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Spatial correlations in vote statistics: a diffusive field model for decision-making

We study the statistics of turnout rates and results of the French elections since 1992. We find that the distribution of turnout rates across towns is surprisingly stable over time. The spatial correlation of the turnout rates, or of the fraction of winning votes, is found to decay logarithmically with the distance between towns. Based on these empirical observations and on the analogy with a two-dimensional random diffusion equation, we propose that individual decisions can be rationalised in terms of an underlying"cultural"field, that locally biases the decision of the population of a given region, on top of an idiosyncratic, town-dependent field, with short range correlations. Using symmetry considerations and a set of plausible assumptions, we suggest that this cultural field obeys a random diffusion equation.

factors (such as prices, reliability of the vaccine, importance of the election, etc.), and, in many cases, the decisions of others play a major role as well. Whether we like it or not, imitation is deeply rooted in living species as a strategy for survival. We, as humans, are influenced by our kindred both at a primitive level (fear of being excluded from the group) and at a rational level (others may possess some information unknown to us). Imitation can lead to collective effects like trends, fashions and bubbles that would be difficult to understand if we were insensitive to the behaviour of others. These imitation induced opinion shifts can be beneficial for a society as a whole (as in the case of vaccination), but can also be detrimental and lead to major catastrophes (crowd panic, financial crashes, economic crises, rise of extremist ideologies, etc.). Developing reliable models for these collective effects is therefore of primary importance, see [1][2][3][4][5][6][7][8]. This requires, in particular, to garner quantitative empirical information about the imitation processes that may induce strong distortions in the final outcome (see for example [9] for a precise experimental set up to measure these effects, and [10] for a theoretical framework.) In order to study the nature of these behavioural correlations, we have studied the space-resolved (town by town) results of the French elections since 1992. To keep the model and the interpretation of the results simple, we re-strict to binary choice situations, i.e, either the turnout rates for each election, or the results of yes/no referendums or second round of presidential elections. We analyse in details the statistics of these outcomes, with special focus on the dependence of the results on the size of the cities, and on the spatial correlations between the different results. While there might be some indications of direct imitation effects within towns, the structure of inter-town correlations strongly suggests the existence of what we propose to call a 'cultural' field, that evolves in time according to a noisy diffusion equation. This cultural field encodes local biases in intentions, convictions or propensities on a given subject, for example to vote or not to vote, or to vote left or right, to respect or not speed limitations, etc. etc. These (subject specific) cultural fields transcend individuals while being shaped, shared, transported and transformed by them. Although the existence of such cultural fields has been anticipated by sociologists, political scientists and geographers (see [11] for an early insight, and [12,13] for more recent discussions), we believe that our empirical results provide the first quantitative evidence for such a concept, and lay forward the possibility of a precise modelisation of its spatio-temporal evolution. Let us emphasise an important conceptual point: these cultural fields should exist independently of any election, or any other occasions where a decision has to be made. These events provide an instantaneous snapshot of the opinion or of the behaviour of individuals, which are in part influenced by these fields, in a way that we will quantify below.
The cultural field has a dynamics of its own, that we will model and elaborate on in section 4 below.
Most of the empirical electoral studies previously reported in the physics literature deal with proportional voting from multiple choice lists, and investigate the distribution of votes; as in Brazil [14][15][16][17], in Brazil and India [18], in India and Canada [19], in Mexico [20,21], in Indonesia [22]. A universal behaviour was reported in [23].
Statistical results of elections for the city mayor are studied in [24], the typology of Russian elections in [25], correlations between electoral results and party members in Germany in [26], and statistics of votes per cabin for three Mexican elections in [27]. Majority and Media effects were investigated for various countries in [28]. Lastly, the spread of Green Party in several states in the USA is analysed by means of epidemiological models in [29], and data from a Finland election is confronted to a Transient Opinion Model [30]. See also [31,32] for Political Science studies of electoral participation.
The specific feature of the present work is the quantitative analysis of the spatial correlations of the voting patterns. We will first describe several striking empirical regularities in the French vote statistics (that we believe are not restricted to French elections). We then turn to simple models that help putting these findings in context, and explain why the idea of a diffusive cultural field, which is the central proposal of this work, naturally accounts for some of our findings.

