Compartmental voter model

TL;DR

This paper introduces a simple agent-based model for opinion dynamics that accounts for heterogeneity across electoral units by allowing agents to change units, successfully replicating empirical rank-size distributions.

Contribution

The paper presents a novel agent-based model that incorporates agent mobility between electoral units to better match observed opinion heterogeneity.

Findings

Model reproduces empirical rank-size distributions.

Agents can change electoral units, capturing heterogeneity.

Model aligns with real-world opinion distribution data.

Abstract

Numerous models in opinion dynamics focus on the temporal dynamics within a single electoral unit (e.g., country). The empirical observations, on the other hand, are often made across multiple electoral units (e.g., polling stations) at a single point in time (e.g., elections). Aggregates of these observations, while quite useful in many applications, neglect the underlying heterogeneity in opinions. To address this issue we build a simple agent-based model in which all agents have fixed opinions, but are able to change their electoral units. We demonstrate that this model is able to generate rank-size distributions consistent with the empirical data.

Equations

$${\lambda }_{i\to j}^{\left(k\right)}=\{\begin{array}{ll}{X}_i^{\left(k\right)}\left({\epsilon }^{\left(k\right)}+X_j^{\left(k\right)}\right)& \text{if }i\ne j\text{ and }{N}_j<C,\\ 0& \text{otherwise}.\end{array}$$

$${\lambda }_{i+}^{\left(k\right)}$$

$${\lambda }_{i-}^{\left(k\right)}$$

$$p\left(X_i^{\left(k\right)}\right){\lambda }_{i+}^{\left(k\right)}\left(X_i^{\left(k\right)}\right)=p\left(X_i^{\left(k\right)}+1\right){\lambda }_{i-}^{\left(k\right)}\left(X_i^{\left(k\right)}+1\right).$$

$$p\left(X_i^{\left(k\right)}+1\right)=\frac{\left[N^{\left(k\right)}-X_i^{\left(k\right)}\right]\left(\alpha +X_i^{\left(k\right)}\right)}{\left(X_i^{\left(k\right)}+1\right)\left(\beta +\left[N^{\left(k\right)}-X_i^{\left(k\right)}+1\right]\right)}p\left(X_i^{\left(k\right)}\right).$$

$$p\left(X_i^{\left(k\right)}\right)=C\frac{B\left(X_i^{\left(k\right)}+\alpha ,N^{\left(k\right)}-X_i^{\left(k\right)}+\beta \right)}{B\left(\alpha ,\beta \right)}$$

$$f_i^{\left(k\right)}=\frac{X_i^{\left(k\right)}}{N_i}.$$

Full text

1 Introduction

Most well–known models of opinion dynamics seem to imply that a stable fixed state, either consensus or polarization, is inevitable [1, 2, 3, 4, 5, 6, 7]. However, local and spatial heterogeneity and ongoing exchange of opinions and cultural traits is a characterizing feature of social systems. Various modifications of the well–known models were proposed to account for these features, such as inflexibility [8, 9] or spontaneous flipping [10, 11]. Some of the models were modified to account for the theories from the social sciences [12, 13]. Effects of these modifications are still being actively reconsidered in context of network theory, non–linearity, complex contagion and applications towards financial markets [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. Nevertheless even these modified models assume that opinion dynamics occur and are observed within single electoral unit (from here on let us also use the terms “compartment” and “spatial unit” interchangeably) over multiple time steps. Here we propose a novel agent–based model, which is extremely simple yet able to replicate spatial heterogeneity, which in this paper is studied purely through socio–demographic distributions over compartments, observed in census [26, 27, 28, 29, 30] and electoral data [31, 32, 33, 34, 35, 36, 37, 38, 39].

Some of the recent approaches in opinion dynamics [31, 32, 33, 34, 35, 36, 37, 38, 39, 40] have combined empirical analysis of the detailed electoral data (opinions being observed over multiple electoral units during a single time step) with numerical modeling. Still in [34, 38] various groups of researchers have made the same underlying assumption: that the electoral units are mutually independent observations from mostly the same stationary distribution of opinions. This effectively means that it is enough to model dynamics of a single electoral unit. All of these approaches were based on the noisy voter model and thus predict the opinions to be Beta distributed over time, which is somewhat consistent with the empirical research conducted over electoral units [41, 42, 43]. In [35] numerical modeling, using scheme similar to the one we just have described, was supplemented by rigorous empirical analysis demonstrating strong spatial (across US states) and temporal (over a century of US presidential elections) correlations. These patterns could be another way of understanding the heterogeneity of voting behavior. In [37, 36] no independency assumption was made when formulating a multiplicative model, but the proposed model did not provide an agent–based reasoning for the vote share heterogeneity over electoral units. Similarly [31, 32] have provided a purely phenomenological fully spatial model, which takes into account actual geospatial topology of the modeled area, for the voter turnout rates across various countries and elections. Only [33] have built an agent–based model with dynamics occurring across multiple electoral units. Notably the agent–based model proposed in [33] and later improved by [39] is rather complicated and was built specifically for the elections in the United States, taking into account topology of electoral units and empirical data of the commuting patterns between them. Rather similar, yet much simpler, model was proposed in [40], which simply assumes that voters can copy opinions of agents in their home and direct neighbor nodes. This model was also able to reproduce spatial correlation patterns considered in [33].

