2.1. Polarization Type 1: Spread

An obvious way to measure polarization is in terms of the breadth of opinions; that is, how far apart are the extremes? DiMaggio, Evans, and Bryson call this ‘dispersion’: “the event that opinions are diverse, ‘far apart’ in content” (DiMaggio, Evans, and Bryson Reference DiMaggio, Evans and Bryson1996, 694). They also outline a dispersion principle: “Other things being equal, the more dispersed opinion becomes, the more difficult it will be for the political system to establish and maintain centrist political consensus” (693).

Polarization in the sense of spread can be measured as the value of the agent with the highest belief value minus the value of the agent with the lowest belief value (sometimes called the ‘range’ of the data). Polarization in the sense of spread is illustrated by the ends of the horizontal bar in figure 1. In higher dimensions it can be captured as the volume of the minimal bounding ellipsoid.

Figure 1. Belief distribution b shows greater polarization in the sense of spread than does belief distribution a. Two separate groups are shown, but that is irrelevant to polarization in the sense of spread.

Polarization in the sense of spread does not consider whether the agents with the minimum and maximum beliefs are extreme case outliers or the edges of large clusters. Spread is indicated in figure 1 using two groups (red and blue). But it should be emphasized that spread is a concept that applies to belief distribution across an entire population, rather than being group defined. Even if the minimum and maximum agents are representative of groups at the ends of the belief spectrum, spread will ignore that group characteristic just as it will ignore any groups in between. This lack of sensitivity to the shape of the distribution makes spread a weak measure of polarization in isolation, but it does capture the oft-reported feature of America political polarization that the extremists are getting more extreme while the overall shape is largely unchanged.

2.2. Polarization Type 2: Dispersion

Another simple way to measure polarization is statistical dispersion (or statistical variation). Unlike spread, which considers only the extremes of the population, dispersion considers only the overall shape of the distribution. Any of various measures of statistical dispersion might be used: mean difference, average absolute deviation, standard deviation, coefficient of variation, or entropy. Here we use average absolute deviation from the mean as a simple example. Polarization differences in the sense of dispersion are illustrated in figure 2. Note that the diagrams in the figure show dispersion increasing as spread is held constant. Like spread, dispersion is a measure across the distribution, without being tied to notions of groups or subpopulations. Note that this also matches dispersive polarization as defined in DiMaggio et al. (Reference DiMaggio, Evans and Bryson1996, 694)—“opinions are diverse, ‘far apart’ in content”—except that it considers all the beliefs rather than just the extremes.

Figure 2. Distribution c shows greater polarization in the sense of dispersion than does belief distribution b, which is greater than distribution a.

Although dispersion does not depend on any notion of groups, increasing the measure beyond a certain point on one dimension does require the formation of two modes in the distribution (as seen in fig. 2c). Bimodality is frequently mentioned as a feature of polarized distributions and sometimes as part of the definition (Fiorina et al. Reference Fiorina, Abrams and Pope2005). However, we can also find bimodality in distributions for other senses of polarization with specific value ranges. We revisit this point below.

2.3. Polarization Type 3: Coverage

The views of polarized factions are often thought of as constituting narrow and tightly packed sets of beliefs. A polarized society is thought of as one with little diversity of opinion, one in which only narrow bands of the opinion space are occupied. A simple way to envisage polarization in this sense is to think of the spectrum of possible beliefs as divided into small bins. The proportion of empty bins will then constitute a measure of polarization as coverage. That discrete measure, although simple, depends on the choice of bins: varying the width or location of bins can alter the measure for the same underlying data. A continuous variation is also possible by using halos of a given radius around each agent’s belief value and summing the lengths of empty space.

Figure 3 illustrates comparative polarization in the sense of coverage. Although it is not sensitive to either the shape of the distribution (e.g., whether occupied areas are close together or at the extremes) or the number of agents who hold each position, polarization in the sense of coverage does capture this basic feature of opinion diversity or variation. Furthermore, coverage works as a measure for categorical data—data for which the location on a spectrum is meaningless—and across any number of dimensions. Because it is a measure of diversity, with more diversity meaning less polarization, we can also apply more sophisticated diversity measures such as the inverse Simpson index to calculate coverage weighted by the number of agents holding the beliefs at issue (see Size Parity below).

Figure 3. Distribution a is more polarized than b in the sense of representing less coverage on the spectrum of potential belief. Although the plots show groups and differences in heights, neither of those features are aspects of polarization in the sense of coverage.

2.4. Polarization Type 4: Regionalization

Coverage represents how much of the belief spectrum is occupied by a society, without accounting for the pattern of areas occupied. ‘Polarization’ can also be used to indicate belief regionalization, without attending to the total area covered over all. In considering small bins of possible belief, for example, we might measure polarization not in terms of how many bins are filled but in terms of how many empty regions there are between filled areas. The area of uncovered opinion spaces corresponds directly to the concept of coverage. The number of uncovered intervals, in contrast, offers a distinct sense in which distributions can be polarized. Figure 4 shows two cases with the same coverage but in which counting empty regions between occupied areas gives us a measure of regionalization polarization in which b is more polarized than a.

Figure 4. Distributions with equal coverage but in which b shows a higher amount of polarization in the sense of regionalization because of a larger number of empty spaces between occupied areas.

Regionalization per se does not distinguish between a case (i) in which bins between 0 and 0.25 and between 0.35 and 0.60 are filled and a case (ii) in which bins between 0 and 0.25 and between 0.75 and 1 are filled; that is, the most basic notion of regionalization does not account for the widths of the gaps. For some cases, however, this may be the intuitive sense of polarization that we want. It can be combined with other measures to get a more refined description of a phenomenon at issue. Cases i and ii might be regionalized and have coverage to the same degree, for example, although the two groups in ii are farther apart in the sense of dispersion and the beliefs in ii also spread across a wider area.

Regionalization counts the completely distinct clusters in the distribution, related to but distinct from the number of groups (see below for polarization in the senses of both Distinctness and Groups). It does not have a simple, useful extrapolation to higher dimensions and cannot operate on categorical data. Finer-grained quantitative measures of group differences are presented below, but all of these depend on the a priori identification of groups within the data.

