Mean Field Voter Model of Election to the House of Representatives in Japan
aa r X i v : . [ phy s i c s . s o c - ph ] F e b Mean Field Voter Model of Election to the House ofRepresentatives in Japan
Fumiaki S ano , Masato H isakado , and Shintaro M ori Department of Physics, Faculty of Science, Kitasato UniversityKitasato 1-15-1, Sagamihara, Kanagawa 252-0373, JAPAN Fintech Lab. LLC, Meguro, Tokyo 153-0051, JAPANE-mail: [email protected] (Received October 31, 2016)In this study, we propose a mechanical model of a plurality election based on a mean field votermodel. We assume that there are three candidates in each electoral district, i.e., one from the rulingparty, one from the main opposition party, and one from other political parties. The voters are clas-sified as fixed supporters and herding (floating) voters with ratios of 1 − p and p , respectively. Fixedsupporters make decisions based on their information and herding voters make the same choice asanother randomly selected voter. The equilibrium vote-share probability density of herding votersfollows a Dirichlet distribution. We estimate the composition of fixed supporters in each electoraldistrict and p using data from elections to the House of Representatives in Japan (43rd to 47th). Thespatial inhomogeneity of fixed supporters explains the long-range spatial and temporal correlations.The estimated values of p are close to the estimates obtained from a survey. KEYWORDS: correlation, Dirichlet distribution, mean field voter model, plurality election data
1. Introduction
Social phenomena are an active research field in econophysics and socio-physics, and many stud-ies have aimed to deepen our understanding of them [1–4]. Opinion dynamics is a central researchtheme and voter models have been studied extensively as a paradigm of opinion dynamics [5–8].Recently, the validity of a model was tested for describing the real opinion dynamics in the U.S.presidential election [9], where it was concluded that a noisy di ff usive process of opinions can repro-duce the statistical features of elections, i.e., the stationary vote-share distributions and long-rangespatial correlation, which decay logarithmically with distance. The model is simple and attractive inthe domain of physics.In the model, the decisions made by all the voters are described by an infection mechanism. Twovoters are selected randomly and one voter’s choice is made the same as another’s choice, with somenoise. The spatial inhomogeneity of the system is considered by using data to define the initial con-ditions. The noise level has to be fine tuned in order to simulate the statistical properties of elections,particularly maintaining the initial spatial inhomogeneity; otherwise, the spatial pattern disappears orthe spatial correlation becomes too strong and the model cannot reproduce the statistical propertiesof the election data. However, it is necessary to avoid making subtle choices for the parameters whenmodeling a social system. If people interact with others and make decisions in a social system, theprocess should be robust and stable. This is the first problem that needs to be addressed.The second problem is estimating the ratio of floating voters who do not vote for a particularpolitical party. In the voter model described above, the opinions of all the voters can be changedby interactions with other voters. However, this approach is too simple to represent the situation inactual elections. Some people prefer a certain political party and it is reasonable to assume that their pinions will not be changed easily by interacting with others. Nowadays, many people recently havebecome dissatisfied with the current status of the political system and they have no particular politicalparty to vote for [11]. They are called floating voters and it is considered that they have a decisive rolein the results of elections. If we assume that social interactions play a crucial role in their decisions[3, 9, 10], then they should be studied in the framework of a voter model.In the present study, we propose a mean field voter model to describe the dynamics of a pluralityelection in Japan. In the model, voters are classified as fixed supporters who have a preference fora specific political party and floating voters whose decision depends on the choices of others. Theinfluence of other voters on floating voters is described by the voter model mechanism. The voteshare distribution of floating voters follows a Dirichlet distribution and the system is stable. We showhow to decompose the vote share into votes by fixed supporters and those by floating voters, and weestimate the spatial inhomogeneity of the electoral system. We study the fluctuations in the vote shareof floating voters and the ratio of floating voters is estimated. We explain the spatial correlation andtemporal correlation in terms of the spatial inhomogeneity in the fixed supporters.