Here we take a similar approach to [33], but instead of commuting we consider internal migration. While commuting may drive the short–term changes in the electoral behavior, internal migration may give shape to the long–term trends. Furthermore in social sciences it is quite common practice to gain insights from the covariance between census data, which changes due to migration, and electoral data [44]. Unlike [33] we do not infer the migration rates from the empirical data, the proposed model assumes that these rates have a specific form. This allows us to keep the model simple and focused on reproducing general patterns of socio–demographic heterogeneity over compartments. In this sense our approach is rather similar to the classical Schelling segregation model [26], but the migration rules we use are continuous and somewhat more complicated. Similar migration rules can be found in the classical human mobility models [45], especially in the gravity model [46, 47, 48, 49, 50, 45] as we assume that system–wide migration rates depend on population of source and destination compartments. Yet these models have somewhat different goal and reproduction of the mobility patterns does not ensure reproduction of socio–demographic heterogeneity over compartments. Though vice versa is also true. Finally as with most models in opinion dynamics comparison could be also drawn to the Ising model [2]. Unlike the most existing approaches our model is similar not to the Metropolis interpretation of the Ising model [51], but to the Kawasaki interpretation of the Ising model [52, 53], which assumes that particle spins are conserved, but the particles themselves exchange places.

This paper is organized as follows. In Section 2 we introduce the compartmental model and discuss its main statistical properties. In Section 3 we fit the compartmental model to a few selected empirical data sets. We finalize with the discussion in Section 4.

2 Compartmental model

Let us consider $N$ agents of $T$ types migrating between $M$ identical compartments, each of which has a capacity of no more than $C$ agents. Depending on the context the agents could be assumed to represent residents or voters, who have certain fixed socioeconomic traits or opinions (types). Likewise the compartments could be assumed to represent residential areas or electoral units between which the agents can migrate.

Let us assume that the migration rate between the compartments $i$ and $j$ for the agents of type $k$ has the following form:

[TABLE]

In the above $X_{i}^{\left(k\right)}$ is the number of agents of type $k$ in the compartment $i$, $N_{j}$ is the number of agents of all types in the compartment $j$. The migration rate is composed of two terms. One of the terms linearly depends on the number of agents present in the source compartment $X_{i}^{\left(k\right)}$. This term represents spontaneous (idiosyncratic) migration and $\varepsilon^{\left(k\right)}$ is the relative spontaneous (idiosyncratic) migration rate for the agents of type $k$. The second term involves the number of agents present in the source compartment $X_{i}^{\left(k\right)}$ and the number of agents present in the target compartment $X_{j}^{\left(k\right)}$. This nonlinear term represents migration induced by the interaction processes such as recruitment or homophily. To keep the migration rates as simple as possible we assume that such interactions are possible only between the agents of the same type. Induced migration is assumed to occur at a unit rate. Note that transition rates of the same form are present in the noisy voter model. This is why we consider the compartmental model to belong to the voter model family, ignoring the fact the compartmental model does not describe any actual opinion dynamics.

Here we would like to draw clear distinction between two different distributions, which could be obtained from the series of $X_{i}^{\left(k\right)}\left(t\right)$ and other related variables. Usually in theoretical sociophysics papers spatial index $i$ is fixed, while the observations are made over multiple different time steps $t$. If the model is ergodic (this model is ergodic as well as many other well–known models) and the series is long enough, then the sample distribution should approach the stationary distribution. Thus we will refer to this sample distribution as the stationary distribution and use the probability mass function (abbr. PMF) or the probability density function (abbr. PDF), $p\left(X_{i}^{\left(k\right)}\right)$, to show the results. Unlike most of the previous approaches this model allows for $i$ to be variable and $t$ to be fixed. In this case the sampling is no longer temporal, but instead is made over compartments. Therefore we refer to this sample distribution as the compartmental distribution and use the compartmental rank–size distribution (abbr. CRSD), $\tilde{X}_{r}^{\left(k\right)}$, to show the results. If compartments would be mutually independent, then the compartmental distribution should also approach the stationary distribution. Yet the compartments in this model are not independent as the total number of agents is fixed and agent leaving one compartment must move to some other compartment.