Defining Groups.—All of the polarization measures so far have been defined in terms of distribution characteristics observable from the whole population. No concept of groups is required for measures of polarization in terms of spread, dispersion, coverage, or regionalization. Other senses of polarization must be explicitly defined in terms of groups. One way to categorize groups is identify them directly from the histogram as collections of individuals categorized by the basins of attraction between local peaks. In this way groups are identified endogenously by the patterns in belief values, distinguishing unimodal from bimodal or trimodal distributions (Downey and Huffman Reference Downey and Huffman2001).

‘Polarization’ in various group-dependent senses is also used for cases in which groups are exogenously defined (e.g., by region, ethnicity, sex, education level, or other categories). Groups might also be defined in terms of network links representing association, influence, or communication. For example, one can first identify network-based groups using community structure algorithms, then use those collections of nodes as exogenously defined groups in order to break down the belief histogram. Exogenously defined groups may be those indicated by distinct colors in figure 5b. As the two panels of figure 5 make clear, exogenously defined groups may overlap in opinion, generating a very different picture than that indicated in the simpler opinion histogram of 5a. The important point is that the application of each of the group-dependent senses of ‘polarization’ below will depend on how the groups involved are identified.

Figure 5. For exogenously defined groups, the histogram for the entire population (a) may be broken into varying numbers of overlapping subpopulations, as in b.

2.5. Polarization Type 5: Community Fracturing

A first group-dependent sense of ‘polarization’ is community fracturing: the degree to which the population can be broken into subpopulations. Because different endogenous and different exogenous senses of ‘group’ will generate differing levels of community fracturing, this sense offers the level of polarization of the specified groups rather than of the population as a whole.

Figure 6 shows two belief distributions with groups endogenously defined using a local minima method: the distribution in 6b shows more groups than 6a. The population’s beliefs in the second case are more fragmented, indicating greater polarization in the sense of community fracturing. A population that moved dynamically from the first pattern to the second would be a community that showed increased polarization of its endogenously defined groups.

Figure 6. Polarization increases from a to b for endogenously defined groups.

In some applications groups can be exogenously defined; for example, one could have opinion data categorized by educational attainment level on the question of what percentage of the federal budget should be devoted to education. Distinct educational attainment groups could then be plotted together on the same belief spectrum. Defining groups endogenously and exogenously often/typically reveals different groupings of the beliefs. As illustrated in figure 7, aggregated communities may produce identical belief distributions overall yet show very different patterns of polarization in terms of endogenously defined a and exogenously defined b groups.

Figure 7. Polarization as network fracturing. Although overall histograms are identical, categorization in terms of networks may reveal either limited (a) or greater (b) group polarization.

Whether groups in the distribution of beliefs are endogenously or exogenously defined, community fracturing as a measure of polarization is about the number of groups. If agents are connected via a network structure, however, subcommunities may be referred to as ‘polarized’ simply in the sense that there is little or no communication between them. A similar phenomenon occurs in spatial models in which the locations of agents, or clusters of agents, are far apart. Such phenomena can be better thought of as separation and segregation: features that may produce or reflect polarized beliefs and that are also quantified by the number of groups but are not themselves senses of opinion polarization because they are not features of belief distributions.

2.6. Polarization Type 6: Distinctness

The belief groups found endogenously or exogenously can form subdistributions that are very clearly separated (e.g., by a swath of empty bins) or very similar (e.g., two peaks on the same mountain) or anywhere in between. We define polarization in the sense of distinctness as the degree to which the group distributions can be separated. As illustrated in figure 8, for both endogenous and exogenous groups a is less distinct, and therefore less polarized, than b. For endogenous groups, the group boundary is the local minima at the center (where they appear to overlap in the diagram); the height of the distribution at that point measures the distinctness (with greater height indicating less distinctness). For exogenous groups, we can instead measure the overlap of the two groups; the greater the overlap of the distributions, the less distinct, and the less polarized, they are.

Figure 8. Distribution b shows greater polarization than a in terms of distinctness.

When there are more than two groups, some aggregation of all pair-wise comparisons must be made. Some care must also be taken in properly normalizing this measure when the number of groups differs across time or data sets. Both endogenous and exogenous measure versions can be extended unchanged to higher dimensions, but other measures (e.g., in terms of distribution density) may be appropriate as well. It is possible (and in some cases natural) to deploy and combine multiple measures of the same sense because they may each pick up different nuances.

What matters for polarization in this sense is how clearly distinct the groups are, regardless of the distance between them, their size, or their levels of internal cohesion. This seems to match what DiMaggio et al. call ‘bimodality’. People are polarized “insofar as people with different positions on an issue cluster into separate camps, with locations between the two modal positions sparsely occupied” (DiMaggio et al. Reference DiMaggio, Evans and Bryson1996, 694); distinctness focuses on the sparse intermodal region. Attitudes toward abortion between 1970 and 1990, for example, show clear and persistent distinctness (fig. 9).

Figure 9. Attitudes toward abortion, distribution by year, from the full sample General Social Survey 1977–94 (DiMaggio et al. Reference DiMaggio, Evans and Bryson1996).

2.7. Polarization Type 7: Group Divergence

While group distinctness captures how distinct the separation is between groups, regardless of how far away those groups are in their beliefs, group divergence captures the opposite: how distant the groups’ characteristic ideas are without attention to potential group overlap. This sense also fits the definition of ‘dispersion’ in DiMaggio et al. (Reference DiMaggio, Evans and Bryson1996), especially when combined with an assumption of bimodality. One simple measure for divergence will be the distance between group means, whether those groups are defined endogenously or exogenously. As figure 10 indicates, group divergence may increase while measures such as distinctness, spread, and coverage remain constant.

Figure 10. Attitude distribution b shows greater polarization than a in the sense of group divergence.

