# Social Dynamics of Terabithia

In the land of Terabithia, king Jesse and princess May Belle have taken over the duty to rule the kingdom. The people of the kingdom are divided into five tribes at the time of their inauguration. These are liberal people, willing to accept and share new ideas and fashion available in each tribe. Jesse has appointed Abul (the court mathematician) to figure out how and at what rate his newly introduced fashion of wearing a Lungi will spread throughout his kingdom. He has also decided to divide his country into different districts based on different styles of wearing lungi.

Now, there are several factors for Abul to consider here. Some of the tribes have accepted Lungi as their choice of official dress code. However, they have different styles of wearing those. Tribe 1 likes wearing it with no shirts on, tribe 2 refused to accept it as their fashion, tribe 3 wears lungi with a kurta, tribe 4 prefers this style

and so on. The fashion designers in each tribe have made appealing designs to impress and encourage people from other tribes to accept their fashion. Abul needs to figure out and forecast the possibility of each tribe’s fashion surviving and/or flourishing over the whole kingdom.

Problem Statement

In sociology, economics, business planning etc, researchers and analysts are always concerned about how competing cultures, languages, fashion etc survive over a period of time. Some cultural trends are short lived, some defeat the existing trends and completely take over. It is impossible to predict what individuals think about each trend, but it is in our reach to predict what the mass trend will be once a steady state is reached.

Individuals have different philosophies. In general we update our idea of fashion and culture based on movies, TV, books, what others around us do etc. Some of us are trend starters. Others follow them. Many times it is not one person, it is a group of people following a trend that motivates others to follow. In the formal terminology of social science and social dynamics, there are three stages of a cultural trend change before it reaches a steady state.

1. Emergence: A new cultural center emerges in a population. Someone or a group of people start using a new colloquial word, begin wearing a new kind of jeans or may be start listening/propagating a new genre of metal songs to others.
2. Broadcast: This new trend is “shown off” to others occasionally. The trend starters use their newly coined colloquial word too often in public, wear the newly designed clothes in parties or release albums and organize concerts to promote the new genre of metal with some frequency.
3. Following and Competition: Others decide whether they will accept or reject the new trend. If someone decides to accept, occasionally he/she also becomes one of the broadcasters. If there are several competing trends available, then a decision is made based on factors like the current number of people following the trend and how long this trend has been followed or simply personal taste.

Every trend usually dies after some time. This can be months, years or even decades. Our general tendency is to accept the newest trend.

In the field of Social Dynamics, agent based modeling has been used by many mathematicians to study the behavior of such chaotic systems. It is usually hard to find closed form solutions of any mathematical model describing such a complex process, so discrete time Monte Carlo simulations or rule based cellular automata systems have been used to predict mass behavior in a society.

To help Abul, we modify and suggest some improvements to a model proposed in this paper and add some more functionality to study the culture of Terabithian people.

Monte Carlo Model

The Monte Carlo simulation is performed on a lattice of size $L \times L$, containing $L^{2}$ agents who interact with their neighbors. A fashion is defined as $I(s,a)$, where s is the cultural center and broadcaster site, and a is the age of the fashion.

The model assumes that newly arrived fashions are likely to be accepted more. A fashion $I(s_{i},a_{i})$ for a particular agent i can change its preference to another fashion from site $s_{j}$ if $a_{j}. Keeping this in mind, the proposed algorithm to take a timestep from t to t + 1 over the whole lattice is

1. Emergence step: With a very small probability $p_{new}$, a new cultural center emerges. $I_{i}(t)=I(i,0)$.
2. Broadcast step: With a probability $p_{repeat}$, the cultural centers restrengthen themselves to affect their neighbors. $I(i,a_{i,t})=I(i,0)$.
3. Spread step: Each site i randomly chooses one of its nearest neighbors j (with equal probability) by whom it will be affected. If $a_{j}, then the current fashion at i is changed to the fashion followed by j. $I(s_{i},a_{i})=I(s_{j},a_{j})$. Otherwise nothing changes for i.
4. All ages in each lattice site is increased by 1.

