# Effect of Heterogeneity in Initial Geographic Distribution on Opinions’ Competitiveness

^{1}

^{2}

^{*}

## Abstract

**:**

_{tr}= 3.30 ± 0.05 update cycles, and then the system evolves in a two-state regime with complementary spatial distributions of two competing opinions. Even so, the initial heterogeneity in the spatial distribution of one of the competing opinions causes a decrease of this opinion competitiveness. That is, the opinion with initially heterogeneous spatial distribution has less probability to win, than the opinion with the initially uniform spatial distribution, even when the initial concentrations of both opinions are equal. We found that although the time to consensus T

_{C}>> T

_{tr}, the opinion’s recession rate is determined during the first 3.3 update cycles. On the other hand, we found that the initial heterogeneity of the opinion spatial distribution assists the formation of quasi-stable regions, in which this opinion is dominant. The results of Monte Carlo simulations are discussed with regard to the electoral competition of political parties.

## 1. Introduction

## 2. Model and Details of Numerical Simulations

#### 2.1. Spatial Distributions of Competing Opinions

_{j}, where j specifies the spatial (geographic) position of the cell (j = 1, 2, …, K). Per definition ${\sum}_{j=1}^{K}{n}_{j}}=N$. Let us assign the spin s = 1 to N

_{1}nodes and spin s = −1 to N

_{−1}nodes, whereas the rest N

_{0}= (N − N

_{1}− N

_{−1}) nodes remain in the neutral state with spin s = 0. A natural measure of heterogeneity in the spatial spin distribution is the entropy of the distribution defined as:

_{sj}is the number of spins s in the cell j with n

_{j}nodes, p

_{sj}= N

_{sj}/n

_{j}is the “concentration” of spin s in the cell j, and p

_{s}= N

_{s}/ N is the overall concentration of spin s on the whole network. Notice that, per definition, p

_{1}+ p

_{−1}+ p

_{0}=1. In the case of random uniform distribution ${\overline{p}}_{sj}=1/K$ and so S

_{s}= ln K, whereas inhomogeneous spatial distributions are characterized by S

_{s}< ln K.

#### 2.2. Details of Monte Carlo Simulations

_{i}= 1, −1, or 0. In the voter network, the adepts of rival opinions (parties) can be associated with non-zero spins (s =1 or −1), whereas the nodes with spin s = 0 can be associated with uncommitted voters. In order to avoid correlations between initial spatial distributions of the opposite spins, in all simulations performed in this work, the positive spins were initially assigned to 10 randomly chosen nodes in each of N cells, such that p

_{1}j (t = 0) = p

_{1}(t = 0) = 0.1 and ${\overline{p}}_{1j}(t=0)=1/K$. Afterward, the negative spin was assigned to M

_{−1}randomly chosen nodes (0 ≤ M

_{−1}≤10) in each of k randomly selected districts, whereas N/10 − kM

_{−1}negative spins were uniformly distributed among the rest K − k districts. Accordingly, the rest 0.8N nodes are initially in the state with s = 0 (see Figure 1a,c).

_{−1}= M

_{−1}/ m

_{−1}is the ratio of the distribution’s median (M

_{−1}) to its mean (m

_{−1}). Therefore, S

_{1}= ln K, whereas the entropy (1) of the negative spin distribution (3) is an univalued function of the heterogeneity index H

_{−1}, as it is shown in Figure 2.

^{6}simulations were performed with each initial spin distribution on lattice of size N = 2500. Additionally, the Monte Carlo simulations were performed for some initial spin distributions on lattices of other sizes.

_{tr}); (2) the system magnetization Δp(t) = p

_{1}− p

_{−1}; (3) the entropy (1) at the transition from the three- to two-state regime; (4) the time to consensus (T

_{C}= t

_{C}/ N); and (5) the consensus probabilities P

_{1}and P

_{−1}=1− P

_{1}.

## 3. Results and Discussion

- What happens with the three-state regime during the majority rule system evolution? How does the number of update steps needed to disappear all the neutral state (t
_{tr}) depend on the lattice size and the initial distributions of opposite spins? Does the system have a non-zero magnetization at the transition from the three- to two-state regime? If yes, how does the magnetization depend on the initial heterogeneity in the spatial distributions of the opposite spins? How are the opposite spins distributed among cells once the last neutral state has disappeared? - How does the heterogeneous system to consensus in the two-state regime? How does the heterogeneity in the initial spatial distributions affect the time to consensus T
_{C}in the different scenarios of evolution? - How does the heterogeneity in the initial spatial distribution of one of two rival opinions affect its own probability to win the competition?