Like distinctness, the measure of divergence works unchanged for higher dimensions with appropriate distance measures, but some aggregation scheme must be chosen for more than two groups. If the groups are fully distinct, then one can also find the distance between the maximum value in one group to the minimum value in the next, that is, the width of the empty space between the groups. Both the distance between means and of extremes (and others) capture this sense of polarization, but they focus on different features of the distribution and thus can work complementarily, as long as researchers are careful to specify the measure used.

2.8. Polarization Type 8: Group Consensus

The beliefs of group members can be highly scattered across the spectrum or extremely focused on the group’s central ideology (fig. 11). The diversity of opinions within groups constitutes another sense in which those groups can be polarized, independent of how far apart their central ideas are and how distinct the groups are. Prima facie, the greater the variance in beliefs within groups, the more likely it would seem that members of one group might move toward common ground with other groups. The more single-minded or unanimous views are within the groups, the greater the polarization between them and the less likely any such conciliation.

Figure 11. Belief distribution b shows greater polarization than a in the sense of group consensus.

A simple and suggestive measure of groups consensus is the absolute deviation within each group, aggregated over all the groups (e.g., the size-weighted average). The smaller the variance within each distinct group, the greater this sense of polarization across the population. As previously mentioned, this is the main sense in which polarization has increased in the US legislature (McCarty et al. Reference McCarty, Poole and Rosenthal2008). It is not the case that the party lines have shifted much over the past few decades but rather that the party members have more consistently voted along the lines of their party.

If the position of the groups’ mean beliefs are fixed and close enough (low enough divergence), then a change that increases consensus also yields an increase in distinctness. But it is possible for the group consensus to change independently of any of the other measures. Depending on the measures used for each sense and the particular distribution being analyzed, consensus and distinctness may or may not be mathematically linked even though they are conceptually distinct.

2.9. Polarization Type 9: Size Parity

A society that has one dominant opinion group with a few small minority outliers seems less polarized than one with a few comparably sized competing groups. Groups are more polarized in this sense if the different clusters of beliefs are held by equal numbers of people (fig. 12). For example, a society in which a small group of 1% of the population with extremist views gains in popularity to capture a third of the population (while holding all the other measures fixed) is a society with increasing polarization.

Figure 12. Groups with comparable sizes are more polarized than a large group with smaller outliers; in the sense of size parity, belief distribution a is more polarized than b.

There are a few obvious measures for capturing size parity. If there are just two groups (which may not exhaust the population) then a direct subtraction of their sizes is perhaps sufficient. For larger numbers of groups it is preferable to consider measures that aggregate more intuitively. One example is the sum of the differences in size of each group from the mean group size. Using the proportions of the population for each group, one can also use an entropy measure or a diversity indicator such as the inverse Simpson index.

2.10. Further Considerations for the Senses of Polarization

The nine senses of ‘polarization’ outlined here offer only a starter set, and the senses can be multiplied quite quickly by a range of other factors.

2.10.3. Multiple Dimensions

As noted initially, we have focused on senses of ‘polarization’ that can obtain in one-dimensional distributions and for which appropriate measures can be formulated for a single-issue histogram. Other senses of the term, incorporating one or more of these, will be appropriate where more than one issue is at stake. One example of a sense of polarization requiring multiple dimensions is belief convergence. Given groups that are polarized on issue A, are these same groups polarized on B, C, and D? The more connected rival beliefs are within rival groups, the greater the polarization across the community. This can be measured using a data-clustering technique. Another example is belief coherence: are changes in one belief accompanied by analogous changes in another belief—a solidarity of ideas? Coherence of beliefs can be measured using correlations of the belief values across dimensions. Figure 13 illustrates beliefs on two axes showing the relative correlation of two opinions on two axes.

Figure 13. Beliefs on two topics, indicated on the axes, are correlated in b and not in a. Greater correlation of belief values across multiple issues marks greater polarization in the sense of belief coherence.

In measuring multiple opinions, all the senses of ‘polarization’ above will return with a new complexity. Groups may have tightly knit sets of opinions that are uniformly polarized in any of the senses noted. But, their opinions on different issues A, B, and C may turn out to be polarized in some of the importantly different ways we have tried to distinguish.

3.1. The Axelrod Tradition

Axelrod (Reference Axelrod1997) proposes that polarization can arise from an intuitive mechanism that would at first sight seem only to promote conformity and cultural convergence. We first review Axelrod’s original version of the model and discuss the senses of polarization it can and cannot produce. We provide the same treatment for three variants of the model that differ from the original in ways that affect polarization.

The basic premise is this: people tend to interact more with those like themselves and tend to become more like those with whom they interact. But if people come to share one another’s beliefs (or other cultural features) over time, why do we not observe complete cultural convergence? Axelrod acknowledges a number of proposed mechanisms for durable cultural differences—active social differentiation, preferences for extreme views, geographical isolation, specialization, and exogenous changes in the environment or technology—but his model relies on none of these. At the model’s core is a spatially instantiated imitative mechanism that produces cultural convergence within local groups but also progressive differentiation and cultural isolation from other groups. He refers to that differentiation as ‘polarization’.

Axelrod’s base model consists of 100 agents arranged on a 10 × 10 lattice such as that illustrated in figure 14. Each agent is connected to four others: top, bottom, left, and right. The exceptions are those at the edges or corners of the array, connected to only three and two neighbors, respectively.Footnote ³ Agents in the model have multiple cultural ‘features’, each of which carries one of multiple possible ‘traits’. One can think of the features as categorical variables and the traits as options or values within each category. For example, the first feature might represent culinary tradition, the second one the style of dress, the third music, and so on. In the base configuration an agent’s ‘culture’ is defined by five features (F = 5) each having one of 10 traits (q = 10): q_f ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. For example, agent x might have a cultural signature specified by traits {8, 7, 2, 5, 4} while agent y has a cultural signature specified by traits {1, 4, 4, 8, 4}. Agents are fixed in their lattice location and hence their interaction partners. Agent interaction and imitation rates are determined by neighbor similarity, where similarity is measured as the percentage of feature positions that carry identical traits. With five features, if a pair of agents share exactly one such element they are 20% similar; if two elements match then they are 40% similar, and so forth. In the example above, agents x and y have a similarity of 20% because they share only one feature, their fifth: x ₅ = y ₅ = 4. For each iteration, the model picks at random an agent to be active and one of its neighbors. With probability equal to their cultural similarity, the two sites interact and the active agent changes one of its dissimilar elements to that of its neighbor. If agent i = {8, 7, 2, 5, 4} is chosen to be active and it is paired with its neighbor agent j = {8, 4, 9, 5, 1}, for example, the two will interact with a 40% probability because they have two elements in common. If the interaction does happen, agent i changes one of its mismatched elements to match that of j, becoming perhaps {8, 7, 2, 5, 1}. This change creates a similarity score of 60%, yielding an increased probability of future interaction between the two.