Although the compartmental model is quite simple, it is not straightforward to obtain closed form expressions for the stationary distribution or the compartmental distribution. If the compartments are able to hold all the agents, $C=N$, it is quite straightforward to derive the total entry and exit rates for agents of type $k$:

[TABLE]

In the above $N^{\left(k\right)}=\sum_{i=1}^{M}X_{i}^{\left(k\right)}$ is the total number of agents of type $k$. These rates are identical to the transition rates of the multi–state noisy voter model [38]. This similarity allows us to conclude that $x_{i}^{\left(k\right)}=X_{i}^{\left(k\right)}/N^{\left(k\right)}$ will be Beta distributed in the asymptotic limit, while as whole the value sets $\vec{x}^{\left(k\right)}$ will be Dirichlet distributed. If $N^{\left(k\right)}$ is finite, then $X_{i}^{\left(k\right)}$ will follow Beta-binomial distribution. We can confirm this intuition from the detailed balance condition (which holds for $X_{i}^{\left(k\right)}\in\left[0,N^{\left(k\right)}-1\right]$):

[TABLE]

Rearranging detailed balance condition gives us a set of recursive equations:

[TABLE]

In the above we have used $\alpha$ in place of $\varepsilon^{\left(k\right)}$ and $\beta$ in place of $\left[M-1\right]\varepsilon^{\left(k\right)}$. We can see that PMF of the Beta-binomial distribution:

[TABLE]

satisfies the recursive equations. In the above $C$ is normalization constant and $B\left(x,y\right)$ is the Beta function. Identical derivation of the stationary PMF for the noisy voter model can be found in [40].

In Fig. 1 we can see that Beta-binomial PMF fits the numerical PMF rather well. While Beta-binomial RSD seems to be a good fit for numerical CRSD as shown in Fig. 2.

If $\frac{N}{M}<C<N$, we can no longer ignore the fact the number of agents within compartment, $N_{i}$, is capped. Consequently we can no longer simplify the sums in the total entry and exit rates, Eqs. (2) and (3), to a tractable form even for the simplest cases. Though it is possible to obtain stationary PMF for the specific cases by treating the model as one dimensional Markov chain or from detailed balance condition. Using the Markov chain approach we were able to find that for $T=1$ and $M=2$ PMF of $X_{i}^{\left(k\right)}$ precisely follows truncated Beta-binomial distribution (see Fig. 3). One can easily confirm this from detailed balance condition Eq. (4), which would now apply to a narrower interval $X_{i}^{\left(k\right)}\in\left[C,N^{\left(k\right)}-C-1\right]$. As the condition itself has not changed, in order to get stationary PMF in this case we just have to appropriately truncate the Beta-binomial PMF. Yet for other, more complicated cases, we were unable to discover a general pattern with either of the discussed approaches. Furthermore size of the Markov chain (the number of states) seems to grow exponentially with both $T$ and $M$. The number of recursive equations to be satisfied also seems to grow exponentially fast. Hence for realistic $N$, $T$ and $M$ it is not feasible to obtain an analytical result.

In this model we have assumed that compartments are identical, while in the real world the compartments will not be identical. In the real world there will be a natural variation in number of people in spatial units, due to historical or geographical reasons. Also number of people in empirical data sets might be larger than numerical simulations could deal with (at least in reasonable computation time). For these reasons instead of making direct comparisons with the raw $X_{i}^{\left(k\right)}$, we introduce use a scaled variable, which we refer to as a population fraction:

[TABLE]

In the empirical context closest match to the population fraction would be the vote share, which in empirical works is defined as fraction of votes case for specific party or candidate during the given election. Obtaining closed form expression for the stationary distribution of $f_{i}^{\left(k\right)}$ is quite problematic, because even in the simplest case $f_{i}^{\left(k\right)}$ is a ratio of a Beta distributed random variable and a sum of correlated Beta distributed random variables. In mathematical statistics some results are known only for a very simple combinations of independent Beta random variables [54, 55]. Nevertheless at least for a few selected cases we see that $f_{i}^{\left(k\right)}$ has a stationary distribution which is well approximated by the Beta distribution (see Fig. 4), although the distribution parameter values must be fitted on case–by–case basis and some discrepancies are noticeable. Though notably the fitted parameter values for $f_{i}^{\left(k\right)}$, we have used Maximum Likelihood Estimation to obtain them, appear to be reasonably close to $\alpha=\varepsilon^{\left(k\right)}$ and $\beta=\left(\sum_{t=1}^{T}\varepsilon^{\left(t\right)}\right)-\varepsilon^{\left(k\right)}$.