2. Mean Field Voter Model
Fig. 1.
Composition of voters. Voters are classified as fixed supporters in the inner circle and floating votersin the area surrounded by inner and outer circles. The number of fixed (floating) voters is N S ( N F ). We assume K = ff ected by the choices of other voters. They only a ff ect the choices of floating supporters. The red arrowshows the influence from a fixed supporter of political party 1 to a floating supporter of political party 2. Bycontrast, the choices of floating supporters are influenced by those made by both fixed supporters and floatingsupporters. The blue (green) arrow shows the influence from a floating supporter of political party 2 (3) to afloating supporter of political party 3 (2). There are K political parties, I electoral districts, and T elections. In each election, K candi-dates fight for a single congress seat in each electoral district. We denote the political parties as k ∈{ , , · · · , K } , electoral districts as i ∈ { , , · · · , I } , and elections as t ∈ { , , · · · , T } . There are N ( t , i )votes in district i for election t , and N ( t , i , k ) voters vote for political party k . N ( t , i ) = P k N ( t , i , k )holds. N ( t , i , k ) are classified as N F ( t , i , k ) floating (herding) voters and N S ( t , i , k ) fixed supporters ho vote for political party k (Figure 1). N ( t , i , k ) = N F ( t , i , k ) + N S ( t , i , k ) and we denote the totalfloating voters and supporters as N F ( t , i ) ≡ P k N F ( t , i , k ) and N S ( t , i ) ≡ P k N S ( t , i , k ), respectively.We write the ratio of floating voter as p ( t , i ), p ( t , i ) ≡ N F ( t , i ) / N ( t , i ). We denote the vote share forpolitical party k as Z ( t , i , k ) ≡ N ( t , i , k ) / N ( t , i ). Z ( t , i , k ) is then decomposed as Z ( t , i , k ) = (1 − p ( t , i )) · N S ( t , i , k ) / N S ( t , i ) + p ( t , i ) · N F ( t , i , k ) / N F ( t , i ) . We introduce µ ( t , i , k ) ≡ N S ( t , i , k ) / N S ( t , i ), which represents the vote share of political party k among N S ( t , i ) fixed supporters. The ratio N F ( t , i , k ) / N F ( t , i ) is denoted as X ( t , i , k ), which represents the voteshare of political party k among N F ( t , i ) floating voters. Z ( t , i , k ) = (1 − p ( t , i )) · µ ( t , i , k ) + p ( t , i ) · X ( t , i , k ) . (1)We assume that the choices of fixed supporters do not change and that the choices of floating votersare described by a voter model. In each turn, a randomly selected floating voter changes his choice tothat of another randomly selected voter. The fixed supporters only a ff ect the choices made by floatingvoters. We also assume that among N S fixed supporters, only φ, ≤ φ ≤ N S supporters can a ff ectthe choices of floating voters. As shown in the following, φ controls the fluctuations in X ( t , i , k ) from µ ( t , i , k ).At equilibrium, K variables X ( t , i , k ) with values in the standard unit interval (0 ,
1) add up to 1, P k X ( t , i , k ) =
1, which constrains the sample space of K dependent variables to a K − X ( t , i , K ) can always be omitted due to X ( t , i , K ) = − P K − k = X ( t , i , k ). Weuse the abbreviated form of ~ X ( t , i ) = ( X ( t , i , , · · · , X ( t , i , K )) and ~µ ( t , i ) = ( µ ( t , i , , · · · , µ ( t , i , K )).The probability density of ~ X ( t , i ) is described by a Dirichlet distribution, which is given by Eq. (2).We employ the alternative parametrization of the Dirichlet distribution introduced by Ferrari andCribari-Neto [12]. The derivation is given in Appendix B. P ( ~ x | ~µ ( t , i ) , φ ) = φ~µ ( t , i )) Y k x µ ( t , i , k ) φ − k . (2)The denominator B(( φ~µ ( t , i ))) in eq.