Figure 14. Typical initial set of “cultures” for a basic Axelrod-style model consisting of 100 agents on a 10 × 10 lattice with five features and 10 possible traits per agent. The marked sight shares two of five traits with the site above it, giving it a cultural similarity score of 40% (Axelrod Reference Axelrod1997).

In the course of approximately 80,000 iterations, the model process produces large areas in which cultures of traits on features are identical: the ‘local convergence’ of Axelrod’s title. It is also true, however, that arrays such as that illustrated do not typically move to full convergence. They instead tend to produce a small number of culturally isolated stable regions—groups of identical agents none of whom share features in common with adjacent groups and so cannot further interact. As an array develops, agents interact with increasing frequency with those with whom they become increasingly similar, interacting less frequently with the dissimilar agents. With only a mechanism of local convergence, small pockets of similar agents emerge that move toward their own homogeneity and away from that of other groups. With the parameters described above, Axelrod reports a median of three stable regions at equilibrium. It is this phenomenon of global separation that Axelrod refers to as ‘polarization’.

Which of the senses of polarization that we have outlined does Axelrod’s model capture? In his primary use of the term, Axelrod’s ‘polarization’ refers to stable, distinct, contiguous, and wholly culturally differentiated regions at equilibrium. An array is polarized in his sense when there are two or more stable regions, although we might also extend his usage by speaking of grids with a greater number of culturally distinct regions as being more polarized. In this way we can see Axelrod’s ‘polarization’ as a form of community fracturing even though measuring the number of groups in this model does not involve separating histograms at their basins of attraction as described above. Although the measure is different, the number of isolated regions in Axelrod’s model is clearly also a way to capture groups of agents with internally similar ‘beliefs’ separated from other groups with opposed beliefs.

A fundamental characteristic of Axelrod’s model is its multidimensionality: each feature represent a different dimension in which agents or groups can be the same or different. Although the categorical nature of the traits means that correlation cannot be applied, it is clear that the model exhibits polarization in the general sense of belief convergence described in section 2.11.3. In the equilibrium state, agents who have the same trait for the first feature will also have the same trait for the second, third, fourth, and fifth traits. That mix of traits will be (almost always)Footnote ⁴ completely distinct from the mix of traits of any other group; thus, data clustering will pick out the different groups cleanly.

Although the multidimensionality of Axelrod’s model fosters analysis through additional measures, the categorical nature of the traits precludes any notion of closeness and with it most of the measures of polarization we described above. One measure that does apply, expanded to higher dimensions, is coverage. By considering the three-dimensional histogram of agents’ cultures on the F × q discrete belief space, we can count the proportion of empty bins and watch the number grow as the population moves toward isolated homogeneous communities.

The result that different abutting communities cannot share any of their cultural traits also sounds much like distinctness, and the result that intragroup traits become homogeneous sounds much like group coherence, but these cannot be measured using histograms of beliefs. A notion of similarity in terms of the number of identical features and using measures based on Hamming distance or a similar alternative might be developed, however. For example, spread could be measured as the highest number of pair-wise dissimilar traits, and dispersion could be taken as average pair-wise dissimilarity. Both of these measures are greatest when there are isolated communities and zero when there is one homogeneous population. Extensions of this sort do seem to offer multidimensional categorical analogies to the senses of polarization outlined.Footnote ⁵ Thus, Axelrod’s model nicely instantiates some of the general concepts of polarization we have described, even though agents are not characterized in terms of opinions with cardinal belief values, and therefore other measures would be required to quantify most senses.

3.2. Bounded Confidence Family Models

In an influential series of articles, Hegselmann and Krause develop a ‘bounded confidence’ model of opinion polarization that functions in terms of mutual influence among those within a specific threshold of similarity (Hegselmann and Krause Reference Hegselmann and Krause2002, Reference Hegselmann and Krause2005, Reference Hegselmann and Krause2006). The primary results of the model are the formation of consensus given certain thresholds for who counts as ‘close enough’ and the formation of polarized groups with narrower thresholds. Furthermore, the number and location of the formation of polarized groups occurs at different points for different thresholds. Stated simply, this is another case of seeming incongruity between a plausible mechanism that can only increase the similarity of agents yet results in social fragmentation into separate ‘polarized’ attitude groups.

Opinions in the Hegselmann-Krause model are mapped onto the [0, 1] interval, with initial opinions spread uniformly at random. Belief updating is done by taking a weighted average of the opinions that are ‘close enough’ to an agent’s own. As agents’ beliefs change, a different set of agents or a different set of values can be expected to influence further updating. One way to think about the Hegselmann-Krause model is that all agents are effectively linked in a complete network, since it is possible for any agent to be influenced by any other. The primary mechanism of the model is then the threshold for what counts as ‘close enough’ for actual influence. Alternatively, one can think of the model as representing a dynamic network in which only those with opinions ‘close enough’ to an agent’s are linked in ways effective for belief updating.

Figure 15 shows the changes in agent opinions over time in single runs with thresholds ε set at 0.01, 0.15, and 0.25 respectively. With a threshold of 0.01, individuals remain isolated in a large number of small local groups. With a threshold of 0.15, the agents form two permanent groups. With a threshold of 0.25, the groups fuse into a single consensus opinion. These are typical representative cases, and runs vary slightly. As might be expected, all results depend on both (a) the number of individual agents and (b) their initial random locations across the opinion space. Given any threshold and sufficient individuals distributed evenly enough, the result of averaging will be inevitable consensus.