Empirical regularities and spatial correlations
We have analysed the turnout rate of all French elections [33] with national choices since 1992 (13 events).
A subset of 6 elections offered a binary choice to voters: 3 referendums and 3 second round of presidential elections (See Table 1 for details and summary statistics). The national results are broken down into ≃ 36, 000 local results, corresponding to communes (towns), of various population sizes. For each voter i, we define S i = 1 to correspond to participation to the vote or belonging to the majority vote, whereas S i = 0 corresponds to not participating, or belonging to the minority vote. For a town α, the total number of potential voters is N α ; the total number of voters is V α = Nα i=1 S i , the turnout rate is p α = V α /N α and the total number of winning votes is W α . We found convenient to work with logarithmic rates for participation or winning votes: τ α = ln(V α /N α − V α ) = ln(p α /1 − p α ) and . Each commune is characterised by the spatial coordinates [34] of its mairie (town-hall), R α .
The distance between two communes, r αβ , is defined as r αβ = |R α − R β | (even if the presence of -say -mountains or rivers in between would make the actual travelling distance much longer).
The issues at stake in all these elections are clearly very different, and so it is not a priori obvious that anything universal (across different elections) can be found in the statistics of votes. However, to our surprise, we found a number of empirical regularities that we now detail, fo- cussing first on turnout rates where these regularities are most robust. Very similar results are also found for winning votes -see below.
The simplest quantity to look at is the pdf of the logarithmic turnout rate τ over different communes, i.e. what is the probability P (τ )dτ that a given commune, irrespective of its size, has a turnout rate τ to within dτ . Although the average turnout rate m = τ varies quite substantially between elections (see Table 1), the shape of the distribution ofτ = τ − m is remarkably constant -even without rescaling by the root mean square σ, see  Table 1). In fact, a Kolmogorov-Smirnov test where one only allows for a relative shift of the distributions does not allow one to reject the hypothesis that P (τ /σ) is indeed the same for all elections (except perhaps 2009 which gives marginal results).
Note that the distribution of τ is clearly non Gaussian, with significant positive skewness and kurtosis.  The statistics of τ does in fact depend on the communesize N . We find that both the mean and variance of the conditional distribution P (τ |N ) decrease with N -see  [35]), but also a larger dispersion around the average. This is of course expected for a simple binomial process, which predicts that σ 2

A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A
whereas Σ 2 describes the 'true' variance of the turnout rate. The simplest assumption is that Σ 2 is N -independent, in which case the observed variance varies significantly faster (by a factor 2 or so) than the simple binomial prediction that assumes independent voters, see Fig. 2, inset.
A possible interpretation is that the votes of different individuals are not independent within the same commune, leading to an effectively smaller value of N in the binomial. One can for example assume that within families, or groups of close friends, the decision to vote or not to vote is exactly the same. If the size distribution of these groups of "clones" is Q(s), then it is easy to show that for N large enough, these correlations in decision amount to . ., which leads to an effective value of N reduced by a factor (1 − z)/(1 + z).
In order to account for a factor of 2 in the variation of σ 2 τ with N , we therefore need to choose z ≈ 1/3. Since z is the probability that the group is larger than s = 1, this looks quite large. In fact, as we will discuss below, there is an alternative interpretation of the excess variance that does not rely on direct imitation.
Let us now turn to the spatial correlations between turnout rates, measured by the following correlation func-tion: where m α is the average of τ over all communes with size in the same "bin" as that of α, so as to remove systematic spatial correlations between town sizes. The central result of our empirical study is that for all elections, C τ (r) is long-range correlated. It is found to decay as the logarithm of the distance (see Fig. 3, left): for 0 < r < L, C τ (r) = −λ 2 ln(r/L), and C τ (r > L) ≈ 0. Whereas the logarithmic slope λ 2 depends on the election and varies by a factor at most 2 (between 1999 and 2007) the cutoff distance L is, remarkably similar for all elections, with L ≈ 300 km. In order to visualise more directly the correlations between neighbouring towns, we have also studied the relation between τ α and the average of τ β over the 16 β towns closest to α. We find a very good linear regression between the two over the whole range of τ 's, with a slope equal to unity (not shown, see [35]).
The same general picture holds for the statistics of winning votes, except that: (a) the distribution of the logarithmic winning rate P (ρ) is much less universal across elections than P (τ ). We have noticed in particular that the skewness of P (ρ) varies significantly between elections, and changes sign between presidential elections, where it is positive, and referendums, where it is negative. Contrarily to τ | N , ρ | N does not show any systematic pattern, whereas σ 2 ρ (N ) is again significantly larger than the simple binomial prediction; (b) the spatial correlation of winning votes C ρ is also logarithmic with r (see Fig. 3, , although the 2002 result shows a more pronounced curvature. The value of the cut-off distance L is similar to the one reported above, whereas the slope λ 2 tends to be larger, except in the 2000 election (see [35] for more details, and also for more statistical regularities in these elections.)