(2) is the multinomial beta function, which serves as a normal-ization constant, and it is defined asB(( φ~µ ( t , i ))) ≡ Γ ( φ ) Y k Γ ( φµ ( t , i , k )) . Γ ( · ) is the gamma function defined as Γ ( x ) = R ∞ t x − e − t dt .We write the random variable ~ X ( t , i ) obeys the Dirichlet distribution in the alternative parametriza-tion ~µ ( t , i ) , φ as ~ X ( t , i ) ∼ D a ( ~µ ( t , i ) , φ ) . Each component X ( t , i , k ) of ~ X ( t , i ) is marginally beta-distributed with α = φ · µ ( t , i , k ) and β = φ · (1 − µ ( t , i , k )). The expected values are defined as E( ~ X ( t , i )) = ~µ ( t , i ), the variances are V( X ( t , i , k )) = µ ( t , i , k )(1 − µ ( t , i , k )) / ( φ + X ( t , i , k ) , X ( t , i , l )) = − µ ( t , i , k ) µ ( t , i , l ) / ( φ + φ is a “precision” parameter to model the dispersion of the variables. In the context of the mean fieldvoter model, a small φ indicates that the influence is weak and the fluctuations in X ( t , i , k ) are large.If φ is large, the influence is strong and the fluctuations in X ( t , i , k ) from µ ( t , i , k ) are suppressed, and X ( t , i , k ) almost coincides with µ ( t , i , k ). able I. House of Representatives (general) Election in Japan. We use data from the 43rd to 47th pluralityelections. t date Districts Regions Voters Votes Ruling Political Party1 2003 / / . × . × LDP2 2005 / /
11 300 2534 1 . × . × LDP3 2009 / /
30 300 2037 1 . × . × LDP4 2012 / /
16 300 1994 1 . × . × DPJ5 2014 / /
14 295 1983 1 . × . × LDP
3. Data Analysis
We employ data from elections to the House of Representatives (general election) in Japan. Thisis a plurality election and there are about 300 electoral districts, where several candidates compete fora single congress seat. We use data of T = t = , , , , and 5 in Table I. During this period, the Liberal Democratic party ofJapan (LDP) and the Democratic party of Japan (DPJ) were elected as the ruling party in t = , , t = k = k =
2. There might be several candidates from other politicalparties (OPP), we combine them together and treat them as one candidate. We label them as k = Fig. 2.
Decomposition of Z ( t , i , k ). Z ( t , i , k ) are decomposed as Z ( t , i , k ) = (1 − p ( t , i )) µ ( t , i , k ) + p ( t , i ) · X ( t , i , k )in Eq. (1). µ K ( k ) is the fitness of political party k and µ K ( k ) + ∆ µ I ( i , k ) is the fitness of a candidate from politicalparty k in region i . ∆ µ T ( t , k ) represents the trend for political party k in election t . We use data from the regions where the above three candidates fought for a single seat five timesin a row due to the necessity to infer µ ( t , i , k ) in the mean field voter model. There are I =
488 regions,which we label as i = , , · · · , I . We denote the vote share for candidate k in region i in election t as Z ( t , i , k ). P k Z ( t , i , k ) = i in election t . In order to decompose Z ( t , i , k ) as described able II. Symbol legend K , k Number of political parties and their index T , t Number of elections and time variable I , i Number of regions and their index N ( t , i ) Number of votes in region i for election tN ( t , i , k ) N ( t , i ) for political party kZ ( t , i , k ) ≡ N ( t , i , k ) / N ( t , i ) Votes share for political party k in region i for election tN F ( t , i ) , N S ( t , i ) Number of floating voters and fixed supporters in region i , election tp ( t , i ) = N F ( t , i ) / N ( t , i ) Ratio of floating voter N S ( t , i , k ) N S ( t , i ) for political party k µ ( t , i , k ) ≡ N S ( t , i , k ) / N S ( t , i ) Vote share of political party k among N S ( t , i ) fixed supporters µ K ( k ) ≡ P i , t Z ( t , i , k ) / IT Overall fitness of political party k ∆ µ T ( t , k ) ≡ P i Z ( t , i , k ) − µ K ( k ) Temporal deviation of fitness from µ K , ∆ µ I ( i , k ) ≡ P t Z ( t , i , k ) − µ K ( k ) Regional deviation of fitness from µ K , N F ( t , i , k ) N F ( t , i ) for political party kX ( t , i , k ) ≡ N F ( t , i , k ) / N F ( t , i ) Vote share of political party k among N F ( t , i ) floating voters φ Number of fixed supporters who can a ff ect the choices of floating voters. in Eq. (1), we write Z ( t , i , k ) as Z ( t , i , k ) = µ ( t , i , k ) + ( Z ( t , i , k ) − µ ( t , i , k )) ,µ ( t , i , k ) ≡ µ K ( k ) + ∆ µ T ( t , k ) + ∆ µ I ( i , k ) ,µ K ( k ) ≡ IT X t X i Z ( t , i , k ) , ∆ µ T ( t , k ) ≡ I X i Z ( t , i , k ) − µ K ( k ) , ∆ µ I ( i , k ) ≡ T X t Z ( t , i , k ) − µ K ( k ) .µ K , ∆ µ T , ∆ µ I shows the di ff erent types of fitness values for the politicians from political party k .Please refer to Figure 2. µ k shows the overall fitness, and ∆ µ T and ∆ µ I show the temporal and regionaldeviation from µ K , respectively. µ ( t , i , k ) shows the fitness of a candidate from political party k inregion i in election t .We summarize the notations in Table II. The decomposition is similar to ANOVA (analysis ofvariance ) in statistics. There are three factors, k , i and t and we assume there is no interaction e ff ectamong them and µ ( t , i , k ) are estimated as the sum of the three factors, µ K ( k ) , ∆ µ T ( t , k ) and ∆ µ I ( i , k ).In the next section, we use the maximum likelihood principle to estimate the model parameters. Basedon the results, we test the validity of the mean field voter model.
4. Results
We assume that the floating voter ratio p ( t , i ) does not depend on region i and write it as p ( t ). Inthe mean field voter model, ~ Z ( t , i ) are decomposed as, ~ Z ( i , t ) = (1 − p ( t )) · ~µ ( t , i ) + p ( t ) · ~ X ( t , i ) ,~ X ( t , i ) ∼ D a ( ~µ ( t , i ) , φ ) . (3)There are T + φ, p , p , · · · , p T ) in the model, which we estimate using the maximumlikelihood principle. When we apply the maximum likelihood principle, some X ( i , t , k ) becomes neg-ative for small values of p . In this case, we assume that p = φ is estimated as 10 . ± . II shows the results for p , · · · , p T in the first row. We also show the number of cases where we em-ployed p = I =
488 cases, at least one of X ( t , i , k ) for each i becomes negative by the choice p ( t ) in election t . Thus, the estimate of µ ( t , i , k ) is not appropriate orthe assumption that p ( t , i ) is independent of i is too crude.We also show the estimates obtained by Japan Broadcasting Corporation (NHK) in the third row,which are based on a public poll of more than 1000 people [14]. According to the poll, 25% to 39%are estimated as floating voters. The estimate obtained by the mean field voter model is in the sameorder. µ vs Z and standardized X R Fig. 3.
Scatter plot of ~µ ( t , i ) vs ~ Z ( t , i ) for all data t = , , , ,
5. The red symbol denotes the plots for k = k =
2, and the green symbol indicates those for k = We show the scatter plots for µ ( t , i , k ) and Z ( t , i , k ) in Figure3. If the decomposition in Eq. (1)is correct, then Z ( t , i , k ) is distributed around µ ( t , i , k ). As shown in Figure 3, Z ( t , i , k ) for k = k = Z ( t , i , k = X ( t , i , k ), whichis denoted as X R ( t , i , k ). If the model is correct and ~ X ( t , i ) ∼ D a ( ~µ ( t , i ) , φ ) holds, then V( X ( t , i , k )) = Table III.