Figure 15. Example changes in opinion across time from single runs with different threshold values ε ∈ {0.01, 0.15, 0.25} in the Hegselmann and Krause (Reference Hegselmann and Krause2002) model.

An illustration of average outcomes for different threshold values appears as figure 16. What is represented here is not change over time but rather the final opinion positions given different threshold values. As the threshold value climbs from 0 to roughly 0.20, there is an increasing number of results with concentrations of agents at the outer edges of the distribution, which themselves are moving inward. Between 0.22 and 0.26 there is a quick transition from results with two final groups to results with a single final group. For values still higher, the two sides are sufficiently within reach that they coalesce on a central consensus, although the exact location of that final monolithic group changes from run to run creating the fat central spike shown.

Figure 16. Frequency of equilibrium opinion positions for different threshold values in the Hegselmann and Krause model scaled to [0, 100] (as original with axes relabeled; Hegselmann and Krause Reference Hegselmann and Krause2002). Slicing along each threshold value produces a histogram from average outcomes of the type used in defining the measures in section 2.

Hegselmann and Krause describe the progression of outcomes with an increasing threshold as going through three phases: “As the homogeneous and symmetric confidence interval increases we transit from phase to phase. More exactly, we step from fragmentation (plurality) over polarisation (polarity) to consensus (conformity)” (2012, 11). Here the term ‘polarization’ is being used to mean bimodality, already mentioned as a common and problematic identification in the literature. There are (at least) two ways to analyze the dynamics of the Hegselmann-Krause mechanism through our various senses of polarization: (1) how a population’s opinions shift as a run progresses (fig. 15) and (2) how the average distribution changes as the threshold increases (fig. 16).

The dynamics of an individual run make a few features immediately clear. First, in this model the groups are defined endogenously in exactly the way described in section 2.4. As a run progresses, the number of groups, and hence polarization in the sense of community fragmentation, decreases. Second, the averaging mechanism forces all the opinions within a group to be eventually identical. Polarization as group consensus therefore necessarily increases through the process. Third, because the opinion-averaging mechanism fuses any groups that are within the threshold of each other, the resulting groups are always completely distinct at equilibrium, another sense in which the mechanism necessarily increases polarization. In addition to these, it is clear from the second two frames of figure 15 that spread decreases as agents at extreme positions are pulled toward the population center, coverage decreases as agents coalesce into single-opinion groups, and the number of empty regions (regionalization) decreases along with the number of groups.

In Hegselmann and Krause’s own description of the results of increasing the threshold, it is clear that the dominant focus is on the number of groups, going from many to two and then to just one as the threshold increases. Although they emphasize the fact that at medium threshold values the population becomes bimodal, we take it that in the sense of community fracturing the polarization steadily decreases with increasing threshold values. The senses that change with increasing thresholds exactly mirror those that the mechanism generates in a single run—increasing the threshold serves to amplify the effects of that mechanism.

The Hegselmann-Krause model gives us the curious result that it is only in the senses of group distinctness and group consensus that polarization increases through model runs. In the population-based measures as well as community fracturing and group divergence polarization decreases. This is a very different polarization profile than the Axelrod family of models, in which most of the senses were shown to increase together. Even though the Hegselmann-Krause model produces distributions that many intuitively recognize as being highly polarized, they are polarized in only a couple of our senses. The opposite is true in many more. We are not proposing that these senses can be directly aggregated into some single measure. The full profile of all the senses is necessary to understand model dynamics both here and elsewhere.

The basic Hegselmann-Krause model, as outlined above, involves thresholds applied symmetrically; agents with opinions either to the left or right within a certain threshold are included in updating an agent’s opinion. Hegselmann and Krause also consider variations in which thresholds are applied asymmetrically, that is, with a different ‘inclusion’ to the left or the right, either (a) with the same bias for all agents or (b) with a bias keyed to the current opinion. The idea in the latter case is that those to the right pay more attention to those to their right; those to the left, more attention to their left. The former, not surprisingly, pushes the distribution patterns to one side, while the latter accentuates bimodality.

Because of boundary effects, applying the same bias to everybody brings size parity back into the picture: it is not just the locations of final groups that change but the sizes of those groups. Although this is still not a sense of polarization that Hegselmann and Krause invoke in describing the model, such an effect could be important in analyzing the application of these mechanistic variations to real social phenomena. Just as the first variant a above increases size parity, the second variant b increases the spread, dispersion, and divergence senses of polarization. All still decrease as the model runs over time, but they decrease less than in the base model. Hegselmann and Krause do provide intuitive explanations of the central mechanism and its effects, but incorporating specific measures of polarization for each sense would offer a precise mathematical and conceptual description of how the distribution of opinions changes.

Deffuant’s Relative Agreement Model.—Deffuant and his collaborators introduce a number of additional mechanisms in what they term a ‘relative agreement’ model. Whereas the bounded confidence mechanism updates agents’ opinions in terms of the average opinion of those within a certain threshold range of current opinion, Deffuant et al. update agents in randomized pair-wise interactions. Any agent may be paired with any other agent to determine influence, reflecting something like a completely connected underlying interaction network (Deffuant et al. Reference Deffuant, Amblard, Weisbuch and Faure2002; Deffuant Reference Deffuant2006; Meadows and Cliff Reference Meadows and Cliff2012).

More significantly, perhaps, the Deffuant model also replaces the sharp cutoff of influence in Hegselmann-Krause with continuous influence values. Agents are again assigned both opinion values and threshold (“uncertainty”) ranges, but the extent to which the opinion of agent i is influential on agent j is proportional to the ratio of the overlap of their ranges (opinion plus or minus threshold) over i’s range. The closer i and j’s opinions, the greater the overlap of their threshold ranges and the greater their influence on each other. The farther apart their opinions, as long as there is any overlap at all, the smaller the overlap and the smaller their influence. Deffuant et al. update both opinion centers and threshold ranges accordingly, resulting in the possibility of individuals with narrower and wider ranges. Given the updating algorithm, influence may be asymmetric: individuals with a narrower range of tolerance, which Deffuant et al. interpret as higher confidence or lower uncertainty, will be more influential on individuals with a wider range than vice versa.

