Partial Differential Equations Model Political Segregation

By Bertram Du¨ring

Political campaigns, elections and referenda are crucial to the democratic process and shape the future development of human societies. In the past few years, we have been thrilled, often shocked, and sometimes delighted by the outcomes of various elections and referenda all over the world. With pollsters struggling to predict results the need for a better understanding of how political and other opinions (think, e.g. consumer product choice) are formed seems more timely than ever.

Mathematicians are striving to build mathematical models of opinion formation, transferring research from sociology and psychology to a more quantitative mathematical language and using advanced mathematical tools, e.g. from kinetic theory and numerical analysis, to analyse and simulate the dynamics of opinion formation in our societies. At the heart of these approaches is the idea of considering the society as a large system of many interacting individuals. One starts modeling at the level of individual interactions and then derives laws and governing equations, which characterise the emergent dynamics of aggregate quantities like the distribution of opinions in a society. Compared to an individual-based simulation, the aim is to obtain an analytically and numerically more tractable model. Another aim is to obtain a deeper understanding of what role various parameters play in such models, and how parameters in the (microscopic) interactions influence patterns in the (macroscopic) opinion distribution.

Figure 1. The Big Sort: Results of the presidential elections in Arizona, U.S. in 1992, 1996, 2000, 2004 and 2016 (from left to right). The colour intensity (blue for Democrats, red for Republicans) reflects the election outcome in percentage. Image credit: Wikimedia Commons, Creative Commons Attribution-Share Alike.

In a minisymposium talk at the SIAM Conference on Computational Science and Engineering in Atlanta, Ga this March, I presented results from a recent research article [3], co-authored with my colleague Dr Wolfram from the University of Warwick. In [3], we try to push the boundaries of existing models and break free from the assumption of a homogeneous society, which often underlies current models. We develop kinetic models for opinion formation when the opinion formation process depends on an additional independent variable and the society is inhomogeneous. This independent variable can represent different features in the dynamics. It can measure leadership qualities to account for the effect of opinion leaders, or it can be a spatial variable, e.g. when considering political segregation.

Our approach leads to an inhomogeneous Boltzmann-type equation for the opinion distribution where the opinion dynamics is encoded through a collision operator while the change of opinion with respect to the additional is modeled in a transport term. We discuss the numer- ical simulation of these equations and derive more tractable, Fokker-Planck type equations in suitable limits.

One of the examples we apply our model to is the so-called Big Sort ,a term coined by Bill Bishop in [1]. According to his findings, U.S. citizens increasingly choose to live among politically like-minded neighbours. Based on county-level election results of U.S. presidential elections in the past four decades, he observed a doubling of so-called ‘landslide counties’, counties in which either candidate won or lost by 20 percentage points or more. Clearly, a segregation of political supporters may result in making political debates more bitter and hamper political decision-making by consensus. It is likely that the observed segregation results from a ‘hidden correlation’ of political opinions with other sociologic and economic factors which drive citizens’ residential preferences. One may consider including such additional factors by coupling opinion dynamics with a kinetic model for wealth distribution [2].

Figure 2. The Big Sort: Distribution of Democrats (left) and Republicans (right) obtained from the numerical solution of the relevant Fokker-Planck type equation. Image credit: Bertram Düring

In [3], we illustrate ‘The Big Sort’ using our new model for the 2004 presidential elections. We consider the state of Arizona in the southwest of the U.S. with its 15 electoral counties. In recent decades, the Republican Party has dominated Arizona’s politics; see, for example the outcome of the presidential elections from the years 1992 to 2016 in Figure 1. The colours red and blue correspond to Republicans and Democrats, respectively. The colour intensity reflects the election outcome in percentage, i.e. dark blue corresponds to Democrats 60–70%, medium blue to Democrats 50–60% and light blue to Democrats 40–50%. Similar colour codes are used for the Republicans. The election results illustrate the clustering trend as the election results per county become more and more pronounced over the years.

To illustrate the segregation process in our opinion formation model we discretize the relevant Fokker-Planck type equation using a finite element method. As an initial distribution, we choose values in accordance of the election result in 1992. Figure 2 shows the spatial distribution of the votes for Democrats and Republicans, respectively. We observe the formation of two larger clusters of democratic voters in the south and the northeast of Arizona. The Republicans cluster towards the northwest as well as the southeast. The simulation results reproduce the qualitative features of the electoral results from 2004 fairly well, except for the second county from the right in the northeast (Navajo).

For all mathematical models of sociological systems there is a large uncertainty, both in parameters and in the model itself. Future research will seek to improve such models by inferring parameters (and parameter distributions) from real data and further improving the models by including more insights from social science research. Research in mathematical models of sociological systems is still a fairly new and fast-emerging field, hence we can expect to see some fascinating new and possibly unexpected results in the future.

This article is based on a minisymposium talk entitled “Inhomogeneous Boltzmann-Type Equations Modelling Opinion Leadership and Political Segregation” at the SIAM Conference on Computational Science and Engineering, held in Atlanta this March.

References

[1] Bishop, B.(2009). The Big Sort: Why the clustering of like-minded America is tearing us apart. Houghton Mifflin Harcourt.
[2] Du¨ring, B., Matthes, D., & Toscani, G. (2008). Kinetic equations modelling wealth redistribution: a comparison of approaches. Phys. Rev. E 78(5), 056103.
[3] Du¨ring, B., & Wolfram, M.T. (2015). Opinion dynamics: inhomogeneous Boltzmann-type equations modelling opinion leadership and political segregation. Proc. R. Soc. Lond. A, 471, 20150345.

Bertram Du¨ring is a reader in mathematics at the University of Sussex, U.K. His research lies in computational and applied partial differential equations. One of his current projects is kinetic and mean-field models modelling socio-economic systems, e.g. wealth distribution and opinion formation in human societies.