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A Dynamical Origin for Consensus and Polarization

By Alessio Franci, Marty Golubitsky, and Naomi Ehrich Leonard

Why do we tend to polarize when it comes to important matters? People's beliefs, opinions, and choices are increasingly polarized within our societies in terms of social, political, and economic decisions [1-2]. The role of social media and bots is evident in polarization [3-4] and information gerrymandering [5]. For our purposes, polarization refers to the situation in which a group of decision-making agents engaged in decision-making processes develop more divergent and extreme opinions than their pre-deliberation preferences. As a consequence, the agents cannot make a collective decision and the group remains in deadlock. Could this phenomenon be inherent to the way collective decisions are made? Are there general, deterministic causes that split people's opinions toward polarization or, conversely, unite them toward consensus?

Dynamical systems theory might help answer these questions. One can model a collective decision-making process as a dynamical opinion network, where the deciding agents’ opinions are the nodes and the agents’ interactions are the edges. We model opinions as continuous variables that describe each agent’s preference for each option as a function of time. In other words, opinions are not binary (or “black-and-white”), but rather nuanced and evolving continuously in time. The evolution in time of each agent’s opinion depends on the agent’s prior preferences (or biases) and the exchange of information with other agents in the network. It is indeed a combination of the agents’ biases and interactions that pushes groups toward either consensus or polarization.

People silhouette vectors courtesy of Freepik. Speech bubbles and figure courtesy of Alessio Franci.

Our approach to dynamical decision networks is novel in that we can analytically characterize all of the feasible (or generic)1 decision-making behaviors—and the likely transitions among them—for an arbitrary number of agents and options. Our method uses the powerful tools of equivariant bifurcation theory to exploit empirical symmetry properties of collective decision-making, namely that agents are equal and options are a priori equally-valued. The resulting predictions are model independent, in the sense that we expect them to be true in any system (mathematical or real) that approximately satisfies those symmetry properties. A basic, model-independent prediction of our theory is that consensus and polarization are the only two non-degenerate decision-making behaviors of an arbitrary dynamical decision network. Moreover, we can derive explicit model equations by plugging the universal sigmoid nonlinearity into the structure imposed by the equivariant bifurcation theory. These equations are amenable to model-dependent quantitative analysis, as well as the systematic means for deriving testable hypotheses on behavior in natural groups and verifiable design of technological networks. We use a realization of these equations to generate the following numerical simulations.

Bifurcation theory is well-suited for the study of opinion network dynamics because one can naturally model the transition from indecision to decision as a bifurcation. A bifurcation diagram provides a concise description of how network decision states change as a function of key system and environmental features (the strength of agent interactions, for instance) that one can model via a scalar bifurcation parameter \(\lambda\) and vector of unfolding parameters \(\beta\). Figure 1 illustrates this idea in the simplest case of a consensus decision between two options: A and B. Continuous and dashed lines represent the evolution of the group’s average opinion as a function of \(\lambda\) (modeling the strength of agent interactions) and for fixed common scalar bias \(\beta\) (e.g., modeling the difference between the value of option A and value of option B). Continuous lines signify stable decision states, while dashed lines signify unstable decision states. 

For small \(\lambda\), the network state is at indecision, roughly midway between options A and B. As \(\lambda\) increases, different things can happen depending on the difference between the values of the two options. If the two options are valued equally \((\beta=0)\), the indecision state becomes unstable at some critical value of \(\lambda\) and bifurcates into two symmetric decision states, one of which favors A and the other B. The actual decision depends on the network initial conditions, noise, and other small random perturbations. In other words, the group flips a coin. If there is a difference in option values \((\beta>0\) or \(\beta<0)\), the decision state smoothly converges toward the higher-valued option. Researchers have observed and modeled these types of collective decision-making behaviors in a number of biological systems and societies, including bees [6], fish [7], neurons [8], molecular regulatory networks [9], and birds [10]. The bifurcation theory approach has been crucial in formalizing how decision-making can be both ultra-sensitive to relevant inputs and robust, with respect to irrelevant inputs [11].

Figure 1.

When one assumes the possibility of polarized decision states and occurrence of more options (as compared to Figure 1), the bifurcation theory of opinion network dynamics is a largely unexplored field. To explore it constructively, we translated the characteristics of a “democratic” decision-making problem—made challenging by indistinguishable agents and options that are a priori equally valued—into symmetry properties of an underlying dynamical system. That is, we assumed that the decision process is equivariant (symmetric) with respect to both agent and option permutations. Under these assumptions, the powerful tools of equivariant bifurcation theory [12] let us uncover the following properties of collective decision making [13], illustrated with numerical simulations in Figure 2:

(i) Polarization and consensus are the only two non-degenerate decision states to which a dynamical opinion network can bifurcate from indecision.

(ii) Polarization typically appears in only two forms: (a) uniform polarization, in which agent opinions are homogeneously spread across the options, and (b) moderate/extremist polarization, in which a small group (the “extremists”) and a large group (the “moderates”) develop diametrically opposed opinions. In the moderate/extremist situation, deadlock is not broken despite the presence of a majority of moderates because the extremists develop a much stronger opinion than the moderates.

(iii) Well-defined parameter modulation, determined by the algebraic structure of an underlying equivariant singularity, can switch opinion networks between different decision-making behaviors (for instance, between consensus and uniform polarization). The animation in Figure 2 was obtained by continuously changing a single parameter, thus representing the balance between agent competition and cooperation. This parameter was constructively singled out in the equations as the key modulator of decision-making behavior. One can observe the model-independent properties of decision-making in the simulation; as anticipated above, these properties are expected for any model that approximately satisfies the symmetry assumptions. By contrast, model-dependent properties of decision-making, such as the stability of solutions, depend on the model specifics. For instance, distinct solution types abruptly lose or gain stability as the modulated parameter is continuously varied in Figure 2, thus inducing abrupt changes in the decision-making behavior.