For initial opinion values that are uniformly random on the [−1, 1] opinion space and initially identical threshold ranges, the results are very much the same as those of Hegselmann and Krause. Figure 17 shows a typical time progression of agents’ opinion dynamics, very similar in outcome despite the important differences in the treatment of thresholds.

Figure 17. Opinion dynamics in a population of 200 agents for an initial uniform threshold of 0.4, with the x-axis indicating the average number of iterations per agent (a proxy for time). Although the relative agreement mechanism differs from the bounded confidence mechanism, with the base parameters the opinion dynamics are very similar. From Deffuant et al. (Reference Deffuant, Amblard, Weisbuch and Faure2002).

One of the major goals of the Deffuant model is the attempt to produce extremism: convergence of opinion at an extreme end of the opinion range. Deffuant et al. note that this is possible in the Hegselmann-Krause model with asymmetric tolerance ranges but that it requires extreme parameters to produce. Related to this, the model is intended to demonstrate that a position that is initially extreme and held by only a few individuals can persuade the whole population to accept concordant extreme opinions. Deffuant et al. are able to produce the effect using ‘stubborn’ or ‘high-confidence’ individuals with narrow uncertainty thresholds placed at both ends of the opinion spectrum.

Although Deffuant et al. thoroughly explore the parameter space and reveal a variety of opinion dynamics in different value ranges, the result in figure 18 is the one most closely associated with ‘polarization’ phenomena and marks the primary departure from the Hegselmann-Krause model. Within this parameter zone the result is always two groups, one at each extreme, because the initially extreme agents at one end will never move to the other end. It is common for one-sided extremism to result even when there are the same number of initial extremists at both ends as in figure 18. Whether the result is most of the agents at one end or a split to both sides depends mainly on the random order of the pair-wise comparison updating.

Figure 18. When high-confidence (low-uncertainty, narrow threshold) agents are used at the ends of the spectrum, more than 98% of the initially moderate (green) agents are pulled to one side (in this case toward 1). Color indicates increasing confidence (lower ability to influence) as the population becomes more extreme and more homogeneous. From Deffuant et al. (Reference Deffuant, Amblard, Weisbuch and Faure2002).

Although the mechanism does not strictly force a large disparity in sizes, and in some runs the population is split nearly in the center, the typical result is a stochastic symmetry breaking such that most of the moderate central population snowballs to one side. Size parity is therefore a sense of polarization captured by the Deffuant et al. mechanism, more interesting precisely because it usually happens but is not obvious from the description of the agent update rules. The authors themselves admit that they did not foresee such a result.

Like the Hegselmann-Krause model, groups here must be defined endogenously by the opinion space. The number of groups, and hence polarization in the sense of community fracturing, decreases through time. Because of the stubbornness of the extreme agents, spread decreases only slightly because of a centrist pull. Coverage, regionalization, group distinctness, and group consensus change similarly as in the Hegselmann-Krause model, with some increasing and some decreasing.

Population dispersion decreases noticeably here as a by-product of the group size disparity. These two are the only senses of polarization that differ between the bounded confidence mechanism and the relative agreement mechanism. Polarization decreases in the sense of population dispersion, but the literature is rather quiet on that point. The most often-cited result is that polarization increases in the sense of size parity, together with group consensus and distinctness. There is something intuitive about bimodal movement of opinions away from each other on a spectrum. Here however the result is one of near uniformity at biased extremes. Deffuant’s intuitions regarding the influence of strongly held minority opinions shaping the population’s ideology are plausible ones but may not be the same phenomenon at issue in opinion polarization on abortion or gun rights, for example. Although the influence mechanism and the resulting opinion dynamics do seem to capture an important social force, it is not clear that ‘polarization’ is the most appropriate way to understand it.

3.3. Evaluating the Bounded Confidence Family of Models

Perhaps the greatest strike against this second family of models is that the central mechanism itself seems distinctly unreal. The main driver of the Hegselmann and Krause results is a mechanism of ‘peeling back from the edges’. The opinion distribution exhibited occurs for the very specific reason that agents at the left and right edges of the opinion space have no one to pull them left or right. They therefore drift toward the center, with either central or separated points of convergence dictated by the ε threshold. Despite the change in treatment of thresholds, the Deffuant et al. models inherit that central mechanism.

All models in this family cite and draw on a long tradition of belief averaging as a simple representation of the important psychological phenomenon of peer influence (French Reference French1956; DeGroot Reference DeGroot1974). But we are aware of virtually no evidence that real polarization occurs ‘from the edges’ as it does in these models or that it crystallizes in virtue of a specific distance from the edges as it does here. Indeed, there is a great body of evidence that the dynamics of developing polarization are quite different. The classic study by Lord et al. (Reference Lord, Ross and Lepper1979) shows groups in laboratory conditions that progressively polarize, increasing the distance between them over time despite balanced bodies of evidence (see also Miller et al. Reference Miller, McHoskey, Bane and Dowd1993; Kuhn and Lao Reference Kuhn and Lao1996). Cooper, Kelly, and Weaver claim that “one of the most robust findings in social psychology is that of attitude polarization following discussion with like-minded others” (Reference Cooper, Kelly, Weaver, Hogg and Tindale2001, 267).

Recall figure 9, showing American attitudes toward abortion between 1977 and 1993. That same figure serves to indicate the real limitations of the family of models considered. The models at issue can neither produce nor preserve the pattern of polarization evident there, that with a smaller central consensus with a heavy and consistent large group at the liberal end. Most noticeably, and in contrast to the reality of polarization, that liberal consensus would inevitably move right in models with a Hegselmann-Krause mechanism, and the large group of moderates would inevitable move to one side or the other using the Deffuant mechanism. Furthermore, such a phenomenon in the Deffuant models requires a careful planting of extremists with manually altered uncertainty thresholds to counteract the Hegselmann-Krause effect. Repeated studies show average attitudes among groups shifting toward extremes in terms of some mechanism not captured in the family of attraction-driven models considered here. The reality of increasing polarization of these familiar types, wherever it occurs, is a dynamic that this family of models is incapable of producing. These models also seem to capture an important and intuitive piece of the story of social opinion dynamics; however, it is not the whole story, and in many particulars it may not be most profitably read as a story of ‘polarization’ at all.