Modifications

1. The original model proposes choosing a site randomly as an emerging cultural center. This limits Abul to concentrate on one tribe only. In other words, the model does not consider competition between several *existing* fashions. We propose to change this scheme and assume each tribe acts as their own ambassador. Thus we assign a cultural center to each tribe as an initial condition and all of them are broadcasters of their own fashion.
2. The paper considers geographic impacts such as closed boundary at the borders of a country. We extend this by considering geographic factors that can slow down the exchange of information, e.g. rivers and seas that divides countries and regions. To include this, we introduce another random variable $p_{cross}$ that determines the probability of information crossing over the given obstacle. With this probability, we exchange information only between the two sides along the obstacle in step 3.
3. Finally, we also propose changing the global $p_{new}$ and $p_{repeat}$ to local competing groups by converting these variables to arrays. This better captures the reality because different groups of people may have different tendencies to interact with others. It is too simple to assume that every competing group has the same level of interaction and motivation to spread their own culture.

Back to Terabithian Tribes

Mathematica was used to program this Monte Carlo simulation. We study the general model and our modifications one by one.

Terabithia is an infinite kingdom, it is a torus. That is, if you leave the kingdom from one side, you will enter through the other side. There is no escape from lungi or Jesse’s rules. We simulate this by assigning a periodic boundary condition on our lattice when choosing neighbors for a particular site.

For one of our examples though, we will use a closed boundary condition.

We model the kingdom by a $50 \times 50$ lattice , each site represents a group of people (presumably a family or a house). We choose the nearest four neighbors with whom each site will interact.

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(b)

Figure 1. (a) Tribe colors. (b) Initial distribution of different groups.

Figure 1 shows an initial random distribution of different groups of people who follow different tribes, colored according to 1(a).

We would like to run the original paper model first. We choose 500 iterations for the simulation. Estimating that only one active interaction (an interaction with a broadcaster that changes someone’s mind to start following a new fashion) occurs in a day, this would mean the simulation is the result after 500 days.

Figure 2. Steady state reached after 500 iterations (days).

Figure 2 shows how fashions of different tribes have clustered over 500 days. Tribe 4 is less dominant for this particular number of iterations. The important point is that trends have emerged over different regions, it is not a random distribution of fashion anymore, as in 1(b).

Since we have imposed a periodic boundary condition, note how the clusters have continuity (wraps around) at the horizontal and vertical ends of the lattice.

We have helped Abul solve the initial problem. He can show this simulation to the king and inform him about possible clustering and number of people following particular fashions in this competing culture war after different number of days. However, there are more modifications on the way.

Rivers and Borders in Terabithia

Often we have borders among regions that stop people from interacting with each other. There are rivers and seas that distance people out (assuming no internet, like in the ancient days) and slow down the exchange of information and culture. Abul wants to incorporate these factors in the model. We help him by changing our neighborhood selection algorithm to include a set of cells in our lattice that stop information exchange.

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Figure 3. (a) A different tribe is included among the cells (off-white cells creating a rectangle) which act as information barrier (a physical border, in a sense). (b) The two separate regions evolve independently, as expected.

Notice in figure 3 that some cultural trends have disappeared, they could not survive for 500 days. However, it is possible that they can come back if we ran the simulation for more iterations.

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Figure 4. (a) A river that splits the kingdom into two equal regions. However, a slight probability of information exchange is allowed across the two river sides. Closed boundary conditions are used. (b) Result after 500 iterations.

Figure 4 shows our modification of the original algorithm to include slow information exchange among two regions. 4(b) shows what we would expect. The same cultures have emerged around the river sides since they have some interaction. Note the consistency of the clusters on both sides. However, they are dominated by the trends that are stronger in the larger regions. Closed boundary conditions were used for this example.

Individual Probabilities for Tribes

As promised, we now help Abul make the model more realistic by assigning individual $p_{new}$ and $p_{repeat}$ variables to each tribe. The algorithm needs to changed slightly for doing this.

Figure 5. Biased and higher probabilities assigned to tribe #5, which is why it cultural trend dominates over the others.

This is much better because it gives us the capability to include real world information in the model. Although the probability distributions for each of these random variables are decided here by us, in reality we can conduct surveys and build realistic discrete distributions from which we can run a much reliable simulation of the future events.

Conclusion

With the proposed modifications, we can expect that the model is useful in a real world setting to predict future outcomes for competing cultural trends in a society. The best way to integrate all the modifications is to create an MCMC (Markov Chain Monte Carlo) sampler to estimate parameters for the joint distribution function for $p_{new}$ and $p_{repeat}$ from real world data.

The resulting parameters will be used in the above simulation and we may get some useful results if we cleverly choose a good neighborhood coverage for each lattice site (e.g. considering information exchanged over internet among distant sites).

Finally, Abul can present these results to the king and expect to get a promotion in return. Math saves the day, once more.