highly fluid,’ meaning that the mixing time of the random walk on the graph describing the social network is small relative to (the inverse of) the relative size of the linkages to stubborn agents, the ergodic beliefs of most of the agents concentrate around a certain common value. We also show that under additional conditions, the ergodic beliefs distribution becomes ‘approximately chaotic,’ meaning that the variance of the aggregate belief of the society vanishes in the large population limit while individual opinions still fluctuate significantly.”
I’ll sketch the meaning of their technical term ergodic. As Caltech alumnus (the other “Tech” with a beaver as mascot) Dr. Eric W.Weisstein puts it:
Ergodic theory had its origins in the work of Boltzmann in statistical mechanics problems where time- and space-distribution averages are equal. Steinhaus gives a practical application to ergodic theory to keeping one’s feet dry (‘in most cases,’ ‘stormy weather excepted’) when walking along a shoreline without having to constantly turn one’s head to anticipate incoming waves. The mathematical origins of ergodic theory are due to von Neumann, [George David] Birkhoff and Koopman in the 1930s. It has since grown to be a huge subject and has applications not only to statistical mechanics, but also to number theory, differential geometry, functional analysis, etc. There are also many internal problems (e.g., ergodic theory being applied to ergodic theory) that are interesting.
So what do the MIT folks mean by “approximately chaotic” in this context? They imply that, if the influence of any of the stubborn agents’ opinions does not dominate the influence of the rest, then the mean square disagreement does not vanish in the large population size. They conjecture that, in highly fluid social networks without a significant presence of stubborn agents, intermediate condition between approximate consensus and chaotic ergodic belief distribution should emerge in the large population limit.
My friend, the late Dr. Isaac Asimov, Professor of Biochemistry at Boston University Medical School, based his “Foundation novels” (originally 3 novels, then integrated with others to make a coherent set of 10 novels, to which there are 3 authorized sequels) on the imaginary future science of “psychohistory.”This is a precise mathematical predictive science of aggregate human behavior. Asimov (who as an undergraduate had trouble deciding between history and chemistry as a major) hypothesized that for sufficiently large numbers of humans (a galactic empire), cooperative effects would occur, and also analogized to the Kinetic Theory of gasses, for which Boltzmann first developed statistical mechanics.I conversed with Asimov about this, as I was only the 2nd active member of Science Fiction Writers of America to have done a doctoral dissertation in enzymology (in retrospect, I see that it was about the connection between enzymology and the not-yet-named field of nanotechnology), and promised him a formal citation to his dissertation in a refereed work, which nobody had done before. Near the end of his prodigiously prolific life (over 500 books), he wrote a story that undercut the assumptions of psychohistory, explicitly referring to chaos theory. The greatest psychohistorian of all time was Asimov’s Hari Seldon: “I quite understand that psychohistory is a statistical science and cannot predict the future of a single man with any accuracy.”
So, whether agents are robots following Asimov’s 3 Laws of Robotics or humans in an internet-connected world of political blogs, we feel that each of us as individual men and women cannot be predicted. And yet these strange chaotic patterns beyond pattern emerge.
I’ll give Asimov the last word, on stubborn belief: “Those people who think they know everything are a great annoyance to those of us who do.”
Daron Acemoglu, Giacomo Como, Fabio Fagnani, Asuman Ozdaglar, “Opinion fluctuations and disagreement in social networks,” download the PDF
Weisstein, Eric W. “Ergodic Theory.” From MathWorld – A Wolfram Web Resource.
Asimov: The Foundation Series is a science fiction series by Isaac Asimov that covers a span of about 550 years. It consists of seven volumes that are closely linked to each other, although they can be read separately. The term “Foundation Series” is often used more generally to include the Robot Series which are set in the same fictional universe, but in earlier time periods. In total, there are fifteen novels and dozens of short stories written by Asimov, and six novels written by other authors after his death, expanding the time spanned by more than twenty thousand years.
This is #3 of a 52-week series, the #1 of which was “Laws of Physics, or Merely Local By-laws?” 9 Sep 2010.
Jonathan Vos Post is co-webmaster, Vice President, and Chief Information Officer of Magic Dragon Multimedia. He has taught art, astronomy, biology, chemistry, computer science, English composition and English Literature at various California colleges. Von Post has collaborated with Isaac Asimov, Ray Bradbury, Richard Feynman, David Brin and Arthur C. Clarke, and has innumerable published articles, short stories and poems to his credit.