Theoretical insights and threshold models
The long-range, logarithmic dependence of the correlation functions C τ (r) and C ρ (r) is the most striking finding of our study, in particular because it is strongly reminiscent of the behaviour of the correlation function of a free diffusion field in two dimensions. More precisely, let φ(R, t) be a two-dimensional field that obeys the following stochastic dynamical equation: where ∆ is the two-dimensional Laplacian, D is a diffusion constant and η a Langevin noise with zero mean, variance σ 2 η and short-range correlations both in time and in space.
It is well known that the equal-time correlation of φ is (in equilibrium) given by: where ℓ c is a short scale cut-off (for example the correlation length of the noise η) and L is the linear size of the system. The behaviour of C φ (r) for r < ℓ c and the logarithmic slope Λ 2 depend on short scale details of the model, but not on the the diffusion constant D. The time to reach equilibrium, beyond which the above result holds, is T eq = L 2 /D. The logarithmic behaviour is a hallmark of the two-dimensional nature of the problem. The striking similarity between this prediction and our empirical findings is the motivation of the theoretical analysis that we present now.
There is quite a large literature on binary decision models (see [1,36] for classical references, and [5,4,7,8,[37][38][39][40] for more recent contributions), although the spatial correlations that we want to include appear to be new. to each individual (for example, one may be ill on an election day, meaning that the propensity to go out and vote is very low), while others are common to people living in the same area (for example, the quality of public goods in a given town, or the unemployment rate, etc.). Finally, an important influencing factor is the decision S j made by others, if it is known before making one's own mind, 1 We are aware of a "spatial theory of political choices" (see [41]), but this is a misnomer: 'spatial' in that case refers to a distance in the abstract space of convictions.
where ǫ i (t) is the instantaneous contribution to the intention that is specific to i, and φ(R, t) is an average of the intentions of the fellow denizens expressed in a recent past. This average can be seen as a space dependent 'cultural' field which encodes all the local, stable features that influence the final decision. In essence, this component 2 Memory effects, or more complicated time dependent effects can be implemented along the lines of [43,7].  Note that without the 'cultural field' φ(R, t) and with a random static idiosyncratic term ǫ i , Eq. (4) boils down to the Random Field Ising Model [44,45], with first applications to social dynamics appearing in [46,47], see also [7,8]. For early studies and/or critical reflections in sociophysics [48], see e.g. [49][50][51][52][53][54].

2009-m
As noted above the intra-commune correlations between votes may be due to direct imitations between members of the same family (between which intentions are often shared). But the long-ranged spatial correlations cannot be due to the imitation of decision term. One reason is, as noted above, that elections are not situations where the actual decision of others can matter, since it of the discussion to follow. In fact, this randomisation can be re-absorbed in an appropriate change of P (ǫ).
is known too late. Interestingly, however, the data itself strongly rejects a model where the field φ(R, t) is shortranged correlated, while assigning the spatial correlations of τ to a coupling term J αβ between nearby communes.
The reason this model cannot be made to work is the following: for long-ranged correlations to emerge, the coupling J must be such that the system is close to its critical point J c , beyond which imitation is so strong that the solution of the coupled equations giving the {S i } becomes multi-valued. But when this is the case, the corresponding distribution of turnout rates becomes very wide, or even bimodal, and negatively skewed in a way that is incompatible with the unimodal, positively skewed and rather narrow distribution observed empirically (see Fig. 1). A detailed discussion of the inadequacy of this model in the case of elections is provided in [35]. The long-ranged correlations of τ should, we believe, be sought in the spatial structure of the cultural field φ(R, t).
From now on, we therefore neglect the direct imitation component in Eq. (4) above. We introduce the cumulative distribution of the instantaneous, idiosyncratic component the theoretical turnout rate of commune α, the realized turnout rate is given by: where ξ a standardised Gaussian noise. We will assume a logistic distribution for P (ǫ), which will make the follow-ing discussion particularly transparent, i.e.: where µ is the average of the ǫ and σ ǫ the width of P (ǫ) that we assume, for simplicity, to be constant in space and in time, and that we set equal to unity in the following. The average µ may itself depend both on space and time, and can be seen as an extra, commune specific spatial noise that adds up to the smooth cultural field φ. 4 In fact, we will subtract from φ(R) any short-range correlated component that we assign to µ(R), such that by definition, the correlation function of the φ field C φ (r) tends smoothly towards 1 when r → 0 + .
With this particular logistic choice above, one finds: Up to a shift and a noise component that vanishes when N α → ∞, τ α and φ(R α ) are now the very same object.
For other choices of the distribution of ǫ, such a strict identification is not warranted, but we expect that both object share similar statistical properties.
Within the above identification, the variance of τ α is given by: where the ratio of variances A is introduced for later con- where the index 'c' means that we take a cumulant average over all pairs of elections and all communes. If the binomial noise was the only source of the N dependence of σ 2 τ , the above covariance of the τ s at different times should be independent of N . This is not what the data shows. There is indeed a residual N -dependence that must be ascribed to the statistics of the average of idiosyncratic intentions, µ(R) (although we have no interpretation for the ∼ N −3/2 dependence that we observe. The assumption that the dispersion of individual biases, σ ǫ , is commune-independent might in fact not be warranted). Interestingly, we have found that the following relation accounts very well for the data: where B ≈ 1.5. The remarkable point of this analysis is that the coefficient of the binomial contribution is found to be exactly unity, meaning that within this interpretation one does not need to invoke the presence of intracommunes "clones", or more precisely that the probability z of herds is too small to be detectable using our data set 5 The coefficient B simply accounts for the average decay of the correlation of φ and µ as a function of time, and its value is compatible with the results shown in Fig. 5 below.