3.4. Structural Balance

Although less heralded in the computational literature, a third family of models for polarization has equal claim to consideration. Structural balance theory, also known as social balance theory, originated in the mid-1940s through the work of Fritz Heider, who studied patterns of belief coherence in individual psychology (Heider Reference Heider1946). In the mid-1950s, Cartwright and Harary generalized and formalized Heider’s theory using basic graph theory (Cartwright and Harary Reference Cartwright and Harary1956; Harary Reference Harary1959). Consider a set of nodes (e.g., people or countries) joined into a network capturing not just whether they have a relationship but also the valence of that relationship. In the base version, if two nodes are joined then they are either friends or enemies; later versions allow valence weights between −1 and 1 (Kulakowski, Gawronski, and Gronek Reference Kulakowski, Gawronski and Gronek2005). In the original analyses, a structure was considered ‘balanced’ whenever all paths—all unique sequences of links connecting each pair of nodes—had an odd number of negative links. Although there is some recent work that returns to the original ‘all paths’ version of balance calculations (Facchetti, Iacono, and Altafini Reference Facchetti, Iacono and Altafini2011), Abell (Reference Abell1968) made the case against longer paths being meaningful for social relationships, and in most structural balance research the fundamental unit of analysis has been a triad of three mutually linked nodes.

A triad is considered unstable if there is social pressure to change one of the relationship links; it is considered stable if there is no social pressure to change. The case of zero enmity is considered a stable triangle of friendship. A stable triad with two enmity links simultaneously captures the slogans ‘the enemy of my friend is my enemy’ and ‘the enemy of my enemy is my friend’. Those slogans also highlight the source of instability in the other possible patterns. If some agent A is friends with both B and C, but B and C are enemies, then A will be under pressure by both friends to choose sides, and B and C will be under pressure to make amends. In a triad with three mutual enemy relationships, there is pressure for two of them to ally against the third. Graphical relationships of these stable and unstable configurations are presented in the left portion of figure 19. Note that changing the valence of any single link in a stable triad will make it unstable, while changing the valence of any single link in an unstable triad will make it stable.

Figure 19. Left, four possible configurations of positive and negative links (ignoring rotation) as well as their classification under stable or unstable triads. Right, four larger structures capturing different types of stable and unstable social networks.

Larger social systems are classified as ‘balanced’ if all triads are stable, ‘unbalanced’ otherwise, with the level of instability typically calculated as the proportion of stable triads in the network as a whole. There are three types of stable configurations in larger networks: universal harmony, factions, and mixed networks, depicted on the right side of figure 19. In balanced networks with universal harmony, all links are friendship links. In factions, the network exhibits groups of mutually friendly nodes separated entirely from each other by enmity links. Mixed networks contain both positive and negative links in such a way that the network is balanced without forming factions.

It is easy to envisage a dynamic through which a network with initially randomly assigned valences can move toward a balanced network step by step. For example, (1) choose a triad at random; (2) if it is stable, do nothing; (3) if unstable, the valence of a random link in that triad is ‘flipped’ from positive to negative or vice versa. Flipping that valence may well make neighboring triads that also used that link unstable, of course, but as the process continues the network can be expected to move toward one of the balanced network configurations. In fact, given enough time any such random walk-like dynamic will eventually reach an equilibrium stable configuration.

A promise for understanding polarization can be seen in that dynamic. It is easy to prove that, for any complete network, progressively flipping valences within unstable triads will necessarily lead to either universal harmony or (with much higher probability) a ‘social mitosis’ resulting in precisely two groups, linked only by friendship internally and only by links of enmity between the two groups (Wang and Thorngate Reference Wang and Thorngate2003; Sack et al. Reference Sack, Flocken, Grim, Bramson and Berger2014). Social networks that are less than fully connected allow for more than two factions and the possibility of mixed networks, but the pattern toward social mitosis of the groups is still very much a dominant one (Hummon and Doreian Reference Hummon and Doreian2003).

In order to use structural balance as a model for opinion polarization, we must first interpret it in terms of opinions, beliefs, or attitudes rather than friends and enemies. The most direct way to achieve this is by assuming that agents hold beliefs on some topic. Agents who agree in their belief have a positive link, while those who disagree have a negative link. This interpretation makes both the stability and instability of triads as well as factional splitting natural: a network is balanced whenever all and only people who hold opposite beliefs disagree. The modeling implication for polarization is this: perhaps we see so much polarization because almost all stable opinion configurations within a population are polarized in this way. There is some empirical evidence for changes in social and political relations operating in roughly this way (Hart Reference Hart1974; Facchetti et al. Reference Facchetti, Iacono and Altafini2011; Kunegis Reference Kunegis2014).

What senses of polarization can a structural balance mechanism produce? As outlined, the basic dynamics are driven by increasing network ‘balance’, but on an interpretation in terms of agreement and disagreement there are implicit belief changes as well. We can assign all nodes of a stable triad with all positive ‘agreement’ links a belief of either p or ¬p. In a stable triad with two positive ‘agreement’ links and one negative ‘disagreement’ link, we can assign p to two nodes and ¬p to one, or ¬p to two and p to one. But there is no way to assign p and ¬p to nodes of unstable triads of agreement and disagreement without some node being assigned both p and ¬p. The basic mechanism of structural balance proceeds by changing link valences, and this is equivalent to changing beliefs so that the relationships of agreement and disagreement change appropriately.