Maximum likelihood estimate for p ( t ). The first row shows the results for p ( t ) and the second rowshows the number of cases where we assume that p = I = t p ( t ) 0 . ± . . ± . . ± . . ± . . ± . p = .
387 0 .
256 0 .
295 0 .
335 0 . ig. 4. Probability density functions for renormalized X R ( t , i , k ) in Eq. (4), (a) k = k = k = . , .
74, and 1 .
59 for k = ,
2, and 3, respectively. µ ( t , i , k )(1 − µ ( t , i , k ) / ( φ + X R ( t , i , k ) is defined as X R ( t , i , k ) ≡ p φ + X ( t , i , k ) − µ ( t , i , k )) p µ ( t , i , k )(1 − µ ( t , i , k )) . (4)Figure 4 shows the probability densities for X R ( t , i , k ) with k = , ,
3, respectively. The variances in X R ( t , i , k ) are estimated as 0 . , .
74, and 1 .
59 for k = , ,
3, respectively. As shown in Figure 4, thedistribution with k = k = , X R does not follow a normal distribution, sothe asymmetrical nature of the distribution is not crucial. However, the variance with k = k = ,
2, which suggests that the decomposition in Eq. (1) is not good. We discussimprovements to the proposed model in the conclusion.
Fig. 5. (a) Spatial vote-share correlations and µ correlations as a function of distance. The solid line showsthe spatial correlation for the vote share Z ( t , i , k ) and the dashed lines indicate the spatial correlations for ∆ µ I ( i , k ). (b) Temporal vote-share correlation for Z ( t , i , k ) and (c) that for X ( t , i , k ) as a function of time. We estimate the spatial and temporal correlations. The spatial correlation function is computedas C ( r , t ) = ( h Z ( t , i , k ) Z ( t , j , k ) i | r ( i , j ) = r − µ T ( t , k ) ) / V( µ ( t , k )) , ( r ) = T T X t = C ( r , t , k ) ,µ T ( t , k ) ≡ X i Z ( t , i , k ) / I , V( µ ( t , k )) ≡ X i ( Z ( t , i , k ) − µ T ( t , k )) / I (5)The first term h Z ( t , i , k ) Z ( t , j , k ) i | r ( i , j ) = r is averaged over pairs of regions separated by a distance r .The spatial correlation decays logarithmically with geographical distance [Figure 5(a)], as reportedin previous studies. The logarithmic decay of the spatial correlations is considered generic for thefluctuations in electoral dynamics [9]. In the figure, we also plot the correlation functions for ∆ µ I ( i , k ),which clearly almost coincide with C ( r ) for Z ( t , i , k ). This suggests that the physical description ofthe spatial correlation is spatial inhomogeneity in ∆ µ I . Every region has certain characteristics, suchas the type of the region and its historical nature. These characteristic do not change rapidly even ifsome people move between regions. If all the voters are floating voters [9], the di ff usion and noiseshould be controlled to maintain the inhomogeneity in ∆ µ I . In our model, the voters are classified asfixed supporters and floating voters, and there is no movement of people among di ff erent regions, soit is not necessary to consider this issue.Figure 5(b) shows the results for the temporal correlation, C ( ∆ t ), which is computed as C ( ∆ t ) = Cov( Z (1 , i , k ) , Z (1 + ∆ t , i , k )) / p V( Z (1 , i , k ))V( Z (1 + ∆ t , i , k )) . (6)Here Cov( A ( i ) , B ( i )) is defined as P i A ( i ) B ( i ) / I − P i A ( i ) / I · P j B ( j ) / I . C ( ∆ t ) does not decay with ∆ t ,which can be explained by the decomposition in Eq. (1). The covariance of Z (1 , i , k ) and Z (1 + ∆ t , i , k )is decomposed into the covariance of µ ( t , i , k ) and µ ( t + ∆ , i , k ), and the