_{tr}= t

_{tr}/ N = 3.30 ± 0.05 cycles of N updates, independently of the lattice size and heterogeneity in the initial spatial distributions of the rival opinions. Furthermore, we found that the magnetization at the transition from the three- to two-state regime Δp

_{tr}= p

_{1}(t

_{tr}) − p

_{−1}(t

_{tr}) increases as the heterogeneity of the negative spin distribution increases, even when the initial magnetization of the system was zero (Δp

_{0}= p

_{1}(0) − p

_{−1}(0) = 0). Specifically, we found that the data of numerical simulations are best fitted by the following relationship:

_{0}= H

_{1}(0) − H

_{−1}(0), while a = 0.17 and β = 1.47 ± 0.07 are the fitting constants (see Figure 3a). Likewise, we found that the heterogeneities of spin distributions at the transition from the three-state to the two-state regime are characterized by entropies S

_{1}(t

_{tr}) ≥ S

_{−1}(t

_{tr}) and heterogeneity indexes H

_{1}(t

_{tr}) ≥ H

_{−1}(t

_{tr}), where the equalities are achieved only if H

_{−1}(0) =1 (see Figure 3b). It is pertinent to point out that these results were reproduced on lattices of different sizes.

_{tr}= (3.30 ± 0.05)N updates in the three-state regime (see Figure 1d) differ from the spin distributions after 3.3N updates in the classic two-state majority rule model. The last is similar to the spin distributions after t

_{0}updates in the three-state system with H

_{−1}(0) = H

_{1}(0) =1 (see Figure 1b).

_{C}= t

_{C}/ N cycles of updates. In the remaining realizations, the system reaches a long-living non-consensus configuration associated with segregation of alternative states (opinions), into two or more nearly straight stripes (see Figure 5 in [20]). These stripes are ultimately unstable and so consensus is always reached, albeit very slowly. The authors of [20] have suggested two ways of automatic detection of stripe configurations. One is based on the analysis of correlation functions for two spins that are located a distance $\sqrt{N}/2$ apart (stripe state arises if the correlation function in the direction(s) parallel to the stripe is greater than a threshold value about 0.5). Alternatively, all realizations with t

_{C}> t

_{S}were counted as reaching a coherent state [20]. Specifically, in simulations on lattices of size N = 2500 with G = 3 it was found that t

_{S}= 600 [20]. Both methods lead to essentially the same results. In this work, we confirm this result in the case of G = 3 and found that in the case of discussion group size G = 5; all realizations with t

_{CS}> 210 updates are characterized by the stripe configurations. In this way, we found that in the case of uniform initial distributions of both positive and negative spins (that is H

_{−1}(0) = H

_{1}(0) =1) non-consensus long-living configurations (stripes) are formed with probability P

_{S}= 30% ± 3%, as this is in the case of the two-state majority rule model [20], as well as in the zero-temperature evolution of the Ising model with Glauber kinetics [22,23]. We also found that the characteristic time to consensus in systems with long-living stripes scales with the system size as:

_{S}= 33% and quickly decreases as the difference between initial spin densities Δp(0) increases (see Figures 8 and 9 from [20]). Controversially, in our simulations we found that the probability to form a non-consensus coherent state increases with the increase of ΔH

_{0}(see Figure 4a) as:

_{tr}is also the increasing function of ΔH

_{0}(see Figure 3a). Accordingly, although the increase of P

_{S}with increase of Δp

_{tr}is rater small, the tendency P

_{S}= 0.4 − 0.023ln(Δp

_{tr}) clearly differs from the quick decrease of P

_{S}as the initial magnetization Δp

_{0}increases in the two-state majority rule model (see Figure 4a). This unexpected reverse behavior of P

_{S}can be associated with the characteristic spatial heterogeneity of the opposite spin distributions at the starting of the two-state regime in the three-state majority rule model (see Figure 3a). This means that the effect of heterogeneity in the spin spatial distributions at the transition point exceeds the effect of magnetization at this point.

_{0}and consistent with the value α = 1.24 found in the numerical simulations of the two-state majority rule model on square lattices with G = 3 [20]. Furthermore, we found that the mean time to consensus is an almost independent of ΔH

_{0}, despite the observation that Δp

_{tr}increases with increase of ΔH

_{0}(see Figure 3a). This suggests, that the effect of magnetization Δp

_{tr}> 0 is compensated by the effect of heterogeneity in spatial distributions of spins at t

_{tr}, which is characterized by S

_{1}(t

_{tr}) ≥ S

_{−1}(t

_{tr}). In this regard, we found that the medians of spin distributions at the transition from the three- to two-state regime M

_{s}(t

_{tr}) are monotonic functions of ΔH

_{0}(see Figure 5a), such that their difference ΔM

_{tr}= M

_{1}(t

_{tr}) − M

_{−1}(t

_{tr}) can be well fitted by following empirical equation:

_{1}= 1.8 and b

_{2}= 15 are fitting constants (see Figure 5b). At the same time, we found that the probabilities to reach the consensus in positive (P

_{1}) or negative (P

_{−1}) spin state, both are linear functions of ΔH

_{0}, such that:

^{1/}

^{β}(see Figure 6b). So, the decrease of opinion competitiveness due to the heterogeneity in the initial spatial distribution is yet determined by the system evolution in the three-state regime during the first 3.3 update cycles.