Here we might suggest a model in which agents may have beliefs 1 or −1 but in which agents in ‘frustrated’ positions (i.e., nodes in unstable triangles) are assigned intermediate belief values that are the average across all neighbors of link valence × belief. If an agent’s only connection is a negative ‘disagreement’ link to a neighbor with belief =1, that agent will have a belief = −1, for example. If an agent has disagreement links to both a neighbor with belief = 1 and a neighbor with belief = −1, his belief will be 0 as the average. If a node has agreement links to two neighbors with beliefs = 1 and an agreement link to a neighbor with belief = −1, his intermediate belief value will be the link valence-weighted average of 1/3. As a final example, a node with an agreement link to a neighbor with belief = 1/3 and a disagreement link to a neighbor with belief = 2/3 will have an intermediate belief = −1/9.

Given this assignment of intermediate beliefs, we can suggest a dynamics for belief revision that uses the basic mechanisms of structural balance theory, although within a model of opinion change. Consider a randomly chosen node, and suppose that the product of link valence × neighbor values used to calculate the current belief of that node is positive—equivalently, whether its current value is positive. We then change the valence of random link, if any, that will result in a recalculation of belief value that moves that value further in the positive direction. If we randomly choose a node for which that product is negative—such that its current value is negative—we change the valence of a random link, if any, that will take its value further in the negative direction. Our conjecture is that a belief dynamics of this sort, appropriate to interpretation in terms of opinion polarization, will characteristically show a pattern of opinion mitosis toward belief extremes of 1 and −1 that parallels social mitosis in the original structural balance model.

What senses of polarization could such a model capture? What counts as ‘polarization’ under structural balance theory is clearly community fracturing, a move from something like the unstable mixed diagram to the stable factions diagram in figure 19. A network that is polarized in opinion through analogous mechanisms will also have factions in which all agents within each faction hold the same belief of either 1 or −1. Multiple groups of nodes holding the same belief can exist as long as there are no direct links between those same-belief factions. The degree of polarization in the sense of community fracturing can then be measured as the number of such groups.

Our conjecture is that final histograms of community-fractured (or factionalized) belief configurations will show beliefs close to 1 and −1; fully balanced systems will converge to the extremes, and only stable mixed configurations would show intermediate values in the final histograms. For these outcomes, the population-based senses of polarization—spread, dispersion, coverage, and regionalization—could not be usefully applied to compare final outcomes. Dispersion and coverage could be used, however, to track movements in population beliefs as the network stabilizes. Coverage would decrease as nodes with intermediate values push to extremes by mechanisms of triad balancing, with an increase in polarization in the sense of section 2.3. Polarization in the sense of the dispersion of beliefs increases as agents move to any stable configuration. Polarization as distinctness will increase along with coverage and dispersion as a population factionalizes, but it should be noted that none of these will necessarily progress monotonically; as in the original structural balance model, a change that increases the balance of a local triad may cause imbalance in other triads sharing one of its links. Size parity can also be measured as an opinion network evolves, but it is neither explicitly nor implicitly part of structural balance.

A fairly glaring limitation of models in the structural balance tradition should also be mentioned. As early as 1964, Abelson notes that computational models of opinion dynamics can easily fail by inevitably producing a radically unrealistic convergence of opinion to the center (Abelson Reference Abelson1964). Computational models that produce ‘polarization’ as inexorable drives to exclusive polar extremes of −1 and 1 will be radically unrealistic in the opposite direction. Structural balance models, whatever their other virtues, are clearly in this category.

3.6. Group Polarization in Social Psychology

Up to now our focus has been on the three major families of computational models applied to phenomena of polarization. The term ‘polarization’ has also appeared in other literature without being operationalized into a formal model. Here we discuss one such case for which something approaching a model can be abstracted from the literature, at least close enough to be worthy of inclusion.

The experimental findings of Lord et al. (Reference Lord, Ross and Lepper1979) on belief polarization show that individuals tend to reinforce and move their beliefs in a more extreme direction when exposed to mixed evidence that, prima facie, would seem to lead to a more centrist position. Extending this finding further, self-categorization theory within social psychology defines ‘group polarization’ as the “conformity to a polarized norm which defines one’s own group in contrast to other groups within a specific social context” (Hogg et al. Reference Hogg, Turner and Davidson1990, 77). The theory predicts that in the context of confrontation with another group or other groups, in-group discussion will yield a “consensual group position that is more extreme than the mean of the individual group members’ prediscussion attitudes in the direction already favored by the group” (78). In the case of a centrist group confronted with groups on both sides, the prediction is a tighter focus on the mean. Although psychological explanations are provided for these attitude movements, they have not been operationalized as formal mechanisms.

Self-categorization theory posits that a group’s norm is actually further away from the population’s mean attitude than the members’ mean attitudes. When confrontation makes the issue salient, the group members adjust their attitudes to the perceived group norm and thus exaggerate their attitudes. This force bootstraps itself over time to produce increasingly extreme positions for members of the groups on the edges of the population. But the same mechanism pushes groups toward the population mean when confronted with groups more extreme than they are. Although the work in social psychology does not provide us models or opinion data, we can still refine our understanding of the claims being made by analyzing them in terms of our senses of polarization.

The theory is already conceived in terms of groups, so the appropriate senses of polarization will be the group-based ones. We can understand the claim that confrontation with an external group pushes the group mean to the perceived group norm as an increase in polarization in the sense of group consensus (decreased variance of in-group attitudes). The specific claim regarding the direction and amount of movement is relative to the external group(s) and thus can best be captured as an increase in divergence. This is also consistent with the case in which the focal group is sandwiched between extreme groups, increasing divergence by narrowing on to the mean value. There is no mention in the theory of a change in the number of groups, or of the focal group splitting, so community fracturing is not a sense relevant to this theory. Although the combined changes in group consensus and divergence are likely to produce an increase in distinctness, spread, dispersion, and coverage as well, there is nothing explicitly said about the dissimilarity of the groups beyond the direction and magnitude of their intragroup and intergroup differences. The ‘group polarization’ of self-categorization theory thus can be fully described in terms of consensus and divergence, although the dynamics of belief change may also show spread, dispersion, coverage, and distinctness.