covariance of X ( t , i , k ) and X ( t + ∆ , i , k ).Cov( Z ( t , i , k ) , Z ( t + ∆ t , i , k )) = (1 − p (1))(1 − p (1 + ∆ t ))Cov( µ (1 , i , k ) , µ (1 + ∆ t , i , k )) + p (1) p (1 + ∆ t )Cov( X ( t , i , k ) , X ( t , i , t + ∆ t )) . We assume that µ ( t , i , k ) and X ( t , i , k ) are independent from each other. If we further assume that thecorrelation between X (1 , i , k ) and X (1 +∆ t , i , k ) decays with ∆ t , as ∆ µ K ( k ) and ∆ µ T ( t , k ) do not dependon i , we have Cov( µ (1 , i , k ) , µ (1 + ∆ t , i , k )) = V( ∆ µ I ( i , k )) . Then, the temporal correlation C ( ∆ t ) for a large value of ∆ t is approximately expressed as C ( ∆ t ) = (1 − p (1))(1 − p (1 + ∆ t )) · V( ∆ µ I ( i , k )) √ V( Z (1 , i , k ))V( Z (1 + ∆ t , i , k )) , (7)which suggests that the physical origin of the temporal correlation is also the spatial inhomogeneityof Z I .We also check the decomposition in Eq. (1) by studying the temporal correlation of X ( t , i , k ).Figure 5(c) shows it, which is defined by replacing Z ( t , i , k ) with X ( t , i , k ) in eq. (6). C ( ∆ t ) decayswith ∆ t and it almost vanishes for ∆ t =
2. However, C ( ∆ t ) becomes negative for ∆ t ≥ k = , Z ( t , i , k ) should be performed.
5. Conclusions
In this study, we proposed a mean field voter model for a plurality election. We assumed thatvoters are classified as fixed supporters and floating voters, where the behavior of the latter can bedescribed by the voter model’s infection mechanism. We decomposed the vote share ~ Z into that fromthe fixed supporters ~µ and that from the floating voters ~ X , ~ Z = (1 − p ) · ~µ + p · ~ X . ~µ has three factors µ K , ∆ ~µ T and ∆ ~µ I . ~ X follows a Dirichlet distribution with the parameters φ, ~µ . We estimated the modelparameter p , φ, ~µ , where we used electoral data from the House of Representatives elections in Japanduring 2003–2014. There we assume that p does not depend on the region and there is no interactione ff ect among the three factors in ~µ . Using the estimated parameters, we decomposed ~ Z and checkedthe validity of the proposed model. • The variances in the standardized ~ X R are 0 . , .
74, and 1 .
59 for k = , ,
3, respectively (Figure4). • The spatial correlation functions for ~ Z are described by those for ∆ µ I ( i , k ) (Figure 5(a)). • The temporal correlation function for ~ Z is explained by the variance in ∆ µ I ( i , k ) (Figure 5(b) andEq. (7)).These results show that the decomposition of ~ Z = (1 − p ) ~µ + p ~ X is e ff ective for simulating thestatistical nature of the electoral data. However, the decomposition should be performed carefully toverify the validity of the mean field voter model. We estimated ~µ using some average values of ~ Z ,which should be replaced with the maximum likelihood estimates. In addition, the assumption of theindependence of p ( t , i ) on i is also excessively crude and it is necessary to consider their regionaldependencies. These issues should be addressed in future studies. Appendix A: Election Data
We provide further information regarding the data discussed in the main text. We used the resultsof House of Representatives elections from 2003 (43rd) to 2014 (47th), which were aggregated intoelectoral districts [13]. In FigureA · · ff erent parties in the regions considered in themain text. There are I =
488 regions. The shares were computed region-by-region, and we thenextracted the averages and standard deviations.