_{−1}(t

_{o}) and probability to win (P

_{−1}) as the heterogeneity in the initial spatial distribution of party adepts increases explains the well known fact that the “electoral strength” of political party depends not only on the number of party members, but also on their distribution among the electoral districts (see [24]). Specifically, a spatial concentration of party members unfavorable affects its global electoral performance. This is consistent with empirical observations [25].

## 4. Conclusions

_{C}>> T

_{tr}is independent of the initial geographic distributions of rival opinions.

## Acknowledgments

**PACS Codes:**89.65.-s; 87.23.Ge; 02.50.-r

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Spatial distributions of positive (black), negative (white), and zero (gray) spins on lattice of size 50× 50 : (

**a**) Initial spatial distributions with M

_{1}= M

_{−1}= 10 and (

**b**) the corresponding distribution of spins at the transition from the three- to two-state regime of majority rule dynamics. (

**c**) Initial spatial distributions with M

_{−1}= 5 < M

_{1}=10 and (

**d**) the corresponding distribution of spins at the transition from the three- to two-state regime of the majority rule dynamics.

**Figure 2.**The normalized entropy S

_{−1}/ ln K of the initial spatial distribution of negative spins on square lattices of different size, versus the heterogeneity index H

_{−1}. From the bottom to top: N = 1600, 2500, 3600, and 4900.

**Figure 3.**(

**a**) System magnetization at the transition from the three- to two-state regime Δp

_{tr}versus ΔH

_{0}; straight line—data fitting with Equation (5). (

**b**) Heterogeneity indexes of spin distribution at the transition from the three- to two-state regime versus the heterogeneity index at t = 0 for positive (full circles) and negative (circles) spins: data are averaged over 10

^{6}realizations on lattice of size N = 2500; curves—data fittings to eye guide.

**Figure 4.**Probability of non-consensus coherent configurations (stripes) P

_{S}versus: (

**a**) Δp

_{tr}(three-state model) and Δp

_{0}(two-state model). Circles and full rhomb—data of Monte Carlo simulations in this work on the lattice of size N = 2500; rhombi—data from Figure 9 of [20] obtained by simulations of the two-state majority rule model on lattice of size N = 2500 and (

**b**) ΔH

_{0}.

**Figure 5.**(

**a**) Normalized medians M

_{s}(t

_{0})/100 of spin distributions versus ΔH

_{0}(circles and squares correspond to positive and negative spin distributions, respectively). (

**b**) Normalized difference ΔM

_{s}(t

_{tr})/100 versus Δp

_{tr}. Data are averaged over 10

^{6}realizations on lattices of different sizes; curve in panel (b)—data fitting with Equation (9).

**Figure 6.**The difference of exit probabilities ΔP versus: (

**a**) ΔH

_{0}and (

**b**) Δp

_{0}for the tree-state MR dynamics without long-living non-consensus configurations. Circles - data of numerical simulations averaged over 10

^{5}realizations; curves—data fittings with Equations (10) and (11) in panels (a) and (b), respectively.

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## Share and Cite

**MDPI and ACS Style**

Balankin, A.S.; Martínez Cruz, M.Á.; Gayosso Martínez, F.; Martínez-González, C.L.; Morales Ruiz, L.; Patiño Ortiz, J.
Effect of Heterogeneity in Initial Geographic Distribution on Opinions’ Competitiveness. *Entropy* **2015**, *17*, 3160-3171.
https://doi.org/10.3390/e17053160

**AMA Style**

Balankin AS, Martínez Cruz MÁ, Gayosso Martínez F, Martínez-González CL, Morales Ruiz L, Patiño Ortiz J.
Effect of Heterogeneity in Initial Geographic Distribution on Opinions’ Competitiveness. *Entropy*. 2015; 17(5):3160-3171.
https://doi.org/10.3390/e17053160

**Chicago/Turabian Style**

Balankin, Alexander S., Miguel Ángel Martínez Cruz, Felipe Gayosso Martínez, Claudia L. Martínez-González, Leobardo Morales Ruiz, and Julián Patiño Ortiz.
2015. "Effect of Heterogeneity in Initial Geographic Distribution on Opinions’ Competitiveness" *Entropy* 17, no. 5: 3160-3171.
https://doi.org/10.3390/e17053160