Figure 2. Time evolution of the opinion state of a group of 17 agents deciding between three options. Each agent state, represented by a colored circle, is projected onto a 2-simplex, which signifies the opinion state-space of an agent (see Figure 3). The closer an agent state to one of the vertices of the 2-simplex, the larger the agent’s preference toward the associated option. The closer an agent state to the simplex center, the more undecided, or neutral, it is. An agent can also reject one of the options and be undecided between the remaining two, in which case its state lies roughly midway between the vertices of the two in question. Options are color-coded to aid visualization; the color of the circle that represents the agent state changes according to its opinion. The animation shows the results of a simulation obtained in a network model. At the beginning of the simulation the agents interact competitively, tending to reject other agent opinions. This results in a rapid convergence to a polarized network state, with three clusters of agents exhibiting sharply different opinions (each cluster strongly favors a different option). As time evolves, a single parameter continuously transforms to slowly change the nature of the agent interactions from competitive to cooperative. Each agent gradually begins to accept other agent opinions. As a consequence, a succession of prototypical decision-making behaviors is observable. Around time \(t=1000\), the three-cluster polarization smoothly changes into a moderate/extremist polarization. The red (option 2) and blue (option 3) clusters merge into a large “moderate” cluster whose members weakly agree to reject option 1, in the sense that the agent state is close to neutral. On the other hand, the small green cluster maintains a strong (“extremist”) preference for option 1. Despite the fact that the majority of agents reject option 1, the extremist group manages to ensure that the deciding network remains in a polarized (deadlock) state. A few extremists move toward the larger moderate group, but this still cannot drive the whole group to consensus. The extremist/moderate situation eventually becomes unstable, and the deciding network reverts to a rather plastic three-cluster configuration in which agents continuously change their favorite options. A small group of agents is close to neutral around time t=2100, while the other agents maintain a three-cluster polarized state. After this highly dynamic phase, the network converges to a rigid three-cluster configuration, but with increasingly weaker agent opinions. When the agent opinions are close to neutral, their cooperation finally brings about a consensus decision in favor of option 2.

Figure 3. A 2-simplex signifying the opinion state-space of an agent (see Figure 2 for more information).
The opportunity to analyze the dynamics of opinion networks in a deterministic, agent-based framework—and for an arbitrary number of agents and an arbitrary number of options—provides new tools for understanding social decisions and translating this knowledge into distributed technological systems. We especially hope to gain a novel deterministic viewpoint on statistic- and probability-oriented modeling efforts that tend to average-out the details of the agent level [14-23]. Considering the agent level dynamics and exploiting the corresponding symmetry properties in a deterministic setting allowed us to discover that familiar—and sometimes alarming—decision-making behaviors (like the appearance of moderates and extremists) are intrinsic properties of collective decision-making. The dynamical mechanisms underlying these behaviors are independent of the specific nature of the agents and their interactions. Similarly, one can engage (in principle) the switch between democratic consensus and group polarization by subtle—but well-determined and predictable—parameter changes in any dynamical decision-making network. 

Our theoretical framework is naturally suited to studying the effect of social attacks on democratic decision networks (such as information gerrymandering, bots, zealots, and contrarians) using the tools of dynamical systems theory. The interaction between social psychology and dynamical systems theory might uncover mechanisms of collective decision-making that provide the opportunity for important insights into our rich—and sometimes frightening—social behavior.

1 In the sense that they are expected to characterize the dynamics of decision-making for all dynamical decision networks except for a small (zero-measure) set of degenerate parameters.

Alessio Franci presented this work during a minisymposium presentation at the 2019 SIAM Conference on Applications of Dynamical Systems, which took place in May in Snowbird, Utah.

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Alessio Franci earned his Master 2 (Laurea Specialistica) degree in theoretical physics from the University of Pisa in 2008 and his Ph.D. in physics and control theory from the University of Paris Sud 11 in 2012. Between 2012 and 2015, he was a postdoctoral researcher at the University of Liege and at INRIA Lille, as well as a long-term visiting researcher at the University of Cambridge. He has been a professor in the Mathematics Department at the National Autonomous University of Mexico since April 2015. His research interests span different fields, but the central focus is on the regulation principles of biological behaviors —particularly rhythmic and excitable neuronal behaviors (from single neurons to brain states) and collective decision-making. 

Marty Golubitsky is a distinguished professor of mathematics at the Ohio State University. He is a past president of SIAM, the founding editor-in-chief of the SIAM Journal on Applied Dynamical Systems, and a former chair of the SIAM Activity Group on Dynamical Systems.
Naomi Ehrich Leonard is Edwin S. Wilsey Professor of Mechanical and Aerospace Engineering and associated faculty in applied and computational mathematics at Princeton University.  She is a MacArthur Fellow and a fellow of SIAM, the American Academy of Arts and Sciences, the Institute of Electrical and Electronics Engineers, the International Federation of Automatic Control, and the American Society of Mechanical Engineers. Leonard received her BSE in mechanical engineering from Princeton and her Ph.D. in electrical engineering from the University of Maryland. Her research is in control and dynamics with application to multi-agent systems, mobile robotic sensor networks, collective animal behavior, and human decision dynamics. 
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