3 Comparison with selected empirical data sets

In this section we use the compartmental model to fit the empirical census and electoral data. We take three subsets of UK census 2011 data (which can be obtained from the NOMIS website111https://www.nomisweb.co.uk/query/select/getdatasetbytheme.asp) and one subset of Lithuanian parliamentary election 1992 data (which can be obtained from a dedicated GitHub repository222https://github.com/akononovicius/lithuanian-parliamentary-election-data). These subsets were selected semi–randomly, namely we have ensured that all considered “groups” of people would be quite well represented and reasonably segregated. We have rejected some other randomly selected subsets, because they were either dominated by a single “group” or were too uniformly spread out.

The first example we consider is the ethnic group distribution over the postal districts (in total $155$ of them) in London (see Fig. 5). We consider three most well represented ethnic groups (White, Asian and Black), while combining less represented groups in to the other group. Members of mixed ethnic groups were assigned to either Asian, Black or the other group. In the simulation we have used $77345$ agents to represent $4856091$ people in the data set (approximately $1:60$ ratio). Other model parameter values are listed in the caption of Fig. 5.

The second example we consider is the religious group distribution over the postal sectors ($109$ of them) in Leicester (see Fig. 6). We consider three groups: Christian, no religion and other religion. These groups were selected, because they were the most well represented. No religion other than Christianity was sufficiently well represented so all their followers were combined into the other religion group. In the simulation we have used $54391$ agents to represent $925071$ people in the data set (approximately $1:17$ ratio). Other model parameter values are listed in the caption of Fig. 6.

The third and final UK census example is based on the National Statistics Socio–Economic Classification (abbr. NS–SEC). UK census 2011 data is available in the eight–class resolution, but we down scale to the three–class resolution to ensure that each class would be well represented. This gives us four groups: higher (class 1), intermediate (class 2), lower occupations (class 3) and long–term unemployed. We have ignored data regarding persons still in education and ones who have retired. In the simulation we have used $97000$ agents to represent $902970$ people in the data set (approximately $1:9$ ratio). In Fig. 7 we have plotted the employment class distribution over the postal sectors ($194$ of them) in Sheffield. Caption of the figure lists other parameter values used in the simulation.

For the last example we take a subset of Lithuanian parliamentary election 1992 data: the vote share distribution over the polling stations ($89$ of them) in Vilnius (see Fig. 8). As was done earlier in [38] we have selected three most successful parties to have their own groups, while all other less successful parties were combined into a single group. In the simulation we have used $44500$ agents to represent $177505$ voters in the data set (approximately $1:4$ ratio). Other model parameter values are listed in the caption of Fig. 8.

As can be seen in the figures the compartmental model despite its simplicity is able to provide a rather good fits for the empirical data. Some deviations are observed due to variety of factors. Namely, we have neglected in–type heterogeneity, while not all people of the same “group” would have the same tendency to migrate independently. Also not all “classifications” of the people are equally important for the migration purposes (e.g., sex would be not important at all) or multiple “classifications” might play a role at the same time (e.g., if the person strongly identifies with both his ethnicity and religious beliefs).

4 Conclusions

In the [33] Fernandez-Garcia and coauthors have asked whether the voter model is a model for voters. Here we have proposed a simple compartmental model based on the noisy voter model with the aim to provide our answer to this question. We have assumed that agents (voters) have fixed types (cultural traits or opinions), but are able to migrate between the compartments (residential areas or electoral districts). In such model there is no actual opinion dynamics only spatial organization (migration). While exploring the general statistical properties of the proposed model we have shown that in some cases the model generates Beta distributed random variables, which is consistent with the empirical observations. To strengthen our argument we have used the compartmental model to provide a rather good fits for the empirical census and electoral data. We believe that this allows us to conclude that the spatial variations in the electoral data can arise purely from the spatial organization patterns. So, while the voter model is assumed to be a model for voters similar patterns can be recovered even without any actual opinion dynamics. Alternatively, the actual opinion exchange process could be described by a different model, maybe even a convergent model.

The proposed model is quite simple and invites variety of further explorations both from numerical and analytical perspectives. From numerical perspective it would reasonable to give the compartments some actual spatial structure and explore the arising spatial correlations. This would also allow to make comparisons to the human mobility models. While from analytical perspective it would be quite useful to establish a technique to derive the compartmental rank–size distributions, which emerge after an almost infinite time. From both perspectives an important future consideration would be to allow the agents to change their “types” and see what dynamics arise from the two competing processes (compartmental organization and opinion exchange).

Acknowledgements

Research was funded by European Social Fund (Project No 09.3.3-LMT-K-712-02-0026).