Fig. A · Japanese general election results. (a) Global trends in the absolute values of di ff erent quantities,such as the turnout, and the votes for LDP, DPJ, and OPP. (b) Changes in the shares associated with the turnoutand votes for di ff erent parties in I =
488 regions where more than three candidates fought for a single seat fivetimes in row. 9 igure A · I =
488 electoral regions during five generalelections.
Fig. A · Schematic representation of the election results for the House of Representatives in 2003–2014.The winning political party is shown for I =
488 regions where candidates from more than three politicalparties fought four a single congress seat in all elections. The political parties are shown in red (LDP), blue(DPJ), and green (OPP).
Appendix B: ~ X ∼ D a ( ~µ, φ ) We derive the Dirichlet distribution for ~ X . There are N F floating voters and N i voters for politicalparty i . N F = P i N i . We write the probability function as Pr( ~ N = ~ n ) = P ( ~ n ). The proportion of fixedsupporters ~µ a ff ects the choices of the floating voters because there are φ~µ voters. The probabilitythat a randomly selected voter is a floating voter with choice i is n i / N F . The probability that anotherrandomly chosen voter makes the choice j is ( φµ j + n j ) / ( φ + N F − n i , n j ) → ( n i − , n j +
1) for any pair ( i , j ) , ≤ i , j ≤ K is given asPr(( n i , n j ) → ( n i − , n j + = n i N F · φµ j + n j N F + φ − . Similarly, the transition probability from state ( n i − , n j + → ( n i , n j ) is given asPr(( n i − , n j + → ( n i , n j )) = n j + N F · φµ i + n i − N F + φ − . he detailed balance condition for the stationary P ( ~ n ) can be solved easily and we obtain P ( ~ n ) = N F ! Q j n j ! Q j ( φµ j ) [ n j ] θ [ N F ] , where x [ j ] ≡ x ( x + · · · ( x + j − N F → ∞ , the probability function P ( ~ n ) becomes aDirichlet distribution. P ( ~ x ≡ ~ n / N F ) = lim N F →∞ P ( ~ n ) N F φ~µ ) Y k x µ k φ − k . References [1] R. N. Mantegna and H. E. Stanley, Introduction to Econophysics: Correlations and Complexity in Finance(Cambridge University Press, Cambridge, 2007).[2] A. Pentland, Social Physics: How good ideas spread (Penguin Press, 2014).[3] P. Ormerod, Positive Linking, (Faber & Faber, 2012).[4] S. Mori, K. Nakayama and M. Hisakado : Phys. Rev. E. (2016)052301.[5] C. Castellano, S. Fortunato, and V. Loreto: Rev. Mod. Phys. (2009)591.[6] S. Mori and M. Hisakado: J. Phys. Soc. Jpn. (2010)034001.[7] M. Hisakado and S. Mori, J. Phys. A (2010)315207.[8] S. Mori, M. Hisakado, and T. Takahashi: Phys. Rev. E. (2012)026109.[9] J. Fernandez-Gracia, K. Suchecki, J. J. Ramasco, M. San Miguel, and V. M. Egu´ıluz: Phys. Rev. Lett (2014)158701.[10] N. A. Arajo, J. S. Andrade, and H. J. Herrmann: PLoS One (2010)e12446.[11] L. Killan, The Swing Vote: The Untapped Power of Independents (St. Martin’s Press, 2012).[12] S. P. L. Ferrari and F. Cribari-Neto: Journal of Statistical Software (2004)799.[13] S. Mizusaki and Y. Mori, JED-M Ver 3.2 28th-47th general elections regional data (LDP press, 2015)[14] Japan Broadcasting Corporation (NHK) conducts public opinion polls by telephone every month to ex-amine political consciousness of the people. The survey target is male and female over the age of 20nationwide and the investigation method is the telephone method (RDD tracking method). One can get thedata from the website, https: // / bunken / research / yoron / political //