The random diffusion equation
We will now argue why it is natural to expect that the cultural field φ(R α , t) should obey a noisy diffusion equation akin to Eq.  t); furthermore, the model must be invariant under a change of scale of φ. In order to satisfy these constraints, the most general term describing the evolution of φ(R α , t) due to the surrounding influences takes the following, linear form: where Γ αβ (r αβ ) ≥ 0 is a symmetric influence matrix, that we assume to decrease over a distance corresponding to typical daily displacements of individuals, say 10 km or so. The above equation means that through human interactions, the cultural differences between nearby cities tend to narrow. As is well known, the continuum limit of the right hand side of Eq. Our final equation, that respects all the symmetries of the model 6 , is therefore (in the continuum limit): where F (t) represents the public information, for example the subject and importance of the election, national TV programs, etc. In principle, different communes may react differently to the same public information, leading to space dependent 'reactivity' ν(R), but we will neglect these spatial fluctuations in the following. 7 Similarly, we assume that the variance and higher cumulants of η are homogeneous and independent of the size of the commune, but this assumption could be relaxed if needed. Because any average trend is described by F (t), the noise η(R, t) is of zero mean. The average trend can be accounted for by a uniform shift of φ: 6 Note that φ → −φ is another symmetry of the model, corresponding to the fact that the propensity to do something is related to minus the propensity not to do the same thing, Therefore, where A is the ratio of the variance of the idiosyncratic and binomial noise to that of the cultural field, defined in Eq. This means that coarse graining on sufficiently large scales, larger than the correlation length of D(R), the effective equation becomes identical to Eq. (2) with an effective diffusion constant D that can be computed using e.g. perturbation theory, or effective medium approximations [55].
What is of interest for our discussion is that the logarithmic dependence of C φ (r), Eq. (3), is still valid on large enough scales.
In order to test these ideas more quantitatively, we have simulated the model using Eq. (11)  ing equilibration time T eq over L = 300 km is therefore quite long, a century or more. But this is compatible with the fact that the voting habits of the different regions seem to be extremely persistent in history -which is in itself a strong argument for the existence of a cultural field that keeps the memory independently of the presence of partic-ular individuals. This justifies the apparent correlation between geography and votes discussed long ago by Siegfried [11] (Le granite voteà droite, le calcaire voteà gauche). Interestingly, a more precise prediction of our model is that the temporal autocorrelation of φ for a given R should behave as: where the slope is exactly one half of the slope of C φ (r) versus − ln r, and T c the correlation time of the noise η.
We have tested this prediction directly on data, by study-  In any case, in the absence of more information on the dynamics of the µ field, we find the overall agreement between the model and the behaviour of these temporal autocorrelations satisfying.

Conclusion
Let us summarise the main messages of this study. First, the statistics of the turnout rates in French elections is found to be surprisingly stable over time, once the average turnout rate is factored in. The size dependence of the turnout rate variance may have suggested some intra city 'herding' effect, but we believe that the data is more consistent with small towns having a larger dispersion in local, idiosyncratic biases. A convincing argument for why this should be so is however lacking. An explanation could be the systematic difference in the socio-cultural background of the population of large cities compared to that of small cities.
Second, for our whole set of elections, the spatial correlation of the turnout rates, or of the fraction of winning votes, is found to decay logarithmically with the distance between towns. This slow decay of the correlations is characteristic of a diffusive random field in two dimensions.
This result is robust against many non essential modifica- the idiosyncratic field, are clearly needed at this stage. We hope that our work will motivate more empirical studies to refine and calibrate the model proposed here.